Approximations for the Steiner Multicycle Problem

TL;DR

This paper introduces improved approximation algorithms for the Steiner Multicycle problem, a complex generalization of TSP, with specific solutions for metric, edge-weighted, and asymmetric cases, advancing both theoretical bounds and practical applications.

Contribution

The paper presents new approximation algorithms with better ratios for the Steiner Multicycle problem, including a 3-approximation for metric cases and an (11/9)-approximation for specific weight conditions.

Findings

Achieved a 3-approximation for the metric Steiner Multicycle problem.

Developed an (11/9)-approximation for weights of 1 or 2.

Designed an O(log n)-approximation for the asymmetric case.

Abstract

The Steiner Multicycle problem consists of, given a complete graph, a weight function on its vertices, 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 4 vertices. Finally, we devise an O(lg n)-approximation algorithm for the asymmetric version of the problem.