NP-hardness of m-dimensional weighted matching problems

Abstract

Given a graph with km vertices and non-negative weights on the edges, the m-dimensional weighted matching problem consists in partitioning the vertices of the graph into k subsets of m vertices. The goal is to either maximize, or minimize the sum of the weights of the edges whose both endpoints are in the same subset. In this paper, we define the pMax-mDWM and pMin-mDWM problems, respectively, by considering the vertices of the graph as points in an arbitrary Euclidean space and defining the edge weights using the ℓp-norm. We present NP-hardness proofs of the maximization problem for m≥3 and p≥2, and of the minimization problem for m≥3 and p≥1. We also show explicitly that the balanced minimum sum-of-squares clustering problem and the minimization version of the m-dimensional weighted matching problem are NP-hard for m≥3.