Hard cases of the multifacility location problem

Abstract

Let μ be a rational-valued metric on a finite set T. We consider (a version of) the multifacility location problem: given a finite set V⊇ T and a function c: V 2 → Z + , attach each element x∈ V− T to an element γ( x)∈ T minimizing ∑ c(xy)μ(γ(x)γ(y)):xy∈ V 2 , letting γ( t)≔ t for each t∈ T. Large classes of metrics μ have been known for which the problem is solvable in polynomial time. On the other hand, Dalhaus et al. (SIAM J. Comput. 23 (4) (1994) 864) showed that if T={ t 1, t 2, t 3} and μ( t i t j )=1 for all i≠ j, then the problem (turning into the minimum 3- terminal cut problem) becomes strongly NP-hard. Extending that result and its generalization in (European J. Combin. 19 (1998) 71), we prove that for μ fixed, the problem is strongly NP-hard if the metric μ is nonmodular or if the underlying graph of μ is nonorientable (in a certain sense).