Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses

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 cooperatively decide 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 we show that if satisfiability for n-variable dCNF formulas has a protocol of cost O(nd) 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 = 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 probab...

• Type: Conference
• Year: 2010
• Where: STOC
• Authors: Holger Dell and Dieter van Melkebeek
• Keywords: Algorithms, NP-hard Vertex Deletion, STOC 2010, Two-player Communication Process, Vertex Cover