A Fast Lower Bound for the Minimum Cost Perfect 2-Matching Linear Program

Abstract

SYNOPTIC ABSTRACTThe minimum cost perfect 2-matching problem is defined on a weighted undirected graph Q = (V,t',c) and is the problem of finding a minimum weight subset of edges £ ~ £ such that each vertex is incident to exactly two edges in £. The problem appears as a relaxation subproblem in routing environments where the distance matrices are symmetric. This paper presents a multiplier adjustment approach for obtaining a quick lower bound on the linear programming relaxation of the minimum cost perfect 2-matching problem. The lower bound is constructed by generating a feasible solution to the dual of the linear programming (LP) relaxation. On a collection of 100 random Euclidean and non-Euclidean problems, the value of the lower bound as a percent of the optimal LP objective value averaged over 97 percent with a standard deviation of only 1.15. The approach is also shown to compare favorably with alternative methods for generating fast lower bounds for the symmetric Traveling Salesman Problem.