In which we present a probabilistic construction of expanders.
In which we give an explicit construction of expander graphs of polylogarithmic degree, state the properties of the zig-zag product of graphs, and provide an explicit construction of a family of constant-degree expanders using the zig-zag product and the polylogarithmic-degree construction.
A family of expanders is a family of graphs , , such that each graph is -regular, and the edge-expansion of each graph is at least , for an absolute constant independent of . Ideally, we would like to have such a construction for each , although it is usually enough for most applications that, for some constant and every , there is an for which the construction applies in the interval , or even the interval . We would also like the degree to be slowly growing in and, ideally, to be bounded above by an explicit constant. Today we will see a simple construction in which and a more complicated one in which .
[In the provincial spirit of Italian newspapers, that often have headlines like “Typhoon in South-East Asia causes widespread destruction; what are the consequences for Italian exports?”, and of men who overhear discussions about women’s issue and say things like “yes, but men have issues too,” I am going to comment on how Babai’s announcement affects me and the kind of problems I work on.]
If someone had told me last week: “a quasi-polynomial time algorithm has been found for a major open problem for which only a slightly subexponential algorithm was known before,” I would have immediately thought Unique Games!
Before Babai’s announcement, Graph Isomorphism had certain interesting properties in common with problems such as Factoring, Discrete Log, and Approximate Closest Vector (for approximation ratios of the order of sqrt (n) or more): no polynomial time algorithm is known, non-trivial algorithms that are much faster than brute force are known, and NP-completeness is not possible because the problem belongs to either or .
But there is an important difference: there are simple distributions of inputs on which Factoring, Discrete Log, and Closest Vector approximation are believed to be hard on average, and if one proposes an efficiently implementable algorithms for such problems, it can be immediately shown that it does not work. (Or, if it works, it’s already a breakthrough even without a rigorous analysis.)
In the case of Graph Isomorphism, however, it is easy to come up with simple algorithms for which it is very difficult to find counterexamples, and there are algorithms that are rigorously proved to work on certain distributions of random graphs. Now we know that there are in fact no hard instances at all, but, even before, if we believed that Graph Isomorphism was hard, we had to believe that the hard instances were rare and strange, rather than common.
It is also worth pointing out that, using Levin’s theory of average-case complexity, one can show that if any problem at all in NP is hard under any samplable distribution, then for every NP-complete problem we can find a samplable distribution under which the problem is hard. And, in “practice,” natural NP-complete problems do have simple distributions that seem to generate hard instances.
What about Small-set Expansion, Unique Games, and Unique-Games-Hard problems not known to be NP-hard, like -approximation of Sparsest Cut? We don’t know of any distribution for which it is plausible to conjecture that such problems are hard, and we have algorithms (Lasserre relaxations of constant degree) with no known counterexample. Many simple distributions of instances are rigorously solved by known algorithms. So, if we want to believe the Unique Games conjecture, we have to believe that there are hard instances, but they are rare and strange.
I am sure that it is possible, under standard assumptions, to construct an artificial problem L in NP that is in average-case-P according to Levin’s definition but not in P. Such a problem would not be polynomial time solvable, but it would be easy to solve on average under any samplable distribution and, intuitively, it would be a problem that is hard even though hard instances are rare and strage.
But can a natural problem in NP exhibit this behavior? Now that Graph Isomorphism is not a plausible example any more, I am inclined to believe (until the next surprise) that no natural problem has this behavior, and my guess concerning the Unique Games conjectures is going to be that it is false (or “morally false” in the sense that a quasipolynomial time algorithm exists) until someone comes up with a distribution of Unique Games instances that are plausibly hard on average and that, in particular, exhibit integrality gaps for Lasserre relaxations (even just experimentally).
Next Tuesday, a week from today, Laci Babai will talk at the University of Chicago about a new algorithm that solves graph isomorphism in quasipolynomial time. There should also be a follow-up talk the following Thursday that, by a lucky coincidence, I will be able to attend, and then report back.
