Strongly refuting random CSPs below the spectral threshold

Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with n variables and m clauses, there is a value of m = Ω(n) beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when m/n = ω(1)). Intuitively, strong refutation should become easier as the clause density m/n grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as k-SAT and k-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, m/n ≥ Ο(nk/2-1), and the clause density at which instances become unsatisfiable with high probability, m/n = ω (1). In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random k-XOR instances with clause density m/n ≥ Ο(n(k/2-1)(1-δ)) in time exp(Ο(nδ)) or in Ο(nδ) rounds of the sum-of-squares hierarchy, for any δ ∈ [0,1) and any integer k ≥ 3. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at δ = 0, and brute-force refutation at the satisfiability threshold when δ = 1. We also leverage our k-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors.