Dieter van Melkebeek at theory lunch, Oct 30

Theory lunch talk, Friday October 30:

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

Dieter van Melkebeek, University of Wisconsin Madison

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 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 $d \geq 3$ and positive real $\epsilon$ we show that if satisfiability for $n$-variable $d$-CNF formulas has a protocol of cost $O(n^{d-\epsilon})$ then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to the third level. The result even holds for conondeterministic protocols, and is tight as
there exists a trivial deterministic protocol for $\epsilon = 0$. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, including 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 the above statement holds for any integer $d \geq 2$. The case $d=2$ implies that no NP-hard vertex deletion problem based on a hereditary graph property can have kernels consisting of $O(k^{2-\epsilon})$
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, bounded-degree
deletion, and feedback vertex set.

Joint work with Holger Dell.