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Author Dell, Holger ♦ Van Melkebeek, Dieter
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2014
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Sparsification ♦ Arithmetic progression free sets ♦ Feedback vertex set ♦ Hereditary graph properties ♦ Kernelization ♦ Parameterized complexity ♦ Probabilistically checkable proofs ♦ Satisfiability ♦ Vertex cover ♦ Vertex deletion problems
Abstract Consider the following two-player communication process to decide a language $\textit{L}:$ The first player holds the entire input $\textit{x}$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $\textit{x};$ their goal is to decide cooperatively whether $\textit{x}$ belongs to $\textit{L}$ 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 $\textit{d} ≥ 3$ and positive real $\textit{ε},$ we show that, if satisfiability for $\textit{n}-variable$ $\textit{d}-CNF$ formulas has a protocol of cost $\textit{O}(\textit{nd} ™ \textit{ε}),$ 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 $\textit{ε} = 0.$ 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 $\textit{n}-vertex$ $\textit{d}-uniform$ hypergraphs, this statement holds for any integer $\textit{d} ≥ 2.$ The case $\textit{d} = 2$ implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of $\textit{O}(\textit{k}2 ™ \textit{ε})$ edges unless coNP is in NP/poly, where $\textit{k}$ denotes the size of the deletion set. Kernels consisting of $\textit{O}(\textit{k}2)$ edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2014-07-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 61
Issue Number 4
Page Count 27
Starting Page 1
Ending Page 27

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Source: ACM Digital Library