The following is a conversation with Richard Karp, a professor at Berkeley and one of the most
important figures in the history of theoretical computer science. In 1985, he received the
Turing Award for his research in the theory of algorithms, including the development of the
admiral's Karp algorithm for solving the max flow problem on networks, Hopcroft's Karp algorithm
for finding maximum cardinality matchings in bipartite graphs, and his landmark paper in
complexity theory called Reduceability Among Combinatorial Problems, in which he proved 21
problems to be NP-complete. This paper was probably the most important catalyst in the
explosion of interest in the study of NP-completeness and the P versus NP problem in general.
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people around the world. And now, here's my conversation with Richard Karp.
You wrote that at the age of 13, you were first exposed to plain geometry and was wonderstruck
by the power and elegance of form of proofs. Are there problems, proofs, properties, ideas,
and plain geometry that from that time that you remember being mesmerized by or just enjoying
to go through to prove various aspects? So Michael Rabin told me this story about an experience
he had when he was a young student who was tossed out of his classroom for bad behavior
and was wandering through the corridors of his school and came upon two older students
who were studying the problem of finding the shortest distance between two non-overlapping
circles. And Michael thought about it and said, you take the straight line between the two centers
and the segment between the two circles is the shortest because a straight line is the shortest
distance between the two centers. And any other line connecting the circles would be on a longer
line. And he thought and I agreed that this was just elegant. The pure reasoning could come up
with such a result. Certainly the shortest distance from the two centers of the circles
is a straight line. Could you once again say what's the next step in that proof?
Well, any segment joining the two circles, if you extend it by taking the radius on each side,
on each side, you get a segment with a path with three edges, which connects the two centers.
And this has to be at least as long as the shortest path, which is the straight line.
The straight line. Yeah. Wow. Yeah, that's quite simple. So what is it about that elegance that
you just find compelling? Well, just that you could establish a fact about geometry
beyond dispute by pure reasoning. I also enjoyed the challenge of solving puzzles in
plain geometry. It was much more fun than the earlier mathematics courses, which were mostly
about arithmetic operations and manipulating them. Was there something about geometry itself,
the slightly visual component of it that you can visualize? Yes, absolutely. Although I lacked
three-dimensional vision. I wasn't very good at three-dimensional vision. You mean being able
to visualize three-dimensional objects? Three-dimensional objects or surfaces, hyperplanes,
and so on. So there, I didn't have an intuition. But for example, the fact that the sum of the
angles of a triangle is 180 degrees is proved convincingly. And it comes as a surprise that
that can be done. Why is that surprising? Well, it is a surprising idea, I suppose. Why is that
proved difficult? It's not. That's the point. It's so easy, and yet it's so convincing.
Do you remember what is the proof that adds up to 180?
You start at a corner and draw a line parallel to the opposite side,
and that line sort of trisects the angle between the other two sides. And you get a
half plane, which has to add up to 180 degrees. And it consists in the angles by the quality of
alternate angles, what's it called? You get a correspondence between the angles created
by the side, along the side of the triangle and the three angles of the triangle.
Has geometry had an impact on, when you look into the future of your work with combinatorial
algorithms, has it had some kind of impact in terms of the puzzles, the visual aspects that were
first so compelling to you? Not Euclidean geometry particularly. I think I use tools like linear
programming and integer programming a lot, but those require high-dimensional visualization,
and so I tend to go by the algebraic properties. Right. You go by the linear algebra and not by
the visualization. Well, the interpretation in terms of, for example, finding the highest point
on a polyhedron, as in linear programming, is motivating. But again, I don't have the high
dimensional intuition that would particularly inform me, so I sort of lean on the algebra.
So to linger on that point, what kind of visualization do you do when you're trying
to think about, we'll get to combinatorial algorithms, but just algorithms in general.
What's inside your mind when you're thinking about designing algorithms,
or even just tackling any mathematical problem?
Well, I think that usually an algorithm involves a repetition of some inner loop,
and so I can sort of visualize the distance from the desired solution
as iteratively reducing until you finally hit the exact solution.
And try to take steps that get you closer to the…
Try to take steps that get closer and having the certainty of converging. So it's basically the
the mechanics of the algorithm is often very simple. But especially when you're trying
something out on the computer, so for example, I did some work on the traveling salesman problem,
and I could see there was a particular function that had to be minimized, and it was
fascinating to see the successive approaches to the optimum.
You mean, so first of all, traveling salesman problem is where you have to visit
every city without ever the only ones? Yeah, that's right. Find the shortest path
through the cities. Yeah, which is sort of a canonical standard, a really nice problem that's
really hard. Exactly. So can you say again, what was nice about being able to think about the
objective function there and maximizing it or minimizing it? Well, just that as the algorithm
proceeded, you were making progress, continual progress, and eventually getting to the optimum
point. So there's two parts, maybe. Maybe you can correct me. But first is like getting an
intuition about what the solution would look like, or even maybe coming up with a solution.
And two is proving that this thing is actually going to be pretty good. What part is harder for
you? Where's the magic happen? Is it in the first sets of intuitions, or is it in the messy details
of actually showing that it is going to get to the exact solution, and it's going to run at a
certain complexity? Well, the magic is just the fact that the gap from the optimum decreases
monotonically, and you can see it happening. And various metrics of what's going on are improving
all along until finally you hit the optimum. Perhaps later we'll talk about the assignment
problem that I can illustrate a little better. Now zooming out again, as you write, Don Knuth
has called attention to a breed of people who derive great aesthetic pleasure from contemplating
the structure of computational processes. So Don calls these folks geeks. And you write that you
remember the moment you realized you were such a person, you were showing the Hungarian algorithm
to solve the assignment problem. So perhaps you can explain what the assignment problem is and what
the Hungarian algorithm is. So in the assignment problem, you have
n boys and n girls. And you are given the desirability of or the cost of matching
the ith boy with the jth girl for all i and j. You're given a matrix of numbers,
and you want to find the one-to-one matching of the boys with the girls
such that the sum of the associated costs will be minimized. So the best way to match the boys
with the girls or men with jobs or any two sets. Any possible matching is possible?
Yeah, all one-to-one correspondences are permissible. If there is a connection that
is not allowed, then you can think of it as having an infinite cost. So what you do is
to depend on the observation that the identity of the optimal assignment, or as we call it,
the optimal permutation is not changed if you subtract
a constant from any row or column of the matrix. You can see that the comparison between
the different assignments is not changed by that. Because if you decrease a particular row,
all the elements of a row by some constant, all solutions decrease by the cost of that,
by an amount equal to that constant. So the idea of the algorithm is to start with a matrix of
non-negative numbers and keep subtracting from rows or entire columns in such a way that you
subtract the same constant from all the elements of that row or column, while maintaining the
property that all the elements are non-negative. Simple. Yeah. And so what you have to do is
find small moves which will decrease the total cost while subtracting constants from rows or
columns. And there's a particular way of doing that by computing the shortest path through
the elements in the matrix. And you just keep going in this way until you finally get a full
permutation of zeros while the matrix is non-negative, and then you know that that has to be the cheapest.
Is that as simple as it sounds? So the shortest path through the matrix part?
Yeah. The simplicity lies in how you find, I oversimplified slightly, what you will end up
subtracting a constant from some rows or columns and adding the same constant back to other rows
and columns. So as not to reduce any of the zero elements, you leave them unchanged.
But each individual step modifies several rows and columns by the same amount,
but overall decreases the cost. So there's something about that elegance that made you
go, aha, this is beautiful. It's amazing that something like this, something so
simple can solve a problem like this. Yeah, it's really cool. If I had mechanical ability,
I would probably like to do woodworking or other activities where you sort of shape something
into something beautiful and orderly. And there's something about the orderly,
systematic nature of that iterative algorithm that is pleasing to me.
