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 Host, 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.