After McMullen’s laudatio on Mirzakhani, it was time for Sanjeev Arora to talk about the work of the Nevanlinna prize winner Subhash Khot. It was also the time that a significant proportion of the audience decided that enough was enough and left the room. The same thing happened in Hyderabad four years ago, and on both occasions I was fairly shocked: I think it shows a striking disrespect, not so much for the speaker and prizewinner, though there is that aspect too, as for theoretical computer science in general. It seems to say, “Right, that’s the interesting prizes over — now we’re on to the ones that don’t really matter.” Because I have always been interested in computational complexity and related areas, my interest in the Nevanlinna prize is comparable to my interest in the Fields medals — indeed, in some ways it is greater because there is more chance that I will properly appreciate the achievements of the winner. And the list of past winners is incredible and includes some of my absolute mathematical heroes.
When the announcement was made a few hours earlier, my knowledge of Subhash Khot could be summarized as follows.
- He’s the person who formulated the unique games conjecture.
- I’ve been to a few talks on that in the past, including at least one by him, and there have been times in my life when I have briefly understood what it says.
- It’s a hardness conjecture that is a lot stronger than the assertion that PNP, and therefore a lot less obviously true.
What I hoped to get out of the laudatio was a return to the position of understanding what it says, and also some appreciation of what was so good about Khot’s work. Anybody can make a conjecture, but one doesn’t usually win a major prize for it. But sometimes a conjecture is so far from obvious, or requires such insight to formulate, or has such an importance on a field, that it is at least as big an achievement as proving a major theorem: the Birch–Swinnerton-Dyer conjecture and the various conjectures of Langlands are two obvious examples.
The unique games conjecture starts with a problem at the intersection of combinatorics and linear algebra.
Suppose you are given a collection of linear equations over the field . Then you can use Gaussian elimination to determine whether or not they have a solution. Now suppose that you find out that they do not have a solution. Then something you might consider doing is looking for an assignment to the variables that solves as many of the equations as possible. If , then a random assignment will solve on average half the equations, so it must be possible to solve at least half the equations. So the interesting thing is to do better than 50%. A famous result of Johan Håstad states that this cannot be done, even when each equation involves just three variables. (Actually, that restriction to three variables is not the surprising aspect — there are many situations where doing something for 2 is easy and the difficulty kicks in at 3. For example, it is easy to determine whether a graph is 2-colourable — you just start at a vertex, colour all its neighbours differently, etc. etc., and since all moves are forced apart from when you start again at a new connected component, if the process doesn’t yield a colouring then you know there isn’t one — but NP-hard to determine whether it is 3-colourable.)
More precisely, Håstad’s result says that for any fixed , if there were a polynomial-time algorithm that could tell you whether it was possible to satisfy at least a proportion of a collection of linear equations over (each equation involving three variables), then P would equal NP. His proof relies on one of the big highlights of theoretical computer science: the PCP theorem.
The unique games conjecture also concerns maximizing the number of linear equations you can solve, but this time we work mod and the equations are very special: they take the form .
To get a little intuition about this, I suppose one should do something I haven’t done until this very moment, and think about how one might go about finding a good algorithm for solving as many equations of this type from some collection as possible. An obvious observation is that once we’ve chosen , the value of is determined if we want to solve the equation . And that may well determine another variable, and so on. It feels natural to think of these equations as a labelled directed graph with the variables as vertices and with an edge from to labelled if the above equation is present in the system. Then following the implications of a choice of variables is closely related to exploring the component of that vertex in the graph. However, since our aim is to solve as many equations as possible, rather than all of them, we have the option of removing edges to make our task easier, though we want to remove as few edges as possible.
Maybe those few remarks will make it seem reasonably natural that the unique games conjecture can be connected with something called the max cut problem. This is the problem of finding a partition of the vertices of a graph into two sets such that the number of edges from one side to the other is as big as possible.
Actually, while browsing some slides of Håstad, I’ve just seen the following connection, which seems worth mentioning. If and all the equal 1, then if and only if the variables and get different assignments. So in this case, solving as many equations as possible is precisely the same as the max cut problem.
However, before we get too carried away with this, let me say what the unique games conjecture actually says. Apparently it has been reformulated a few times, and this version comes from 2004, whereas the original version was 2002. It says that even if 99% of the equations (of the form over ) can be simultaneously satisfied, then it is still NP hard to determine whether 1% of them can be simultaneously satisfied. Note that it is important to allow to be large here, since the random approach gives you a proportion straight away. Also, I think 99% and 1% are a friendly way of saying and for an arbitrary fixed .
In case the statement isn’t clear, let me put it slightly more formally. The unique games conjecture says the following. Suppose that for some there exists a polynomial-time algorithm that outputs YES if a proportion of the equations can be solved simultaneously and NO if it is impossible to solve more than a proportion of them, with no requirements on what the algorithm should output if the maximum proportion lies between and . Then P=NP.
At this point I should explain why the conjecture is called the unique games conjecture. But I’m not going to because I don’t know. I’ve been told a couple of times, but it never stays in my head, and when I do get told, I am also told that the name is something of a historical accident, since the later reformulations have nothing to do with games. So I think the name is best thought of as a strange type of label whose role is simply to identify the conjecture and not to describe it.
