Résumé
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 ${u,v} \in E$. We establish the 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 P when $k = 3$, but the NP-hardness for all fixed $k \ge 4$ if one vertex per cluster is fixed. Lastly, we present a natural greedy algorithm with an approximation ratio better than k/2, and show that our analysis is tight.