Sum-Max Graph Partitioning Problem

  • R. Watrigant
  • M. Bougeret
  • R. Giroudeau
  • J. -C. König
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

In this paper we consider the classical combinatorial optimization graph partitioning problem, with Sum-Max as objective function. Given a weighted graph G = (V,E) and a integer k, our objective is to find a k-partition (V 1,…,V k ) of V that minimizes \(\sum_{i=1}^{k-1}\sum_{j=i+1}^{k}\) \(\max_{u \in V_i, v \in V_j} ~w(u, v)\), where w(u,v) denotes the weight of the edge {u,v} ∈ E. We establish the \(\mathcal{NP}\)-completeness of the problem and its unweighted version, and the W[1]-hardness for the parameter k. Then, we study the problem for small values of k, and show the membership in \(\mathcal{P}\) when k = 3, but the \(\mathcal{NP}\)-hardness for all fixed k ≥ 4 if one vertex per cluster is fixed. Lastly, we present a natural greedy algorithm with an approximation ratio better than \(\frac{k}{2}\), and show that our analysis is tight.

Keywords

Greedy Algorithm IEEE Computer Society Approximation Ratio Edge Weight Maximum Weight 
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 2012

Authors and Affiliations

  • R. Watrigant
    • 1
  • M. Bougeret
    • 1
  • R. Giroudeau
    • 1
  • J. -C. König
    • 1
  1. 1.LIRMM-CNRS-UMR 5506MontpellierFrance

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