Sum-Max Graph Partitioning Problem

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 (V1,…,Vk) 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.

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