Hard cases of the multifacility location problem

• Authors: Alexander V. Karzanov
• Venue: Discrete Applied Mathematics
• Year: 2004

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 Σ(c(xy)µ(γ(x)γ(y)): xy ∈ (V 2 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 = {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).