Approximations for the Steiner Multicycle problem

Abstract

The Steiner Multicycle problem consists in, given a complete graph, a weight function on its edges, and a collection of pairwise disjoint non-unitary sets called terminal sets, finding a minimum weight collection of vertex-disjoint cycles in the graph such that, for every terminal set, all of its vertices are in a same cycle of the collection. This problem generalizes the Traveling Salesman problem and, therefore, is hard to approximate in general. On the practical side, it models a collaborative less-than-truckload problem with pickup and delivery locations. Using an algorithm for the Survivable Network Design problem and T-joins, we obtain a 3-approximation for the metric case, improving on the previous best 4-approximation. Furthermore, we present an (11/9)-approximation for the particular case of the Steiner Multicycle in which each edge weight is 1 or 2. This algorithm can be adapted to obtain a (7/6)-approximation when every terminal set contains at least four vertices. Finally, we devise an $O(\lg n)$-approximation algorithm for the asymmetric version of the problem.