Okay. So what do you think about this idea of geeks as Don Knuth calls them?
What do you think, is it something specific to a mindset that allows you to discover the
elegance in computational processes or is this all of us? Can all of us discover this beauty?
Are you born this way? I think so. I always like to play with numbers. I used to
amuse myself by multiplying four digit decimal numbers in my head and putting myself to sleep
by starting with one and doubling the number as long as I could go and testing my memory,
my ability to retain the information. And I also read somewhere that you wrote that you enjoyed
showing off to your friends by, I believe, multiplying four digit numbers, a couple of
four digit numbers. Yeah, I had a summer job at a beach resort outside of Boston.
And the other employee, I was the barker at a ski ball game. I used to sit at a microphone
saying, come one, come all, come in and play, ski ball, five cents to play, nickel to win,
and so on. That's what a barker. I wasn't sure if I should know, but barker, you're the charming,
outgoing person that's getting people to come in. Yeah, well, I wasn't particularly charming,
but I could be very repetitious and loud. And the other employees were sort of juvenile delinquents
who had no academic bent, but somehow I found that I could impress them by performing this mental
arithmetic. Yeah, there's something to that. One of some of the most popular videos on the
internet is there's a YouTube channel called Numberphile that shows off different mathematical
ideas. There's still something really profoundly interesting to people about math, the beauty of
it. Something, even if they don't understand the basic concept of even being discussed,
there's something compelling to it. What do you think that is? Any lessons you drew from your
early teen years when you were showing off to your friends with the numbers? What is it that
attracts us to the beauty of mathematics, do you think? The general population, not just the computer
scientists and mathematicians? I think that you can do amazing things. You can test whether
large numbers are prime. You can solve little puzzles about cannibals and missionaries.
Yeah. And there's a kind of achievement. It's puzzle solving. And at a higher level,
the fact that you can do this reasoning that you can prove in an absolutely ironclad way that some
of the angles of the triangle is 180 degrees. Yeah, it's a nice escape from the messiness
of the real world where nothing can be proved. And we'll talk about it, but sometimes the ability
to map the real world into such problems where you can prove it is a powerful step.
Yeah. It's amazing that we can do it. Of course, another attribute of geeks is they're not necessarily
endowed with emotional intelligence so they can live in a world of abstractions without having to
master the complexities of dealing with people.
So just to link on the historical note, as a PhD student in 1955, you joined the
Computational Lab at Harvard where Howard Agen had built the Mark I and the Mark IV computers.
Just to take a step back into that history, what were those computers like?
The Mark IV filled a large room much bigger than this large office that we were
talking in now. And you could walk around inside it. There were rows of relays. You could just
walk around the interior and the machine would sometimes fail because of bugs, which literally
meant flying creatures landing on the switches. So I never used that machine for any practical
purpose. The lab eventually acquired one of the earlier commercial computers.
This is already in the 60s? No, in the mid 50s. Or late 50s.
There was already commercial computers in it. Yeah, we had a Univac with 2,000 words of storage.
And so you had to work hard to allocate the memory properly to also the
access time from one word to another depended on the number of the particular words. And so
there was an art to sort of arranging the storage allocation to make fetching data rapid.
Were you attracted to this actual physical world implementation of mathematics? So it's
a mathematical machine that's actually doing the math physically?
No, not at all. I was attracted to the underlying algorithms.
But did you draw any inspiration? What did you imagine was the future of these giant computers?
Could you have imagined that 60 years later, we'd have billions of these computers all over the
world? I couldn't imagine that, but there was a sense in the laboratory that this was the wave
of the future. In fact, my mother influenced me. She told me that data processing was going to be
really big and I should get into it. She's a smart woman. Yeah, she was a smart woman.
And there was just a feeling that this was going to change the world. But I didn't think of it
in terms of personal computing. I had no anticipation that we would be walking
around with computers in our pockets or anything like that.
Did you see computers as tools, as mathematical mechanisms to analyze
the theoretical computer science, or as the AI folks, which is an entire other community of
dreamers, as something that could one day have human level intelligence?
Well, AI wasn't very much on my radar. I did read Turing's paper about the
the Turing test computing and intelligence. Yeah, the Turing test.
What did you think about that paper? Was that just like science fiction?
I thought that it wasn't a very good test because it was too subjective. So I didn't
feel that I didn't feel that the Turing test was really the right way to calibrate
how intelligent an algorithm could be. To link on that, do you think it's pot
because you've come up with some incredible tests later on, tests on algorithms that are
like strong, reliable, robust across a bunch of different classes of algorithms.
But returning to this emotional mess that is intelligence, do you think it's possible
to come up with a test that's as ironclad as some of the computational complexity work?
Well, I think the greater question is whether it's possible to achieve human level intelligence.
Right. So first of all, let me, at the philosophical level, do you think it's possible to create
algorithms that reason and would seem to us to have the same kind of intelligence as human beings?
It's an open question. It seems to me that most of the achievements have
quite operate within a very limited set of ground rules and for a very limited precise task,
which is a quite different situation from the processes that go on in the minds of humans,
which where they have to sort of function in changing environments, they have emotions, they have
physical attributes for exploring their environment, they have intuition, they have desires,
emotions. And I don't see anything in the current achievements of what's called AI
that come close to that capability. I don't think there's any computer program
which surpasses a six-month-old child in terms of comprehension of the world.
Do you think this complexity of human intelligence, all the cognitive abilities
we have, all the emotion, do you think that could be reduced one day or just fundamentally
can be reduced to a set of algorithms or an algorithm? So can a Turing machine
achieve human level intelligence?
I am doubtful about that. I guess the argument in favor of it is that the human brain seems to
achieve what we call intelligence, cognitive abilities of different kinds.
And if you buy the premise that the human brain is just an enormous interconnected set of switches,
so to speak, then in principle, you should be able to diagnose what that interconnection
structure is like, characterize the individual switches and build a simulation outside.
But why that may be true in principle, that cannot be the way we're eventually going to
tackle this problem. That does not seem like a feasible way to go about it. So there is,
however, an existence proof that if you believe that the brain is just a network of neurons
operating by rules, I guess you could say that that's an existence proof of the ability to
build the capabilities of a mechanism. But it would be almost impossible to acquire the information
unless we got enough insight into the operation of the brain.
But there's so much mystery there. Do you think or what do you make of consciousness,
for example, as an example of something we completely have no clue about,
the fact that we have this subjective experience, is it possible that this network of the circuit of
switches is able to create something like consciousness? To know its own identity.
Yeah, to know the algorithm, to know itself. I think if you try to define that rigorously,
you'd have a lot of trouble. So I know that there are many who
believe that general intelligence can be achieved, and there are even some who feel certain
that the singularity will come and we will be surpassed by the machines which will then learn
more and more about themselves and reduce humans to an inferior breed. I am doubtful that this
will ever be achieved. Just for the fun of it, could you linger on what's your intuition,
why you're doubtful? So there are quite a few people that are extremely worried about this
existential threat of artificial intelligence of us being left behind by the super-intelligent
new species. What's your intuition, why that's not quite likely?
Just because none of the achievements in speech or robotics or natural language processing or
creation of flexible computer assistants or any of that comes anywhere near close
to that level of cognition. What do you think about ideas if we look at Moore's law and exponential
improvement to allow us that would surprise us? Our intuition fall apart with exponential
improvement because we're not able to kind of think in linear improvement. We're not able to
imagine a world that goes from the Mark I computer to an iPhone X. We could be really surprised by
the exponential growth or on the flip side, it is possible that also intelligence is actually
way, way, way, way harder even with exponential improvement to be able to crack.
