Sum-Max Graph Partitioning Problem (original) (raw)

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, . . . , V k ) of V that minimizes k-1 i=1 k j=i+1 maxu∈V i ,v∈V j w(u, v), where w(u, v) denotes the weight of the edge {u, v} ∈ E. We establish the N P-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 N P-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 k 2 , and show that our analysis is tight.