No sublogarithmic-time approximation scheme for bipartite vertex cover

König’s theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every ϵ > 0 there exists a constant-time distributed algorithm that finds a ( 1 + ϵ ) -approximation of a maximum matching on bounded-degr...

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Vydáno v:Distributed computing Ročník 27; číslo 6; s. 435 - 443
Hlavní autoři: Göös, Mika, Suomela, Jukka
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014
Springer Nature B.V
Témata:
Algorithms
Approximation
Computer Communication Networks
Computer Hardware
Computer Science
Computer Systems Organization and Communication Networks
Constants
Graphs
Matching
Mathematical analysis
Running
Software Engineering/Programming and Operating Systems
Texts
Theory of Computation
Distributed graph algorithms
Vertex cover
Lower bounds
Local approximation
ISSN:0178-2770, 1432-0452
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Shrnutí:König’s theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every ϵ > 0 there exists a constant-time distributed algorithm that finds a ( 1 + ϵ ) -approximation of a maximum matching on bounded-degree graphs. In this work, we show—somewhat surprisingly—that no sublogarithmic-time approximation scheme exists for the dual problem: there is a constant δ > 0 so that no randomised distributed algorithm with running time o ( log n ) can find a ( 1 + δ ) -approximation of a minimum vertex cover on 2-coloured graphs of maximum degree 3. In fact, a simple application of the Linial–Saks (Combinatorica 13:441–454, 1993 ) decomposition demonstrates that this run-time lower bound is tight. Our lower-bound construction is simple and, to some extent, independent of previous techniques. Along the way we prove that a certain cut minimisation problem, which might be of independent interest, is hard to approximate locally on expander graphs.
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ISSN:0178-2770
1432-0452
DOI:10.1007/s00446-013-0194-z