Hard cases of the multifacility location problem

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. [4] showed that if T = {t1, t2, t3} and μ(titj) = 1 for all i 6= j, then the problem (turning into the minimum 3-terminal cut problem) becomes strongly NP-hard. Extending that result and its generalization in [5], 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).