A Characterization of Minimizable Metrics in the Multifacility Location Problem

In the minimum 0-extension problem (a version of the multifacility location problem), one is given a metric m on a subset X of a finite set V and a non-negative function c on the unordered pairs of elements of V. The objective is to find a semimetric m′ on V that minimizes the inner product c·m′, provided that m′ coincides with m within X and each element of V is at zero distance from X. For m fixed, this problem is solvable in strongly polynomial time if m is minimizable, which means that for any superset V and function c, the minimum objective value is equal to that in the corresponding linear relaxation. In , Karzanov showed that the path metric of a graph H is minimizable if and only if all isometric cycles of H have length four and the edges of H can be oriented so that non-adjacent edges in each 4-cycle have opposite orientations along the cycle (such graphs are called frames in ). Extending this result to general metrics m, we show that m is minimizable if and only if m is modular and its underlying graph is a frame.