I don't think any constant factor improvement could change things. Given our
current comprehension of what cognition requires, it seems to me that
multiplying the speed of the switches by a factor of a thousand or a million
will not be useful until we really understand the organizational principle behind the network of
switches. Well, let's jump into the network of switches and talk about combinatorial algorithms
if we could. Let's step back for the very basics. What are combinatorial algorithms and what are
some major examples of problems they aim to solve? A combinatorial algorithm is one which
deals with a system of discrete objects that can occupy various states or take on various
values from a discrete set of values and need to be arranged or selected in such a way as to
achieve some, to minimize some cost function or to prove the existence of some combinatorial
configuration. An example would be coloring the vertices of a graph. What's a graph?
Let's step back. It's fun to ask one of the greatest computer scientists of all time,
the most basic questions in the beginning of most books. For people who might not know,
but in general, how you think about it, what is a graph? A graph? That's simple. It's a set of
points, certain pairs of which are joined by lines called edges. They represent the,
in different applications, represent the interconnections between
discrete objects. They could be the interconnections between switches in a digital circuit
or interconnections indicating the communication patterns of a human community.
And they could be directed or undirected. And then, as you've mentioned before, might have
costs. They can be directed or undirected. You can think of them as, if a graph were
representing a communication network, then the edge could be undirected, meaning that information
could flow along it in both directions or it could be directed with only one way communication.
A road system is another example of a graph with weights on the edges. And then a lot of problems
of optimizing the efficiency of such networks or learning about the performance of such networks
are the object of a combinatorial algorithm. So it could be scheduling classes at a school
where the vertices, the nodes of the network are the individual classes and the edges indicate
the constraints which say that certain classes cannot take place at the same time or certain
teachers are available only for certain classes, etc. Or I talked earlier about the assignment
problem of matching the boys with the girls, where you have there a graph with an edge from
each boy to each girl with a weight indicating the cost. Or in logical design of computers,
you might want to find a set of so-called gates switches that perform logical functions,
which can be interconnected to realize some function. So you might ask how many gates do
you need in order to, for a circuit to give a yes output if at least a given number of
its inputs are ones and no if fewer are present. My favorite is probably all the work with
network flows. So anytime you have, I don't know why it's so compelling, but there's
something just beautiful about it. It seems like there's so many applications and communication
networks in traffic flow that you can map into these. And then you could think of pipes and
water going through pipes and you could optimize it in different ways or something always visually
and intellectually compelling to me about it. And of course, you've done work there.
Yeah. So there the edges represent channels along which some commodity can flow. It might be
gas, it might be water, it might be information. Maybe supply chain as well, like products
being- Products flowing from one operation to another. And the edges have a capacity,
which is the rate at which the commodity can flow. And a central problem is to determine,
given a network of these channels. In this case, the edges are communication channels.
The challenge is to find the maximum rate at which the information can flow along these
channels to get from a source to a destination. And that's a fundamental combinatorial problem
that I've worked on. Jointly with the scientist Jack Edmonds, I think we're the first to give a
formal proof that this maximum flow problem through a network can be solved in polynomial time.
Which I remember the first time I learned that, just learning that in maybe even grad school.
I don't think it was even undergrad. No. Algorithm, yeah. Do network flows get taught in
in basic algorithms courses? Yes, probably. Okay. So yeah, I remember being very surprised
that max flow is a polynomial time algorithm. Yeah. That there's a nice fast algorithm that
solves max flow. So there is an algorithm named after you and Edmonds, the Edmonds-Karp algorithm
for max flow. So what was it like tackling that problem and trying to arrive at a polynomial
time solution? And maybe you can describe the algorithm. Maybe you can describe what's the
running time complexity that you showed. Yeah. Well, first of all, what is a polynomial time
algorithm? Yeah. Perhaps we could discuss that. So yeah, let's actually just even, yeah,
what is algorithmic complexity? What are the major classes of algorithm complexity?
So in a problem like the assignment problem or scheduling schools or any of these applications,
you have a set of input data which might, for example, be a set of vertices connected by edges
you're given for each edge the capacity of the edge. And you have algorithms which are,
I think of them as computer programs with operations such as addition, subtraction,
multiplication, division, comparison of numbers and so on. And you're trying to construct an
algorithm based on those operations which will determine in a minimum number of computational
steps the answer to the problem. In this case, the computational step is one of those operations.
And the answer to the problem is, let's say, the configuration of the network that carries
the maximum amount of flow. And an algorithm is said to run in polynomial time if, as a function
of the size of the input, the number of vertices, the number of edges and so on. The number of
basic computational steps grows only as some fixed power of that size. A linear algorithm would
execute a number of steps linearly proportional to the size. Quadratic algorithm would be steps
proportional to the square of the size and so on. And algorithms whose running time is bounded by
some fixed power of the size are called polynomial algorithms.
And that's supposed to be relatively fast class of algorithms.
That's right. Theoreticians take that to be the definition of an algorithm being
efficient and we're interested in which problems can be solved by such efficient algorithms.
One can argue whether that's the right definition of efficient because you could have an algorithm
whose running time is the 10,000th power of the size of the input and that wouldn't be really
efficient. In practice, it's oftentimes reducing from an n squared algorithm to an n log n or a
linear time is practically the jump that you want to make to allow a real-world system to
solve a problem. Yeah, that's also true because especially as we get very large networks,
the size can be in the millions and then anything above n log n where n is the size
would be too much for practical solution. Okay, so that's polynomial time algorithms.
What other classes of algorithms are there? So that usually they designate polynomials
of the letter P. There's also NP, NP complete and NP hard. Yeah. So can you try to disentangle
those by trying to define them simply? Right. So a polynomial time algorithm
is one which whose running time is bounded by a polynomial and the size of the input.
Then there's the class of such algorithms is called P. In the worst case, by the way,
we should say. Yeah, that's very important that in this theory, when we measure the complexity
of an algorithm, we really measure the growth of the number of steps in the worst case. So you
may have an algorithm that runs very rapidly in most cases, but if there is any case where it
gets into a very long computation, that would increase the computational complexity by this
measure. That's a very important issue because as we may discuss later, there are some very
important algorithms which don't have a good standing from the point of view of their worst
case performance and yet are very effective. So theoreticians are interested in P, the class
of problem solvable in polynomial time. Then there's NP, which is the class of problems
which may be hard to solve, but when confronted with a solution,
you can check it in polynomial time. Let me give you an example there.
So if we look at the assignment problem, so you have n-boys, you have n-girls,
so the number of numbers that you need to write down to specify the problem instance is n-squared.
And the question is how many steps are needed to solve it? And Jack Edmonds and I were the
first to show that it could be time n cubed earlier algorithms required into the fourth.
So as a polynomial function of the size of the input, this is a fast algorithm.
Now to illustrate the class NP, the question is how long would it take to
verify that a solution is optimal? So for example, if the input was a graph,
we might want to find the largest clique in the graph or a clique is a set of vertices such that
any vertex, each vertex in the set is adjacent to each of the others. So the clique is a complete
subgraph. Yeah, so if it's a Facebook social network, everybody's friends with everybody else,
it's a close clique of friends. That would be what's called a complete graph, it would be.
No, I mean within that clique. Within that clique, yeah. They're all friends.
So a complete graph is when everybody's friends with everybody.
So the problem might be to determine whether in a given graph there exists a clique of a certain
size. That turns out to be a very hard problem. But if somebody hands you a clique and asks you
to check whether it is, hands you a set of vertices and asks you to check whether it's a clique,
you could do that simply by exhaustively looking at all of the edges between the vertices in the
clique and verifying that they're all there. And that's a polynomial time.
So the problem of finding the clique
appears to be extremely hard. But the problem of verifying a clique,
to see if it reaches a target number of vertices, is easy to verify. So finding the
clique is hard. Checking it is easy. Problems of that nature are called
non-deterministic polynomial time algorithms. And that's the class NP.
And what about NP complete and NP hard?
Okay. Let's talk about problems where you're getting a yes or no answer rather than a numerical
value. So either there is a perfect matching of the boys with the girls or there isn't.
