Extended Formulation Lower Bounds for Refuting Random CSPs

Abstract

Random constraint satisfaction problems (CSPs) such as random 3-SAT are conjectured to be computationally intractable. The average case hardness of random 3-SAT and other CSPs has broad and far-reaching implications on problems in approximation, learning theory and cryptography.In this work, we show subexponential lower bounds on the size of linear programming relaxations for refuting random instances of constraint satisfaction problems. Formally, suppose P : {0, 1} k → {0, 1} is a predicate that supports a t −1-wise uniform distribution on its satisfying assignments. Consider the distribution of random instances of CSP P with m ∆n constraints. We show that any linear programming extended formulation that can refute instances from this distribution with constant probability must have size at least Ω exp n t−2For example, this yields a lower bound of size exp(n 1/3 ) for random 3-SAT with a linear number of clauses. We use the technique of pseudocalibration to directly obtain extended formulation lower bounds from the planted distribution. This approach bypasses the need to construct Sherali-Adams integrality gaps in proving general LP lower bounds. As a corollary, one obtains a self-contained proof of subexponential Sherali-Adams LP lower bounds for these problems. We believe the result sheds light on the technique of pseudocalibration, a promising but conjectural approach to LP/SDP lower bounds. of refutation is widely believed to have super-polynomial computational complexity for this entire range of clause densities.Linear and Semidefinite Programming Relaxations for CSPs. With even the worst-case complexity of P NP still out of reach, the average case complexity of random CSPs is well beyond the realm of current techniques to conclusively settle. Therefore, an approach to gather evidence towards hardness of refuting random CSPs is to consider restricted classes of algorithms. Linear and semidefinite programming relaxations are the prime candidates to pursue, since they yield the best known algorithms for CSPs in many worst-case settings. Specific linear and semidefinite programming relaxations such as the Sherali-Adams LP hierarchy and the Sum-of-Squares SDP hierarchy have been extensively studied, and a fairly detailed and clear picture has emerged in the literature. In Section 1.2 we briefly survey the lower bound results for the sum-of-squares SDP (also known as the Lasserre/Parrilo SDP hierarchy), which is the most powerful of the specific LP/SDP hierarchies (see [KMOW17] for a detailed survey).Extended Formulations. In a foundational work, Yannakakis [Yan91] laid out the model of LP extended formulations to capture the notion of a general linear program for a problem. Roughly speaking, an extended formulation for a problem Π consists of a family of linear programs for every input size n with the key restriction being that the feasible region of the linear program depends solely on the input size n, and is independent of the specific instance of the problem (analogous to having a single circuit independen...

Lifting sum-of-squares lower bounds: degree-2 to degree-4

“…Subsequently, the pseudocalibration technique was used in [17] to show SoS lower bounds for Tensor PCA and Sparse PCA. Building on ideas from pseudocalibration, Hopkins and Steurer [18] recovered conjectured computational thresholds in community detection, while [19] use it towards showing LP extended formulation lower bounds for Random 3SAT.…”

Lifting Sum-of-Squares Lower Bounds: Degree-$2$ to Degree-$4$

“…The problem of certifying the size of maximum clique on Erdos-Renyi random graphs (closely related to the planted clique problem) has received much attention lately. Following a series of works [DM15, HKP + 18] that obtained the tight lower bounds for degree four, the breakthrough tour-de-force of Barak et [HS17] recovered conjectured computational thresholds in community detection, while [IBCR20] use it towards showing LP extended formulation lower bounds for Random 3SAT.…”