Meanwhile, if you have any gossip on the proof, then, by any means, go ahead and share it in the comments.
Back around 2010, the Simons Institute for the Theory of Computing at Berkeley offered to organize FOCS in 2013, 2015 and 2017. So far, the IEEE technical committee on mathematical foundations of computing has taken us up on this offer in 2013 and 2015, and, unless a competing bid is presented, FOCS will come again to Berkeley in 2017.
Unfortunately there is no hotel in downtown Berkeley that is able to accommodate FOCS. The Shattuck hotel almost but not quite is. (There are two conference rooms, but they are of very different size, and the space to hang out for coffee breaks is much too small for 200+ people, and it’s outdoors, which is potentially bad because rain in October is unlikely but not impossible in Berkeley.)
This leaves us with the Doubletree hotel in the Berkeley Marina, which has some advantages, such as views of the bay and good facilities, and some disadvantages, such as the isolated location and the high prices. The location also forces us to provide lunches, because it would be inconvenient for people to drive to lunch places and then drive back during the lunch break. Being well aware of this, the hotel charges extortionate fees for food.
This is to say that, planning for FOCS 2017, there is nothing much different that we can do, although there are lots of little details that we can adjust, and it would be great to know how people’s experience was.
For example, did the block of discounted hotel rooms run out too soon? Would you have liked to have received something else with your registration than just the badge? If so, what? (So far, I have heard suggestions for FOCS-branded hats, t-shirts, and teddy bears.) Wasn’t it awesome to have a full bar at the business meeting? Why did nobody try the soups at lunch? The soups were delicious!
This is an odd-numbered year, and FOCS is back in Berkeley. The conference, whose early registration deadline is coming up, will be held on October 18-20 at the Double Tree hotel near the Berkeley marina, the same location of FOCS 2013, and it will be preceded by a day-long conference in honor of Dick Karp’s 80th birthday.
Early registration closes next Friday, so make sure that you register before then.
The weekend before FOCS there will be the Treasure Island Music Festival; Treasure Island is halfway along the Bay Bridge between Oakland and San Francisco, and from the Island there are beautiful views of the Bay Area.
If you arrive on Friday the 16th and you want to spend an afternoon in San Francisco, at the end of the day you can find your way to the De Young Museum in Golden Gate park, which stays open until 8:30pm on Fridays, and it has live music and a bar in the lobby from 5:30 to 8:30.
Did I mention that the early registration deadline is coming up? Don’t forget to register.
Cui Peng of Renmin University in Beijing has recently released two preprints, one claiming a proof of P=NP and one claiming a refutation of the Unique Games Conjecture; I will call them the “NP paper” and the “UG paper,” respectively.
Of all the papers I have seen claiming a resolution of the P versus NP problem, and, believe me, I have seen a lot of them, these are by far the most legit. On Scott Aronson’s checklist of signs that a claimed mathematical breakthrough is wrong, they score only two.
Unfortunately, both papers violate known impossibility results.
The two papers follow a similar approach: a certain constraint satisfaction problem is proved to be approximation resistant (under the assumption that PNP, or under the UGC, depending on the paper) and then a Semidefinite Programming approximation algorithm is developed that breaks approximation resistance. (Recall that a constraint satisfaction problem is approximation resistant if there is no polynomial time algorithm that has a worst-case approximation ratio better than the algorithm that picks a random assignment.)
In both papers, the approximation algorithm is by Hast, and it is based on a semidefinite programming relaxation studied by Charikar and Wirth.
The reason why the results cannot be correct is that, in both cases, if the hardness result is correct, then it implies an integrality gap for the Charikar-Wirth relaxation, which makes it unsuitable to break the approximation resistance as claimed.
