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.
This work has been funded by grant ANR 2010 BLAN 021902.
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Watrigant, R., Bougeret, M., Giroudeau, R., König, J.C. (2012). Sum-Max Graph Partitioning Problem. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_27
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DOI: https://doi.org/10.1007/978-3-642-32147-4_27
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