Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses

Abstract

Consider the following two-player communication process to decide a language L : The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x ; their goal is to decide cooperatively whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d  ≥ 3 and positive real ε , we show that, if satisfiability for n -variable d -CNF formulas has a protocol of cost O ( nd  −  ε ), then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ε  = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on n -vertex d -uniform hypergraphs, this statement holds for any integer d  ≥ 2. The case d  = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O ( k 2 −  ε ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O ( k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.