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Primal LP and dual LP admit local approximation schemes
- Publisher
- Springer Journals
- Copyright
- Copyright © 2013 by Springer-Verlag Berlin Heidelberg
- Subject
- Computer Science; Computer Communication Networks; Computer Hardware; Computer Systems Organization and Communication Networks; Software Engineering/Programming and Operating Systems; Theory of Computation
- ISSN
- 0178-2770
- eISSN
- 1432-0452
- DOI
- 10.1007/s00446-013-0194-z
- Publisher site
- See Article on Publisher Site
Abstract
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 $$\epsilon > 0$$ ϵ > 0 there exists a constant-time distributed algorithm that finds a $$(1+\epsilon )$$ ( 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 $$\delta > 0$$ δ > 0 so that no randomised distributed algorithm with running time $$o(\log n)$$ o ( log n ) can find a $$(1+\delta )$$ ( 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.
Journal
Distributed Computing – Springer Journals
Published: Sep 17, 2013