It's clear that every problem in P is also in NP. If you can solve the problem exactly,
then you can certainly verify the solution. On the other hand, there are problems in the class
NP. This is the class of problems that are easy to check, although they may be hard to solve.
It's not at all clear that problems in NP lie in P. So for example, if we're looking at scheduling
classes at a school, the fact that you can verify when handed a schedule for the school,
whether it meets all the requirements, that doesn't mean that you can find the schedule
rapidly. So intuitively, NP, non-deterministic polynomial, checking rather than finding,
is going to be harder than is going to include is easier, checking is easier.
And therefore, the class of problems that can be checked appears to be much larger than the
class of problems that can be solved. And then you keep adding appears to and
sort of these additional words that designate that we don't know for sure yet.
We don't know for sure. So the theoretical question, which is considered to be the most
central problem in theoretical computer science or at least computational complexity theory,
combinatorial algorithm theory, question is whether P is equal to NP. If P were equal to NP,
it would be amazing. It would mean that every problem where a solution can be rapidly checked
can actually be solved in polynomial time. We don't really believe that's true.
If you're scheduling classes at a school, we expect that if somebody hands you a
satisfying schedule, you can verify that it works. That doesn't mean that you should be able
to find such a schedule. So intuitively, NP encompasses a lot more problems than P.
So can we take a small tangent and break apart that intuition? So do you first of all think that
the biggest sort of open problem in computer science, maybe mathematics,
is whether P equals NP? Do you think P equals NP or do you think P is not equal to NP?
If you had to bet all your money on it.
I would bet that P is unequal to NP simply because there are problems that have been around for
centuries and have been studied intensively in mathematics and even more so in the last 50 years
since the P versus NP was stated. No polynomial time algorithms have been found for these easy
to check problems. One example is a problem that goes back to the mathematician Gauss who was
interested in factoring large numbers. We know what a number is prime if it cannot be written as
the product of two or more numbers unequal to one. So if we can factor the number like 91,
it's seven times 13. But if I give you 20-digit or 30-digit numbers, you're probably going to
be at a loss to have any idea whether they can be factored. So the problem of factoring very
large numbers does not appear to have an efficient solution. But once you have found the factors,
expressed the number as a product to smaller numbers, you can quickly verify that they are
factors of the number. Your intuition is a lot of people have tried to find algorithms. For this
one particular problem, there's many others like it that are really well studied and it would be
great to find an efficient algorithm for. Right. And in fact, we have some results that I was
instrumental in obtaining following up on work by the mathematician Stephen Cook
to show that within the class NP of easy-to-check problems, there's a huge number that are equivalent
in the sense that either all of them or none of them lie in P. And this happens only if P is equal
to NP. So if P is unequal to NP, we would also know that virtually all the standard combinatorial
problems, if P is unequal to NP, none of them can be solved in polynomial time.
Can you explain how that's possible to tie together so many problems in a nice bunch
that if one is proven to be efficient, then all are?
The first and most important stage of progress was a result by Stephen Cook
who showed that a certain problem called the satisfiability problem of propositional logic
is as hard as any problem in the class P. So the propositional logic problem is expressed
in terms of expressions involving the logical operations and or and not operating on variables
that can be either true or false. So an instance of the problem would be some formula involving and
or and not. And the question would be whether there is an assignment of truth values to the
variables in the problem that would make the formula true. So for example, if I take the formula
A or B and A or not B and not A or B and not A or not B and take the conjunction of all four of
those so-called expressions, you can determine that no assignment of truth values to the variables A
and B will allow that conjunction of what are called clauses to be true. So that's an example of a
formula in propositional logic involving expressions based on the operations and or and not.
That's an example of a problem which is not satisfiable. There is no solution that satisfies
all of those constraints. And that's like one of the cleanest and fundamental problems in computer
science. It's like a nice statement of a really hard problem. It's a nice statement of a really
hard problem. And what Cook showed is that every problem in NP can be re-expressed as an instance
of the satisfiability problem. So to do that, he used the observation that a very simple abstract
machine called the Turing machine can be used to describe any algorithm. An algorithm for any
realistic computer can be translated into an equivalent algorithm on one of these Turing
machines which are extremely simple. So a Turing machine, there's a tape and you can walk along
that tape. You have data on a tape and you have basic instructions, a finite list of instructions
which say if you're reading a particular symbol on the tape and you're in a particular state,
then you can move to a different state and change the state of the element that you were looking
at, the cell of the tape that you were looking at. And that was like a metaphor and a mathematical
construct that Turing put together to represent all possible computation. All possible computation.
Now, one of these so-called Turing machines is too simple to be useful in practice. But for
theoretical purposes, we can depend on the fact that an algorithm for any computer can be translated
into one that would run on a Turing machine. And then using that fact, he could describe
any possible non-deterministic polynomial time algorithm, any algorithm for a problem in
P could be expressed as a sequence of moves of the Turing machine described in terms of
reading a symbol on the tape while you're in a given state and moving to a new state and
leaving behind a new symbol. And given that fact that any non-deterministic polynomial time algorithm
can be described by a list of such instructions, you could translate the problem into the language
of the satisfiability problem. Is that amazing to you, by the way? If you take yourself back
when you were first thinking about the space of problems, how amazing is that?
It's astonishing. When you look at Cook's proof, it's not too difficult to
figure out why this is so, but the implications are staggering. It tells us that of all the
problems in NP, all the problems where solutions are easy to check, they can all be rewritten
in terms of the satisfiability problem. Yeah, it's adding so much more weight
to the P equals NP question, because all it takes is to show that one algorithm in this class.
So the P versus NP can be re-expressed as simply asking whether the satisfiability problem of
propositional logic is solvable in polynomial time. But there's more. I encountered Cook's paper
when he published it in a conference in 1971. Yeah, so when I saw Cook's paper and saw this
reduction of each of the problems in NP by a uniform method to the satisfiability problem
of propositional logic, that meant that the satisfiability problem was a universal combinatorial
problem. And it occurred to me through experience I had had in trying to solve other combinatorial
problems that there were many other problems which seemed to have that universal structure.
And so I began looking for reductions from the satisfiability to other problems.
One of the other problems would be the so-called integer programming problem of
solving a determining whether there's a solution to a set of linear inequalities
involving integer variables. Just like linear programming, but there's a constraint that the
variables must remain integers. Integers, in fact, must be the zero or one that could only
take on those values. And that makes the problem much harder. Yes, that makes the problem much
harder. And it was not difficult to show that the satisfiability problem can be restated
as an integer programming problem. So can you pause on that? Was that one of the first
mappings that you tried to do? And how hard is that mapping? You said it wasn't hard to show,
but that's a big leap. It is a big leap, yeah. Well, let me give you another example.
Another problem in NP is whether a graph contains a clique of a given size.
And now the question is, can we reduce the propositional logic problem to the problem of
whether there's a clique of a certain size? Well, if you look at the propositional logic
problem, it can be expressed as a number of clauses, each of which is of the form
A or B or C, where A is either one of the variables in the problem or the negation of one of the
variables. And an instance of the propositional logic problem can be rewritten using operations of
Boolean logic, can be rewritten as the conjunction of a set of clauses, the and of a set of ors,
where each clause is a disjunction on or of variables or negated variables.
So the question in the satisfiability problem is whether those clauses can be simultaneously
satisfied. Now, to satisfy all those clauses, you have to find one of the terms in each clause,
which is going to be true in your truth assignment, but you can't make the same variable both true
and false. So if you have the variable A in one clause and you want to satisfy that clause by
making A true, you can't also make the complement of A true in some other clause.
And so the goal is to make every single clause true if it's possible to satisfy this,
and the way you make it true is at least one term in the clause must be true.
