Hard cases of the multifacility location problem Original Research Article

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:V2→Z+, attach each element x∈V−T to an element γ(x)∈T minimizing ∑c(xy)μ(γ(x)γ(y)):xy∈V2, 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={t1,t2,t3} and μ(titj)=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).