Suppose that we have a constraint satisfaction problem in which every constraint is satisfied by a fraction of assignment. Then for such a problem to not be approximation resistant, we have to devise an algorithm that, for some fixed positive , returns a solution whose cost (the number of constraints that it satisfies) is at least times the optimum. The analysis of such an algorithm needs to include some technique to prove upper bounds for the true optimum; this is because if you are given an instance in which the optimum satisfies at most a fraction of constraints, as is the case for a random instance, then the algorithm will satisfy at most a fraction of constraints, but then the execution of the algorithm and the proof of correctness will give a (polynomial-time computable and polynomial-time checkable) certificate that the optimum satisfies at most a fraction of constraints.
For algorithms that are based on relaxations, such certificates came from the relaxation itself: one shows that the algorithm satisfies a number of constraints that is at least times the optimum of the relaxation, and the optimum of the relaxation is at least the optimum of the constraint satisfaction problem. But if there are instances for which the optimum is and the optimum of the relaxation is , then one cannot use such a relaxation to design an algorithm that breaks approximation-resistance. (Because on, such instances, the algorithm will not be able to satisfy a number of constraint equal to times the optimum of the relaxation.)
In the UG paper, the approximation resistance relies on a result of Austrin and Håstad. Like all UGC-based inapproximability results that I am aware of, the hardness results of Austrin and Håstad are based on a long code test. A major result of Raghavendra is that for every constraint satisfaction problem one can write a certain SDP relaxation such that the integrality gap of the relaxation is equal to the ratio between soundness and completeness in the best possible long code test that uses predicates from the constraint satisfaction problem. In particular, in Section 7.7 of his thesis, Prasad shows that if you have a long code test with soundness and completeness for a constraint satisfaction problem, then for every there is an instance of the problem in which no solution satisfies more than fraction of constraints, but there is a feasible SDP solution whose cost is at least a fraction of the number of constraints. The SDP relaxation of Charikar and Wirth is the same as the one studied by Prasad. This means that if you prove, via a long code test, that a certain problem is approximation resistant, then you also show that the SDP relaxation of Charikar and Wirth cannot be used to break approximation resistance.
The NP paper adopts a technique introduced by Siu On Chan to prove inapproximability results by starting from a version of the PCP theorem and then applying a “hardness amplification” reduction. Tulsiani proves that if one proves a hardness-of-approximation result via a “local” approximation-reduction from Max 3LIN, then the hardness-of-approximation result is matched by an integrality gap for Lasserre SDP relaxations up to a super-constant number of rounds. The technical sense in which the reduction has to be “local” is as follows. A reduction from Max 3LIN (the same holds for other problems, but we focus on starting from Max 3LIN for concreteness) to another constraint satisfaction problems has two parameters: a “completeness” parameter and a “soundness” parameter , and its properties are that:
- (Completeness Condition) the reduction maps instances of 3LIN in which the optimum is to instances of the target problem in which the optimum is at least ;
- (Soundness Condition) the reduction maps instances of 3LIN in which the optimum is to instances of the target problem in which the optimum is at most .
Since we know that it’s NP-hard to distinguish Max 3LIN instances in which the optimum is from instances in which the optimum is , such a reduction shows that, in the target problem, it is NP-hard to distinguish instances in which the optimum is from instances in which the optimum is . The locality condition studied by Tulsiani is that the Completeness Condition is established by describing a mapping from solutions satisfying a fractions of the Max 3LIN constraints to solutions satisfying a fraction of the target problem constraints, and the assignment to each variable of the target problem can be computed by looking at a sublinear (in the size of the Max 3LIN instance) number of Max 3LIN variables. Reductions that follows the Chan methodology are local in the above sense. This means that if one proves that a problem is approximation-resistant using the Chan methodology starting from the PCP theorem, then one has a local reduction from Max 3LIN to the problem with completeness and soundness , where, as before, is the fraction of constraints of the target problem satisfied by a random assignment. In turn, this implies that not just the Charikar-Wirth relaxation, but that, for all relaxations obtained in a constant number of rounds of Lasserre relaxations, there are instances of the target problem that have optimum and SDP optimum , so that the approximation resistance cannot be broken using such SDP relaxations.