So now we, to convert this problem to something called the independent set problem, where you're
just sort of asking for a set of vertices in a graph such that no two of them are adjacent,
sort of the opposite of the clique problem. So we've seen that we can now express that as
finding a set of terms one in each clause without picking both the variable and the negation of
that variable. Because if the variable is assigned the truth value, the negated variable has to have
the opposite truth value. And so we can construct the graph where the vertices are the terms in
all of the clauses. And you have an edge between two occurrences of terms, either if they're both
in the same clause because you're only picking one element from each clause, and also an edge
between them if they represent opposite values of the same variable because you can't make a variable
both true and false. And so you get a graph where you have all of these occurrences of variables,
you have edges, which mean that you're not allowed to choose both ends of the edge,
either because they're in the same clause or they're con negations of one another.
Right. And that's a, first of all, sort of to zoom out, that's a really powerful idea that you
could take a graph and connect it to a logic equation somehow and do that mapping for all
possible formulations of a particular problem on a graph. I mean, that still is hard for me to
believe that that's possible. What do you make of that that there's such a union of, there's such
a friendship among all these problems across that somehow are akin to combinatorial algorithms
that they're all somehow related. I know it can be proven, but what do you make of it that that's
true? Well, that they just have the same expressive power. You can take any one of them and translate
it into the terms of the other. The fact that they have the same expressive power also somehow
means that they can be translatable. Right. And what I did in the 1971 paper was to take
21 fundamental problems, commonly occurring problems of packing, covering, matching and so
forth, lying in the class NP and show that the satisfiability problem can be re-expressed as any
of those, that any of those have the same expressive power. And that was like throwing
down the gauntlet of saying there's probably many more problems like this. Right. But that's just
saying that, look, they're all the same. They're all the same, but not exactly. They're all the
same in terms of whether they are rich enough to express any of the others. But that doesn't mean
that they have the same computational complexity. But what we can say is that either all of these
problems or none of them are solvable in polynomial time.
Yes, so where does NP completeness and NP hard as classes fit?
Oh, that's just a small technicality. So when we're talking about decision problems,
that means that the answer is just yes or no. There is a clique of size 15 or there's not a
clique of size 15. On the other hand, an optimization problem would be asking,
find the largest clique. The answer would not be yes or no, it would be 15. So when you're asking
when you're putting a valuation on the different solutions and you're asking for the one with
the highest valuation, that's an optimization problem. And there's a very close affinity
between the two kinds of problems. But the counterpart of being the hardest decision problem,
the hardest yes-no problem, the counterpart of that is to minimize or maximize an objective
function. And so a problem that's hardest in the class when viewed in terms of optimization,
those are called NP hard rather than NP complete. And NP complete is for decision problems.
And NP complete is for decision problems. So if somebody shows that P equals NP,
what do you think that proof will look like if you were to put on yourself,
if it's possible to show that as a proof or to demonstrate an algorithm?
All I can say is that it will involve concepts that we do not now have and approaches that we
don't have. Do you think those concepts are out there in terms of inside complexity theory,
inside of computational analysis of algorithms? Do you think those concepts are totally outside
of the box who haven't considered yet? I think that if there is a proof that P is equal to NP
or that P is not equal to NP, it'll depend on concepts that are now outside the box.
Now, if that's shown either way, P equals NP or P not, well, actually P equals NP, what impact,
you kind of mentioned a little bit, but can you can you linger on it? What kind of impact would it
have on theoretical computer science and perhaps software systems in general?
Well, I think it would have enormous impact on the world in either case. If P is unequal to NP,
which is what we expect, then we know that for the great majority of the combinatorial problems
that come up, since they're known to be NP complete, we're not going to be able to solve them by
efficient algorithms. However, there's a little bit of hope in that it may be that we can solve
most instances. All we know is that if a problem is not in P, then it can't be solved efficiently
on all instances. But basically, if we find that P is unequal to NP, it will mean that
we can't expect always to get the optimal solutions to these problems. And we have to depend on
heuristics that perhaps work most of the time or give us good approximate solutions.
So we would turn our eye towards the heuristics with a little bit more acceptance and comfort
on our hearts. Exactly. Okay, so let me ask a romanticized question. What to you is one of the
most or the most beautiful combinatorial algorithm in your own life or just in general in the field
that you've ever come across or have developed yourself? I like the stable matching problem,
well, the stable marriage problem very much. What's the stable matching problem? Yeah. Imagine
that you want to marry off n-boys with n-girls. And each boy has an ordered list of his preferences
among the girls, his first choice, his second choice through her n-th choice. And each girl
also has an ordering of the boys, his first choice, second choice, and so on. And we'll say
that a matching, a one-to-one matching of the boys with the girls is stable if there are
no two couples in the matching such that the boy in the first couple
prefers the girl in the second couple to her mate and she prefers the boy to her current mate. In
other words, the matching is stable if there is no pair who want to run away with each other
leaving their partners behind. Gosh, yeah. Actually, this is relevant to matching
residents with hospitals and some other real-life problems, although not quite in the form that
I described. So it turns out that there is, for any set of preferences, a stable matching exists.
And moreover, it can be computed by a simple algorithm in which each boy starts making proposals
to girls. And if the girl receives a proposal, she accepts it tentatively, but she can
drop it later if she gets a better proposal from her point of view. And the boys start going down
their lists proposing to their first, second, third choices until stopping when a proposal is
accepted. But the girls meanwhile are watching the proposals that are coming into them,
and the girl will drop her current partner if she gets a better proposal.
And the boys never go back through the list? They never go back. Yeah. So once they've been
denied, they don't try again because the girls are always improving their
status as they receive better and better proposals. The boys are going down their lists
starting with their top preferences. And one can prove that the process will come to an end
where everybody will get matched with somebody and you won't have any pair that want to
abscond from each other. Do you find the proof or the algorithm itself
beautiful? Or is it the fact that with the simplicity of just the two marching,
I mean, the simplicity of the underlying rule of the algorithm, is that the beautiful part?
Both, I would say. And you also have the observation that you might ask who is better
off the boys who are doing the proposing or the girls who are reacting to proposals.
And it turns out that it's the boys who are doing the best, that as each boy is doing at least as
well as he could do in any other staple matching. So there's a sort of lesson for the boys that
you should go out and be proactive and make those proposals go for broke.
I don't know if this is directly mappable philosophically to our society, but certainly
seems like a compelling notion. And like you said, there's probably a lot of actual real
world problems that this could be mapped to. Yeah, well, you get complications. For example,
what happens when a husband and wife want to be assigned to the same hospital? So you
have to take those constraints into account. And then the problem becomes NP-hard.
Why is it a problem for the husband and wife to be assigned to the same hospital?
No, it's desirable. Or at least go to the same city. So if you're assigning residents to hospitals.
And then you have some preferences for the husband and wife or for the hospitals?
The residents have their own preferences.
Because residents, both male and female, have their own preferences.
The hospitals have their preferences. But if resident A, the boy, is going to Philadelphia,
then you'd like his wife also to be assigned to a hospital in Philadelphia.
So which step makes it a NP-hard problem that you mentioned?
The fact that you have this additional constraint, that it's not just the preferences of individuals,
but the fact that the two partners to a marriage have to be assigned to the same place.
I'm being a little dense. The perfect matching, no, not the stable matching,
is what you referred to. That's when two partners are trying to...
Okay, what's confusing you is that in the first interpretation of the problem,
I had boys matching with girls. In the second interpretation, you have humans matching with
institutions. And there's a coupling between within the, gotcha, within the humans.
Yeah. Any added little constraint will make it an NP-hard problem. Well, yeah.
Okay. By the way, the algorithm you mentioned wasn't one of yours?
