On the p-Median polytope

Abstract.

The p-Median problem defined on a complete directed graph with n nodes ($\vec{K}$ _n (V,A)) asks for a subset T⊆V of cardinality p and such that the weight of n-p arcs going from T to every node of V∖T, is minimized. The p-Median polytope M _n-p ($\vec{K}$ _n (V,A)) is the convex hull of the incidence vectors of all the subsets of n-p arcs in A leaving a set T⊆V of cardinality p and entering in every node of V∖P.

In this paper we show that the polytope M _n-p ($\vec{K}$ _n (V,A)) is an “integral slice” of the Stable Set polytope STAB(G _n ) associated with a suitable graph G _n (X,J) (i.e. is the convex hull of the integral points in the intersection of STAB(G _n ) with the hyperplane ∑ _i∈X x _i =n-p). This allows us to define a very general class of facet-defining valid inequalities of M _n-p ($\vec{K}$ _n (V,A)), called W-2 inequalities, which are also facet-defining for STAB(G _n ) and have a very compact representation in terms of suitable subgraphs of $\vec{K}$ _n (V,A).

We also define a very basic class of facet-defining inequalities of M _n-p ($\vec{K}$ _n (V,A)), called Cover inequalities, which are not valid for STAB(G _n ).

The importance and the role of the above classes is testified by the observation that they provide the complete description of M _n-p ($\vec{K}$ _n (V,A)) if p=n-2.

Cover inequalities can be strengthened by exploiting optimality. We introduce a new class of inequalities, called I ^*-Cover inequalities, which have a non-standard nature: they are not valid for M _n-p ($\vec{K}$ _n (V,A)), but do not cut-off the optimal solution.

A preliminary computational experience shows that the inequalities introduced in this paper are very effective in the solution of test instances.