Sufficiently many to start a soccer team.
Some constraint satisfaction problems are approximation resistant, in the sense that, unless P=NP, there is no polynomial time algorithm that achieves a better approximation ratio than the trivial algorithm that picks a random assignment. For example, a random assignment satisfies (on average) a fraction of the clauses of a given Max 3SAT instance, and, for every , it is NP-hard to achieve approximation . Max 3LIN is the variant of Max 3SAT in which every constraint is a XOR instead of an OR of variables; it is another example of an approximation resistant problem, because a random assignment satisfies (on average) half of the constraints, and approximation is NP-hard for every . (These, and more, hardness of approximation results were proved by Håstad in 1997, in a paper with a notably understated title.)
In 2000, Håstad proved that if we restrict constraint satisfaction problems to instances in which every variable occurs in (at most) a fixed constant number of constraints, then the problem is never approximation resistant. If we have a constraint satisfaction problem in which each constraint is satisfied with probability by a random assignment, and each variable appears in at most constraint, then there is a simple polynomial time algorithm that achieves an approximation ratio . The following year, I showed that if we have a constraint satisfaction problem that is NP-hard to approximate within a factor of , then it is also NP-hard to approximate within a factor , where is a constant (whose value depends on the specific constraint satisfaction problem), when restricted to instances in which every variable occurs at most times.
Thus, for example, if we restrict to instances in which every variable occurs in at most constraints, Max 3SAT can be approximated within a factor but not , and Max 3LIN can be approximated within a factor but not in polynomial time, unless , where are constants.
Last Fall, Prasad Raghavendra and I worked for a while on the problem of bridging this gap. The difficulty with Max 3SAT is that there are instances derived from Max Cut such that every variable occurs in at most clauses, there is no “trivial contradiction” (such as 8 clauses over the same 3 variables, which have a fixed contribution to the cost function and can be eliminated without loss of generality), and every assignment satisfies at most clauses. If we want an approximation ratio , we need our algorithm to certify that such instances are unsatisfiable. It is probably possible to show that there are simple LP or SDP relaxations of Max 3SAT such that a polynomial time algorithm can find assignments that satisfies a number of clauses which is at least a times the optimum of the relaxation, but we could not find any way to reason about it, and we gave up. Also, we wrongly assumed that there was the same issue with Max 3LIN.
Meanwhile, Farhi, Goldstone and Gutmann, who had successfully developed a quantum algorithm to approximate Max Cut on bounded degree graphs, were looking for another problem to tackle, and asked Madhu Sudan what was known about NP-hard and Unique Games-hard problems on bounded degree instances. They were particularly interested in Max 3SAT in bounded-degree instances. Madhu referred them to me, Sam Gutmann happened to be in the Bay Area, and so we met in November and I pointed them to the known literature and suggested that they should try Max 3LIN instead.
A month later, I heard back from them, and they had a approximate algorithm for Max 3LIN. That sounded amazing, so I went looking into the paper for the section in which they discuss their upper bound technique, and there is none. They show that, for every instance that does not have trivial contradictions (meaning two constraints that are the negation of each other), there is an assignment that satisfies a fraction of constraints, and they describe a distribution that, on average, satisfies at least as many. The distribution is samplable by a quantum computer, so the approximation, in their paper, is achieved by a quantum algorithm.
After realizing that we had been wrong all along on the need for non-trivial upper bounds for Max 3LIN, Prasad and I tried to find a way to replicate the result of Farhi et al. with a classical algorithm, and we found a way to satisfy a fraction of constraints in instances of constraint satisfaction problems “without triangles” (a result of this form is also in the paper of Farhi et al.), and then a fraction of constraints in all Max 3LIN instances.