No, no, that was due to Gail and Shapley. And my friend, David Gail,
passed away before he could get part of a Nobel Prize. But his partner, Shapley,
shared in a Nobel Prize with somebody else for economics, for ideas stemming from this
stable matching idea. So you've also developed yourself some elegant, beautiful algorithms,
again, picking your children. So the Robin-Karp algorithm for string searching, pattern matching,
Edmund-Karp algorithm for max flows, we mentioned, Hopcroft-Karp algorithm for finding
maximum cardinality matchings in bipartite graphs. Is there ones that stand out to you,
ones you're most proud of, or just whether it's beauty, elegance, or just being the right discovery
development in your life that you're especially proud of?
I like the Robin-Karp algorithm because it illustrates the power of randomization.
So the problem there is to decide whether a given long string of symbols from some alphabet
contains a given word, whether a particular word occurs within some very much longer word. And so
the idea of the algorithm is to associate with the word that we're looking for, a fingerprint,
some number or some combinatorial object that describes that word, and then to look for an
occurrence of that same fingerprint as you slide along the longer word. And what we do is we
associate with each word a number. So first of all, we think of the letters that occur in a word
as the digits of, let's say, decimal or whatever base here, whatever number of different symbols
there are. That's the base of the numbers, yeah. Right. So every word can then be thought of as a
number with the letters being the digits of that number. And then we pick a random prime number
in a certain range and we take that word viewed as a number and take the remainder on dividing
that number by the prime. So coming up with a nice hash function. It's a kind of hash function.
Yeah. It gives you a little shortcut for that particular word.
So that's the… It's very different than any other algorithms of its kind that we're trying
to do search, string matching. Yeah, which usually are combinatorial and don't involve the idea of
taking a random fingerprint. And doing the fingerprinting has two advantages. One is that as
we slide along the long word, digit by digit, we keep a window of a certain size, the size of the
word we're looking for. And we compute the fingerprint of every stretch of that length.
And it turns out that just a couple of arithmetic operations will take you from the fingerprint
of one part to what you get when you slide over by one position. So the computation of all the
fingerprints is simple. And secondly, it's unlikely if the prime is chosen randomly from a certain
range, that you will get two of the segments in question having the same fingerprint.
Right. And so there's a small probability of error which can be checked after the fact.
And also the ease of doing the computation because you're working with these fingerprints,
which are remainders modulo some big prime. So that's the magical thing about randomized
algorithms is that if you add a little bit of randomness, it somehow allows you to take a pretty
naive approach, a simple looking approach, and allow it to run extremely well. So can you maybe
take a step back and say what is a randomized algorithm, this category of algorithms?
Well, it's just the ability to draw a random number from some range or to associate a random
number with some object or to draw that random from some set. So another example is very simple
if we're conducting a presidential election. And we would like to pick the winner.
In principle, we could draw a random sample of all of the voters in the country.
And if it was of substantial size, say a few thousand, then the most popular candidate
in that group would be very likely to be the correct choice that would come out of counting
all the millions of votes. Of course, we can't do this because first of all, everybody has to
feel that his or her vote counted. And secondly, we can't really do a purely random sample from
that population. And I guess thirdly, there could be a tie in which case we wouldn't have a
significant difference between two candidates. But those things aside, if you didn't have all
that messiness of human beings, you could prove that that kind of random picking would solve the
problem with a very low probability of error. Another example is testing whether a number is
prime. So if I want to test whether 17 is prime, I could pick any number between
1 and 17 and raise it to the 16th power, modulo 17, and you should get back the original number.
That's a famous formula due to Fermat's little theorem that
if you take any A, any number A in the range 0 through n minus 1 and raise it to the n minus
1th power modulo n, you'll get back the number A if A is prime. So if you don't get back the
number A, that's a proof that a number is not prime. And you can show that
suitably define the probability that you will get a value unequal. You will get a violation of
Fermat's result is very high. And so this gives you a way of rapidly proving that a number is not
prime. It's a little more complicated than that because there are certain values of n where
something a little more elaborate has to be done. But that's the basic idea,
taking an identity that holds for primes. And therefore, if it ever fails on any
instance for a non-prime unit, you know that the number is not prime. It's a quick choice,
a fast choice, fast proof that a number is not prime.
Can you maybe elaborate a little bit more of what's your intuition why randomness works so well
and results in such simple algorithms?
Well, the example of conducting an election where in theory you could take a sample and
depend on the validity of the sample to really represent the whole is just the basic fact of
statistics, which gives a lot of opportunities. And I actually exploited that sort of random
sampling idea in designing an algorithm for counting the number of solutions that satisfy a
particular formula and propositional logic.
A particular, so some version of the satisfiability problem?
A version of the satisfiability problem.
Is there some interesting insight that you want to elaborate on, like what some aspect of that
algorithm that might be useful to describe? So you have a collection of formulas and you want to
count the number of solutions that satisfy at least one of the formulas.
And you can count the number of solutions that satisfy any particular one of the formulas,
but you have to account for the fact that that solution might be counted many times if it solves
more than one of the formulas.
And so what you do is you sample from the formulas according to the number of solutions
that satisfy each individual one. In that way, you draw a random solution, but then you correct
by looking at the number of formulas that satisfy that random solution.
And don't double count. So you can think of it this way. So you have a matrix of zeros and ones,
and you want to know how many columns of that matrix contain at least one one.
And you can count in each row how many ones there are. So what you can do is draw from the rows
according to the number of ones. If a row has more ones, it gets drawn more frequently.
But then if you draw from that row, you have to go up the column and looking at
where that same one is repeated in different rows and only count it as a success or a hit
if it's the earliest row that contains the one. And that gives you a robust statistical estimate
of the total number of columns that contain at least one of the ones. So that is an example of
the same principle that was used in studying random sampling. Another viewpoint is that
if you have a phenomenon that occurs almost all the time, then if you sample one of the
occasions where it occurs, you're most likely to, and you're looking for an occurrence,
a random occurrence is likely to work. So that comes up in solving identities, solving
algebraic identities. You get two formulas that may look very different. You want to know if
they're really identical. What you can do is just pick a random value and evaluate the formulas
at that value and seeing if they agree. And you depend on the fact that if the formulas are
distinct, then they're going to disagree a lot. And so, therefore, a random choice will exhibit
the disagreement. If there are many ways for the two to disagree, and you only need to find one
disagreement, then random choice is likely to yield it. And in general, so we've just talked
about randomized algorithms, but we can look at the probabilistic analysis of algorithms.
And that gives us an opportunity to step back. And as we said, everything we've been talking
about is worst case analysis. Because you maybe comment on the usefulness and the power of worst
case analysis versus best case analysis, average case probabilistic. How do we think about the
future of theoretical computer science, computer science in the kind of analysis we do of algorithms?
Does worst case analysis still have a place, an important place, or do we want to try to move
forward towards kind of average case analysis? And what are the challenges there?
So, if worst case analysis shows that an algorithm is always good, that's fine. If worst case analysis
is used to show that the problem, that the solution is not always good,
then you have to step back and do something else to ask, how often will you get a good solution?
Just to pause on that for a second. That's so beautifully put, because I think we tend to
judge algorithms. We throw them in the trash the moment their worst case is shown to be bad.
Right. And that's unfortunate. I think a good example is going back to the
satisfiability problem. There are very powerful programs called SAT solvers,
which in practice fairly reliably solve instances with many millions of variables that arise in
digital design or in proving programs correct in other applications.
And so, in many application areas, even though satisfiability as we've already discussed is
NP complete, the SAT solvers will work so well that the people in that discipline tend to think
of satisfiability as an easy problem. So, in other words, just for some reason that we don't
entirely understand, the instances that people formulate in designing digital circuits or
other applications are such that satisfiability is not hard to check.
And even searching for a satisfying solution can be done efficiently in practice.
And there are many examples. For example, we talked about the traveling salesman problem.