To give an idea of why the UGC is important, Arora took us back to an important paper of Goemans and Williamson from 1993 concerning the max cut problem. The simple random approach tells us that we can find a partition such that the size of the resulting cut is at least half the number of edges in the graph, since each edge has a 50% chance of joining a vertex in one half to a vertex in the other half. (Incidentally, there are standard “derandomization” techniques for converting observations like this into algorithms for finding the cuts. This is another beautiful idea from theoretical computer science, but it’s been around for long enough that people have got used to it.)
Goemans and Williamson were the first people to go beyond 50%. They used semidefinite programming to devise an algorithm that could find a cut for which the number of edges was at least 0.878 times the size of the max cut. I don’t know what that 0.878 really is — presumably some irrational number that came out of the proof — but it was sufficiently unnatural looking that there was a widespread belief that the bound would in due course be improved further. However, a check on that belief was given in 2004 by Khot, Kindler, Mossel and O’Donnell and in 2005 by Mossel, O’Donnell and Oleskiewicz (how they all contributed to the result I don’t know), who showed the very surprising result that if UGC is true, then the Goemans-Williamson bound is optimal. From what I understand, the proof is a lot more than just a clever observation that max cut can be reduced to unique games. If you don’t believe me, then try to explain to yourself how the constant 0.878 can arise in a simple way from a conjecture that involves only the constants “nearly 0” and “nearly 1”.
In general, it turns out that UGC implies sharp thresholds for approximability for many problems. What this means is that there is some threshold, below which you can do what you want with a polynomial-time algorithm and above which doing what you want is NP hard. (So in the max cut example the threshold is 0.878: getting smaller than that proportion can be done in polynomial time, and getting above that proportion is NP hard — at least if you believe UGC.)
Almost as interesting is that the thresholds predicted by UGC all come from rather standard techniques such as semidefinite programming and linear programming. So in some sense it is telling us not just that a certain bound is best possible but that a certain technique is best possible. To put it a bit crudely and inaccurately, it’s saying that for one of these problems, the best you can do with semidefinite programming is the best you can do full stop.
Arora said something even stronger that I haven’t properly understood, but I reproduce it for completeness. Apparently UGC even tells us that the failure of a standard algorithm to beat the threshold on a single instance implies that no algorithm can do better. I suppose that must mean that one can choose a clever instance in such a way that if the standard algorithm succeeds with that instance, then that fact can be converted into a machine for solving arbitrary instances of UGC. How you get from one instance of one problem to lots of instances of another is mysterious to me, but Arora did say that this result came as a big surprise.
There were a couple of other things that Arora said at the end of his talk to explain why Khot’s work was important. Apparently while the UGC is just a conjecture, and not even a conjecture that is confidently believed to be true (indeed, if you want to become famous, then it may be worth trying your hand at finding an efficient algorithm for it, since there seems to be a non-negligible chance that such an algorithm exists), it has led to a number of non-obvious predictions that have then been proved unconditionally.
Soon after Arora’s laudatio, Khot himself gave a talk. This was an odd piece of scheduling, since there was necessarily a considerable overlap between the two talks (in their content, that is). I’ll end by mentioning a reformulation of UGC that Khot talked about and Arora didn’t.
A very important concept in graph theory is that of expansion. Loosely speaking, a graph is called an expander if for any (not too large) set of vertices, there are many edges from that set to its complement. More precisely, if is a -regular graph and is a set of vertices, then we define the expansion of to be the number of edges leaving divided by (the latter being the most such edges there could possibly be). Another way of looking at this is that you pick a random point and a random neighbour of , and define the expansion of to be the probability that is not in .
The expansion of the graph as a whole is the minimum expansion over all subsets of size at most (where is the number of vertices of ). If this quantity is high, it is saying that is “highly interconnected”.
Khot is interested in small-set expansion. That is, he picks a small and takes the minimum over sets of size at most rather than at most .
The precise reformulation I’m about to give is not in fact the one that Khot gave but rather a small modification that Boaz Barak, another well-known theoretical computer scientist, gave in his invited lecture a day later. The unique games conjecture is equivalent to the assertion that it is NP hard to distinguish between the following two classes of graphs.
- Graphs where there exists a set of size at most with small expansion.
- Graphs where every set of size at most has very big expansion.
I think for the latter one can take the expansion to be at least 1/2 for each such set, whereas for the former it is at most for some small that you can probably choose.
What is interesting here is that for ordinary expansion there is a simple characterization in terms of the size of the second largest eigenvalue of the adjacency matrix. Since eigenvalues can be approximated efficiently, there is an efficient method for determining whether a graph is an expander. UGC is equivalent to saying that when the sets get small, their expansion properties can “hide” in the graph in a rather strong way: you can’t tell the difference between a graph that has very good small-set expansion and a graph where there’s a set that fails very badly.
I had lunch with Boaz Barak on one of the days of the congress, so I asked him whether he believed UGC. He gave me a very interesting answer (a special case of the more general proposition that Boaz Barak has a lot of very interesting things to say about complexity), which I have unfortunately mostly forgotten. However, my rapidly fading memory is that he would like it to be true, because it would be a beautiful description of the boundary of what algorithms can do, but thinks it may very well be false. He thought that one possibility was that solving the problems that UGC says are NP hard is not in fact NP hard, but not possible in polynomial time either. It is perfectly possible for a problem to be of intermediate difficulty.
Although it wouldn’t directly contradict NP hardness, it would be very interesting to find an algorithm that solved the small-set expansion problem in a time that was only modestly superpolynomial: something like , say. That would probably get you an invitation to speak at an ICM.