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)


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.


Greedy Algorithm IEEE Computer Society Approximation Ratio Edge Weight Maximum Weight 
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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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