Optimizing over Semimetric Polytopes

called the triangle inequalities. The paper deals with the problem of finding efficient algorithms to optimize a linear function over such a polytope. We briefly mention two reasons to seek for an efficient way to optimize a linear function over M(G). 1. The polytope M(G), for |V | > 4, properly contains the cut polytope associated with G (see, e.g., [5] for the details). Thus, if an edge cost function c ∈ R is given, maximizing c over M(G) produces an upper bound on the maximum c-value cut of G, which can be exploited in branch and bound or branch and cut schemes for solving max-cut to optimality. Actually, in all the computational studies concerning instances of max-cut for very large sparse graphs based on a branch and cut scheme, the only relaxation exploited is M(G), or, more precisely, its projection that will be described shortly later. Consequently, most of the computation time for finding a maximum cut is spent into a (possibly long) series of linear optimizations over M(G). 2. If an edge capacity function c ∈ R and an edge demand function d ∈ R are given, the existence of a feasible multiflow in the network defined by G, c, and d is established by the Japanese Theorem (see, e.g., [7]). According to this theorem, a feasible multiflow exists if and only if μ · (c − d) ≥ 0 holds for every metric μ on V , i.e., for every point of the cone defined by all the homogeneous equations of (1). It is not hard to see that this is equivalent to the condition min{(c − d)x | x ∈ M(G)} ≥ 0. In some approaches to network design problems [2, 4] such a feasibility problem has to be solved several times. Again, this calls for an effective solution algorithm.