On Refutation of Random Constraint Satisfaction Problems

Given a k-ary predicate P , a random instance of a constraint satisfaction problem (CSP) is a set of m constraints on n variables. Each constraint is P applied to k random literals. Such an instance is unsatisfiable with high probability when m n. Although this fact can be proven via a simple probabilistic argument, certifying that a given randomly chosen instance is unsatisfiable, a task called refutation, remains a challenge. Refutation, besides being a natural problem in its own right, has applications in cryptography, hardness of approximation, and learning theory. Our objective is to determine how many constraints are required for both polynomial-time and nondeterministic refutation, in which we show only the existence of a polynomial-size certificate of unsatisfiability. We propose exploring this question in the following specific settings. • First, we study refutation based on semidefinite programming (SDP) relaxations, a class of algorithms that contains most known polynomial-time refutation algorithms. For SDP algorithms, we conjecture that the number of constraints required to refute a random instance of a CSP with k-ary predicate P is nC(P )/2, where C(P ) is the minimum t for which there is no t-wise uniform distribution over {0, 1}k supported on satisfying assignments to P . With Allen and O’Donnell, we proved that Õ(nC(P )/2) constraints are sufficient. Next, we want show that at least nC(P )/2 constraints are necessary, a result that would prove our conjecture and essentially complete our understanding of SDP-based refutation. • We would also like to determine the number of constraints required for nondeterministic refutation of random CSPs. We propose approaches for obtaining nondeterministic refutation for some specific predicates with fewer constraints than current techniques allow. In addition, we consider the question of proving lower bounds on the number of constraints required for nondeterministic refutation in specific proof systems. • We ask whether refutation is possible for semirandom instances constructed by applying random perturbations to an arbitrary instance. One route toward this goal is to develop a strong refutation algorithm for semirandom instances of kXOR. That is, we would like to certify that only about 1/2 of the constraints of a semirandom k-XOR instance can be simultaneously satisfied. January 27, 2016 Proposal Draft