So, just to refresh our memories, the problem is you've got a set of cities. You have
pairwise distances between cities, and you want to find a tour through all the cities that
minimizes the total cost of all the edges traversed, all the trips between cities.
The problem is NP hard, but people using integer programming codes together with some
other mathematical tricks can solve geometric instances of the problem where the cities are,
let's say, points in the plane and get optimal solutions to problems with tens of thousands
of cities. Actually, it'll take a few computer months to solve a problem of that size, but for
problems of size 1,000 or two, it'll rapidly get optimal solutions, provably optimal solutions,
even though, again, we know that it's unlikely that the traveling salesman problem can be
solved in polynomial time. Are there methodologies like rigorous
systematic methodologies for, you said in practice. In practice, this algorithm sounds
pretty good. Are there systematic ways of saying in practice, this one's pretty good?
So, in other words, average case analysis. Or you've also mentioned that average case
kind of requires you to understand what the typical cases, typical instances, and that might be
really difficult. That's very difficult. So, after I did my original work on showing all these
problems through NP complete, I looked around for a way to shed some positive light on
combinatorial algorithms. And what I tried to do was to study problems, behavior on the average,
or with high probability. But I had to make some assumptions about what's the probability space,
what's the sample space, what do we mean by typical problems. That's very hard to say. So,
I took the easy way out and made some very simplistic assumptions. So, I assumed, for example,
that if we were generating a graph with a certain number of vertices and edges,
then we would generate the graph by simply choosing one edge at a time at random until we got the
right number of edges. That's a particular model of random graphs that has been studied
mathematically a lot. And within that model, I could prove all kinds of wonderful things,
I and others who also worked on this. So, we could show that we know exactly how many edges
there have to be in whatever there be a so-called Hamiltonian circuit. That's a cycle that visits
each vertex exactly once. We know that if the number of edges is a little bit more than n log n,
where n is the number of vertices, then where such a cycle is very likely to exist. And we can
give a heuristic that will find it with high probability. And we got the community in which
I was working got a lot of results along these lines. But the field tended to be rather lukewarm
about accepting these results as meaningful because we were making such a simplistic assumption
about the kinds of graphs that we would be dealing with. So, we could show all kinds of
wonderful things. It was a great playground. I enjoyed doing it. But after a while, I concluded
that it didn't have a lot of bite in terms of the practical application.
Oh, okay. So, there's too much into the world of toy problems. But is there a way to find
nice representative real world impactful instances of a problem on which demonstrate
that an algorithm is good? So, the machine learning world is kind of what they
at its best tries to do is find a data set from the real world and show the performance. All the
conferences are all focused on beating the performance of on that real world data set.
Is there an equivalent in complexity analysis? Not really. Don Knuth started to collect
examples of graphs coming from various places. So, he would have a whole zoo of different
graphs that he could choose from and he could study the performance of algorithms on different
types of graphs. But there, it's really important and compelling to be able to define a class of
graphs. The actual act of defining a class of graphs that you're interested in, it seems to be
a non-trivial step before talking about instances that we should care about in the real world.
There's nothing available there that would be analogous to the training set for supervised
learning, where you sort of assume that the world has given you a bunch of examples to work with.
We don't really have that for problems, for combinatorial problems on graphs and networks.
There's been a huge growth, a big growth of data sets available. Do you think
some aspect of theoretical computer science might be contradicting my own question while
saying it, but will there be some aspect, an empirical aspect of theoretical computer science
that will allow the fact that these data sets are huge will start using them for analysis?
If you want to say something about a graph algorithm, you might take a social network
like Facebook and looking at subgraphs of that and prove something about the Facebook graph
and be respected. At the same time, be respected in the theoretical computer science community.
That hasn't been achieved yet, I'm afraid. Is that P equals NP? Is that impossible?
Is it impossible to publish a successful paper in the theoretical computer science community
that shows some performance on a real-world data set? Or is that really just those of two different
worlds? They haven't really come together. I would say that there is a field of experimental
algorithmics where people sometimes they're given some family of examples. Sometimes they just
generate them at random and they report on performance, but there's no convincing evidence
that the sample is representative of anything at all.
Let me ask, in terms of breakthroughs and open problems, what are the most compelling open
problems to you and what possible breakthroughs do you see in the near term in terms of theoretical
computer science? Well, there are all kinds of relationships among complexity classes that can
be studied. Just to mention one thing, I wrote a paper with Richard Lipton in 1979 where we asked
the following question. If you take a combinatorial problem in NP, let's say, and you pick
the size of the problem. I say it's a traveling salesman problem, but of size 52. You ask,
could you get an efficient, a small Boolean circuit tailored for that size 52 where you
could feed the edges of the graph in as Boolean inputs and get as an output the question of
whether or not there's a tour of a certain length. In other words, briefly, what you would say in
that case is that the problem has small circuits, polynomial-sized circuits.
Now, we know that if P is equal to NP, then in fact, these problems will have small circuits.
But what about the converse? Could a problem have small circuits, meaning that an algorithm
tailored to any particular size could work well and yet not be a polynomial-time algorithm,
that is, you couldn't write it as a single uniform algorithm good for all sizes?
Just to clarify, small circuits for a problem of particular size or even further constraint,
small circuit for a particular... No, for all the inputs of that size.
Of that size. Is that a trivial problem for a particular instance? So coming up
an automated way of coming up with a circuit, I guess that's...
That would be hard, yeah. But there's the existential question. Everybody talks nowadays
about existential questions, existential challenges. You could ask the question,
does the Hamiltonian circuit problem have a small circuit for every size, for each size,
a different small circuit? In other words, could you tailor solutions
depending on the size and get polynomial size?
Even if P is not equal to NP.
Right. That would be fascinating if that's true.
Yeah. What we proved is that if that were possible, then something strange would happen
in complexity theory, some high-level class which I could briefly describe,
something strange would happen. So I'll take a stab at describing what I mean.
Sure. Let's go there.
We have to define this hierarchy in which the first level of the hierarchy is P
and the second level is NP. What is NP? NP involves statements of the form
there exists a something such that something holds. For example,
there exists a coloring such that a graph can be colored with only that number of colors
or there exists a Hamiltonian circuit.
There's a statement about this graph.
Yeah. So the NP deals with statements of that kind that there exists a solution. Now,
you could imagine a more complicated expression which says for all x, there exists a y such that
some proposition holds involving both x and y. So that would say, for example, in game theory,
for all strategies for the first player, there exists a strategy for the second player
such that the first player wins. That would be at the second level of the hierarchy.
The third level would be there exists an A such that for all B, there exists a C
that something holds and you can imagine going higher and higher in the hierarchy.
And you'd expect that the complexity class, the classes that correspond to those
different cases would get bigger and bigger or they would get harder and harder to solve.
And what Lifton and I showed was that if NP had small circuits, then this hierarchy would
collapse down to the second level. In other words, you wouldn't get any more mileage by
complicating your expressions with three quantifiers or four quantifiers or any number.
I'm not sure what to make of that exactly. Well, I think it would be evidence that
NP doesn't have small circuits because something so bizarre would happen.
But again, it's only evidence, not proof. Well, yeah, that's not even evidence because
you're saying P is not equal to NP because something bizarre has to happen. I mean,
that's proved by the lack of bizarreness in our science. But it seems like
just the very notion of P equals NP would be bizarre. So any way you arrive at it,
there's no way you have to fight the dragon at some point.
For whatever it's worth, that's what we proved.
Awesome. So that's a potential space of open interesting problems.
Yeah. Let me ask you about this other world of machine learning, of deep learning.
What's your thoughts on the history and the current progress of machine learning field
that's often progressed sort of separately as a space of ideas and space of people
than the theoretical computer science or just even computer science world?
Yeah, it's really very different from the theoretical computer science world because the
results about algorithmic performance tend to be empirical. It's more akin to the world of
sat-solvers where we observe that for formulas arising in practice, the solver does well.
