Let μ be a metric on a finite set T. We consider a version of the multifacility location problem (also called the minimum 0-extension problem): given a finite set VT and a function c: (v/2) →Z+, attach each element x∈V-T to some γ(x)∈T minimizing ∑(c(xy)μ(γ(x)γ(y)): xy ∈ (v/2), letting γ(t)=t for t∈T. It is known that the problem is solvable in polynomial time for any modular metric μ whose underlying graph is hereditary modular and orientable (in a certain sense), as well as for any median metric μ. On the other hand, it was shown that for a fixed metric μ, the problem is NP-hard if μ is nonmodular or if its underlying graph is nonorientable. Decreasing the gap between these tractable and intractable cases, we describe a wide class of modular metrics μ (including the median metrics as a special case) for which the problem is solvable in polynomial time. This result follows from our theorem on the existence of retractions of certain modular graphs, which provides an efficient uncrossing method for the class of modular metrics in question. We also present other results and raise open questions.
