Sum-max graph partitioning problem

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 ∑ i=1k-1 ∑ j=i+1k max u∈vi, v∈vj w(u, v), where w(u, v) denotes the weight of the edge {u,v} ∈ 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 ≥ 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. © 2012 Springer-Verlag.