TY - JOUR

T1 - Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms

AU - Parnas, Michal

AU - Ron, Dana

N1 - Funding Information:
We would like to thank Oded Goldreich for helpful conceptual discussions. We would also like to thank Asaf Shapira for a helpful comment. We thank Luca Trevisan for telling us about the lower bound described in Section 6, and for suggesting that we present a sketch of this lower bound in the paper. Finally we thank the anonymous referees of this paper for their helpful comments and suggestions. The second author’s research was supported by the Israel Science Foundation (grant number 89/05).

PY - 2007/8/22

Y1 - 2007/8/22

N2 - For a given graph G over n vertices, let OPTG denote the size of an optimal solution in G of a particular minimization problem (e.g., the size of a minimum vertex cover). A randomized algorithm will be called an α-approximation algorithm with an additive error for this minimization problem if for any given additive error parameter ε{lunate} > 0 it computes a value over(OPT, ̃) such that, with probability at least 2 / 3, it holds that OPTG ≤ over(OPT, ̃) ≤ α {dot operator} OPTG + ε{lunate} n. Assume that the maximum degree or average degree of G is bounded. In this case, we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem. This reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms, such as the dominating set problem. We also show that for the minimum vertex cover problem, the query complexity of such approximation algorithms must grow at least linearly with the average degree over(d, ̄) of the graph. This lower bound holds for every multiplicative factor α and small constant ε{lunate} as long as over(d, ̄) = O (n / α). In particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o (n).

AB - For a given graph G over n vertices, let OPTG denote the size of an optimal solution in G of a particular minimization problem (e.g., the size of a minimum vertex cover). A randomized algorithm will be called an α-approximation algorithm with an additive error for this minimization problem if for any given additive error parameter ε{lunate} > 0 it computes a value over(OPT, ̃) such that, with probability at least 2 / 3, it holds that OPTG ≤ over(OPT, ̃) ≤ α {dot operator} OPTG + ε{lunate} n. Assume that the maximum degree or average degree of G is bounded. In this case, we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem. This reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms, such as the dominating set problem. We also show that for the minimum vertex cover problem, the query complexity of such approximation algorithms must grow at least linearly with the average degree over(d, ̄) of the graph. This lower bound holds for every multiplicative factor α and small constant ε{lunate} as long as over(d, ̄) = O (n / α). In particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o (n).

KW - Distributed algorithms

KW - Minimum vertex cover

KW - Sublinear approximation algorithms

UR - http://www.scopus.com/inward/record.url?scp=34547153784&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2007.04.040

DO - 10.1016/j.tcs.2007.04.040

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:34547153784

SN - 0304-3975

VL - 381

SP - 183

EP - 196

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1-3

ER -