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An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs

Author Meike Neuwohner



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Meike Neuwohner
  • Research Institute for Discrete Mathematics, Universität Bonn, Germany

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Meike Neuwohner. An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 53:1-53:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.53

Abstract

In this paper, we consider the task of computing an independent set of maximum weight in a given d-claw free graph G = (V,E) equipped with a positive weight function w:V → ℝ^+. Thereby, d ≥ 2 is considered a constant. The previously best known approximation algorithm for this problem is the local improvement algorithm SquareImp proposed by Berman [Berman, 2000]. It achieves a performance ratio of d/2+ε in time 𝒪(|V(G)|^(d+1)⋅(|V(G)|+|E(G)|)⋅(d-1)²⋅ (d/(2ε)+1)²) for any ε > 0, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of d/2-(1/63,700,992)+ε for any ε > 0 at the cost of an additional factor of 𝒪(|V(G)|^(d-1)²) in the running time. In particular, our result implies a polynomial time d/2-approximation algorithm. Furthermore, the well-known reduction from the weighted k-Set Packing Problem to the Maximum Weight Independent Set Problem in k+1-claw free graphs provides a (k+1)/2 -(1/63,700,992)+ε-approximation algorithm for the weighted k-Set Packing Problem for any ε > 0. This improves on the previously best known approximation guarantee of (k+1)/2 + ε originating from the result of Berman [Berman, 2000].

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • d-Claw free Graphs
  • independent Set
  • local Improvement
  • k-Set Packing
  • weighted

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References

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