Toward a 6/5 Bound for the Minimum Cost 2-Edge Connected Subgraph Problem

Abstract

Given a complete graph Kn=(V,E) with non-negative edge costs c∈RE, the problem 2EC is that of finding a 2-edge connected spanning multi-subgraph of Kn of minimum cost. The integrality gap α2EC of the linear programming relaxation 2ECLP for 2EC has been conjectured to be 65, although currently we only know that 65≤α2EC≤32. of solutions for 2ECLP and the concept of convex combination to obtain improved approximation algorithms for 2EC and bounds for α2EC. We focus our efforts on a family J of half-integer solutions that appear to give the largest integrality gap for 2ECLP. We successfully show that the conjecture α2EC=65 is true for any cost functions optimized by some x⁎∈J. Our methods are constructive and thus also provide a 65-approximation algorithm for 2EC for these special cases.