A New Bound for the 2-Edge Connected Subgraph Problem

Abstract

Given a complete undirected graph with non-negative costs on the edges, the 2-Edge Connected Subgraph Problem consists in finding the minimum cost spanning 2-edge connected subgraph (where multi- edges are allowed in the solution). A lower bound for the minimum cost 2-edge connected subgraph is obtained by solving the linear programming relaxation for this problem, which coincides with the subtour relaxation of the traveling salesman problem when the costs satisfy the triangle inequality.

The simplest fractional solutions to the subtour relaxation are the $\frac{1} {2} z$-integral solutions in which every edge variable has a value which is a multiple of $\frac{1} {2}$. We show that the minimum cost of a 2-edge connected subgraph is at most four-thirds the cost of the minimum cost $\frac{1} {2}$-integral solution of the subtour relaxation. This supports the long-standing $\frac{4} {3}$ Conjecture for the TSP, which states that there is a Hamilton cycle which is within $\frac{4} {3}$ times the cost of the optimal subtour relaxation solution when the costs satisfy the triangle inequality.

Supported by NSF grant DMS9509581 and DOE contract AC04-94AL85000.

Research supported in part by an NSF CAREER grant CCR-9625297.