The day before submitting our paper to ICALP (from which it would have been rejected without consideration anyways), I saw a comment by Boaz Barak on Scott Aronson’s blog announcing the same results, so we got in contact with Boaz, who welcomed us to the club of people who had, meanwhile, gotten those results, which also included Ankur Moitra, Ryan O’Donnell, Oded Regev, David Steurer, Aravindan Vijayaraghavan, David Witmer, and John Wright. Later, Johan Håstad also discovered the same results. If you kept count, that’s eleven theoreticians.
The paper is now online (with only 10 authors, Johan
may write posted a separate note); we show that a fraction of constraints can be satisfied in all Max kLIN instances, with odd , and a advantage over the random assignment can be achieved in all “triangle-free” instances of constraint satisfaction problems. It remains an open problem to improve Håstad’s approximation for Max 3SAT.
The argument for Max 3LIN is very simple. Khot and Naor prove that, given a Max 3LIN instance , one can construct a bipartite Max 3LIN instance such that an assignment satisfying a fraction of constraints in can be easily converted into an assignment satisfying a fraction of constraints of ; furthermore, if every variable occurs in at most constraints of , then every variable occurs in at most constraints of .
An instance is bipartite if we can partition the set of variables into two subsets and , such that each constraint involves two variables from and one variable from . The reduction creates two new variables and for each variable of ; every constraint of is replaced by the three constraints
Given an assignment to the and variables that satisfies a fraction of the constraints of , Khot and Naor show that either , or , or an assignment obtained by choosing to be with probability or with probability , satisfies at least a fraction of constraints of .
It remains to show that, given a bipartite instance of Max 3LIN in which every variable occurs in at most constraints (and which does not contain two equations such that one is the negation of the other), we can find an assignment that satisfies a fraction of constraints.
The idea is to first pick the variables at random, and then to pick the variables greedily given the choice of the variables.
When we pick the variables at random, the instance reduces to a series of constraints of the form . Each variable belongs to (at most, but let’s assume exactly, which is actually the worst case for the analysis) such constraints; on average, half of those constraints will be and half will be . If the fixings of clauses of were mutually independent (which would be the case in “triangle-free” instances), then we would expect that the difference between 0s and 1s be about , so the greedy fixing has a advantage over the random assignment.
In general instances, although we do not have mutual independence, we do have pairwise independence and “almost four-wise independence.” Fix a variable , and let us call the set of pairs such that constraint is part of the instance, for some , and let us call the random variable which is if and otherwise, for a random choice of the variables. We want to argue that, with constant probability, .
First, we see that the are unbiased, and they are pairwise independent, so . The fourth moment of is plus the number of 4-cycles in the graph that has vertices and the edges in . Now, contains edges, a four-cycle is completely described by the first and third edge of the cycle, so the fourth moment is . Finally, it is a standard fact that if we have a sum of unbiased random variables, and the second moment of their sum is and the fourth moment of their sum is , then the absolute value of the sum is, on average (and with constant probability) .
The algorithm for general CSPs on triangle-free instances is similarly based on the idea or randomly fixing some variables and then greedily fixing the remaining variables. Without the reduction to the “bipartite” case, which does not hold for problems like Max 3SAT, it is more technically involved to argue the advantage over the random assignment.
Is there a polynomial time algorithm that achieves a approximation ratio for Max 3SAT in instances such that each variable occurs at most times? This remains an open question.
This year, the chair of ICALP decided to play an April Fool’s prank three weeks early, and I received the following message:
“Dear author, we regret to inform you that the margins in your submission are too small, and hence we are rejecting it without review”
I was almost fooled. In my defense, the second time that I applied for a position in Italy, the hiring committee judged all my publications to be non-existent, because the (multiple) copies I had sent them had not been authenticated by a notary. So I am trained not to consider it too strange that a paper could be evaluated based on the width of its margins (or the stamps on its pages) rather than on the content of its theorem.