So it's of that type. We're moving into the empirical evaluation of algorithms.
Now, it's clear that there have been huge successes in image processing, robotics,
natural language processing a little less so. But across the spectrum of
game playing is another one. There have been great successes. And one of those
effects is that it's not too hard to become a millionaire if you can get a reputation in
machine learning and there'll be all kinds of companies that will be willing to offer you
the moon because they think that if they have AI at their disposal, then they can solve all
kinds of problems. But there are limitations. One is that the solutions that you get to
supervise learning problems through convolutional neural networks
seem to perform amazingly well even for inputs that are outside the training set.
But we don't have any theoretical understanding of why that's true.
Secondly, the solutions, the networks that you get are very hard to understand,
and so very little insight comes out. So yeah, they may seem to work on your training set,
and you may be able to discover whether your photos occur in a different sample of inputs or not.
But we don't really know what's going on. We don't know the features that distinguish the
photographs or the objects are not easy to characterize.
Well, it's interesting because you mentioned coming up with a small circuit
to solve a particular size problem. It seems that neural networks are kind of small circuits.
But they're not programs, sort of like the things you've designed are algorithms,
programs, algorithms. Neural networks aren't able to develop algorithms to solve a problem.
Well, they are algorithms. It's just that they're...
It could be a semantic question, but there's not a algorithmic style manipulation of the input.
Perhaps you could argue there is. Well, it feels a lot more like a function of the input.
It's a function. It's a computable function. Once you have the network, you can
simulate it on a given input and figure out the output. But if you're trying to recognize images,
then you don't know what features of the image are really being
determinant of what the circuit is doing. The circuit is sort of very intricate and
it's not clear that the simple characteristics that you're looking for, the edges of the objects
or whatever they may be, they're not emerging from the structure of the circuit.
Well, it's not clear to us humans, but it's clear to the circuit.
Yeah. Well, right. I mean, it's not clear to the elephant how the human brain works,
but it's clear to us humans. We can explain to each other our reasoning and why the cognitive
science and psychology field exists. Maybe the whole thing of being explainable to humans
is a little bit overrated. Oh, maybe, yeah. I guess you could say the same thing about our brain
that when we perform acts of cognition, we have no idea how we do it really. We do, though. I mean,
for at least for the visual system, the auditory system and so on, we do get some understanding
of the principles that they operate under, but for many deeper cognitive tasks, we don't have that.
That's right. Let me ask. Yeah.
You've also been doing work on bioinformatics. Does it amaze you that the fundamental building
blocks, if we take a step back and look at us humans, that building blocks used by evolution
to build us intelligent human beings is all contained there in our DNA?
It's amazing, and what's really amazing is that we are beginning to learn how to edit
DNA, which is very, very, very fascinating. This ability to
take a sequence, find it in the genome, and do something to it.
That's really taking our biological systems towards the worlds of algorithms.
But it raises a lot of questions. You have to distinguish between doing it on an individual
or doing it on somebody's germline, which means that all of their descendants will be affected.
So, that's an ethical…
Yeah. So, it raises very severe ethical questions.
Even doing it on individuals, there's a lot of hubris involved that you can assume that
knocking out a particular gene is going to be beneficial because you don't know what the
side effects are going to be. So, we have this wonderful new world of gene editing,
which is very, very impressive. It could be used in agriculture. It could be used in medicine in
various ways, but very serious ethical problems arise.
What are, to you, the most interesting places where algorithms…
Sort of the ethical side is an exceptionally challenging thing that I think we're going
to have to tackle with all of genetic engineering. But on the algorithmic side,
there's a lot of benefit that's possible. So, is there areas where you see exciting possibilities
for algorithms to help model optimized study biological systems?
Yeah. I mean, we can certainly analyze genomic data to figure out
which genes are operative in the cell and under what conditions and which proteins affect one
another, which proteins physically interact. We can sequence proteins and modify them.
Is there some aspect of that that's a computer science problem,
or is that still fundamentally a biology problem?
Well, it's a big data, it's a statistical big data problem for sure. So, the biological data
sets are increasing. Our ability to study our ancestry, to study the tendencies towards disease,
to personalize treatment according to what's in our genomes and what tendencies for disease we have,
to be able to predict what troubles might come upon us in the future and anticipate them to
understand whether you, for a woman, whether her perclivity for breast cancer is so strong enough
that you would want to take action to avoid it. You dedicate your 1985 Turing Award lecture to
the memory of your father. What's your fondest memory of your dad?
Seeing him standing in front of a class at the blackboard drawing perfect circles
by hand and showing his ability to attract the interest of the motley collection of
eighth grade students that he was teaching. When did you get a chance to see him draw the perfect
circles? On rare occasions, I would get a chance to sneak into his classroom and observe it.
I think he was at his best in the classroom. I think he really came to life and had fun,
not only teaching, but engaging in chit chat with the students and ingratiating himself
with the students. What I inherited from that is a great desire to be a teacher.
I retired recently and a lot of my former students came, students with whom I had done
research or who had read my papers or who had been in my classes. When they talked about
me, they talked not about my 1979 paper or 1992 paper, but about what came away in my classes
and not just the details, but just the approach and the manner of teaching. I take pride in the
at least in my early years as a faculty member at Berkeley. I was exemplary in preparing my
lectures and I always came in prepared to the teeth and able, therefore, to deviate according
to what happened in the class and to really provide a model for the students.
Is there advice you could give out for others on how to be a good teacher? Preparation is
one thing you've mentioned being exceptionally well-prepared, but are there other pieces of
advice that you can impart? Well, the top three would be preparation, preparation, and preparation.
Why is preparation so important, I guess? It's because it gives you the ease to deal with any
situation that comes up in the classroom. If you discover that you're not getting through one way,
you can do it another way. If the students have questions, you can handle the questions.
Ultimately, you're also feeling the crowd, the students of what they're struggling with,
what they're picking up, just looking at them through the questions, but even just through
their eyes. Yeah, that's right. Because of the preparation, you can dance.
You can dance. You can say it another way or give it another angle.
Are there, in particular, ideas and algorithms of computer science that you find were big aha
moments for students? Were they, for some reason, once they got it, it clicked for them and they
fell in love with computer science? Or is it individual? Is it different for everybody?
It's different for everybody. You have to work differently with students. Some of them just
don't need much influence. They're just running with what they're doing and they just need an
ear now and then. Others need a little prodding. Others need to be persuaded to collaborate
among themselves rather than working alone. They have their personal ups and downs, so
you have to deal with each student as a human being and bring out the best.
Humans are complicated. Perhaps a silly question. If you could relive a moment in your life outside
of family because it made you truly happy or perhaps because it changed the direction of your life in
a profound way, what moment would you pick? I was kind of a lazy student as an undergraduate
and even in my first year in graduate school and I think it was when I started doing research.
I had a couple of summer jobs where I was able to contribute and I had an idea
and then there was one particular course on mathematical methods and operations research
where I just gobbled up the material and I scored 20 points higher than anybody else in the class
then came to the attention of the faculty and it made me realize that I had some ability
that was going somewhere. You realize you're pretty good at this thing.
I don't think there's a better way to end it, Richard. It was a huge honor.
Thank you for decades of incredible work. Thank you for talking to us.
Thank you. It's been a great pleasure and you're a superb interviewer.
Thanks for listening to this conversation with Richard Karp and thank you to our sponsors,
and to the faculty and the staff for supporting us.
It really is the best way to support this journey I'm on.
If you enjoy this thing, subscribe on YouTube, review it with 5 stars on Apple Podcast,
support it on Patreon, connect with me on Twitter at Lex Freedman if you can figure out
how to spell that. And now let me leave you with some words from Isaac Asimov.
I do not fear computers. I fear lack of them. Thank you for listening and hope to see you next time.