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Recoverable Values for Independent Sets

  • Uriel Feige
  • Daniel Reichman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

The notion of recoverable value was advocated in work of Feige, Immorlica, Mirrokni and Nazerzadeh [Approx 2009] as a measure of quality for approximation algorithms. There this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has recoverable value ρ, if for every graph it recovers an independent set of size at least max I  ∑  v ∈ I min [1,ρ/(d(v) + 1)], where d(v) is the degree of vertex v, and I ranges over all independent sets in G. Hence, in a sense, from every vertex v in the maximum independent set the algorithm recovers a value of at least ρ/(d v  + 1) towards the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. It easily follows from known results that some simple algorithms for MIS ensure ρ ≥ 1. We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least ρ ≥ 7/3. In addition, we show that approximating MIS in graphs with a given k-coloring within a ratio larger than 2/k is unique games hard. This rules out a natural approach for obtaining ρ ≥ 2.

Keywords

Approximation Algorithm Greedy Algorithm Average Degree Approximation Ratio Minimum Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Uriel Feige
    • 1
  • Daniel Reichman
    • 1
  1. 1.Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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