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

TL;DR

This paper investigates the integrality gap of the linear programming relaxation for the 2-edge connected spanning subgraph problem, providing evidence supporting the conjecture that the gap is exactly 6/5 under certain conditions.

Contribution

The authors demonstrate that the conjectured 6/5 integrality gap holds for cost functions optimized by specific half-integer solutions, advancing understanding of the problem's bounds.

Findings

Supports the conjecture that the integrality gap is 6/5 for certain solutions.

Identifies a family of solutions that achieve the largest gap.

Provides a method to verify the conjecture for specific cost functions.

Abstract

Given a complete graph $K_{n}=(V, E)$ with non-negative edge costs $c\in {\mathbb R}^{E}$, the problem $2EC$ is that of finding a 2-edge connected spanning multi-subgraph of $K_{n}$ of minimum cost. The integrality gap $\alpha\text{2EC}$ of the linear programming relaxation $\text{2EC}^{\text{LP}}$ for $2EC$ has been conjectured to be $\frac{6}{5}$, although currently we only know that $\frac{6}{5}\leq\alpha\text{2EC}\leq\frac{3}{2}$. In this paper, we explore the idea of using the structure of solutions for $\text{2EC}^{\text{LP}}$ and the concept of convex combination to obtain improved bounds for $\alpha\text{2EC}$. We focus our efforts on a family $J$ of half-integer solutions that appear to give the largest integrality gap for $\text{2EC}^{\text{LP}}$. We successfully show that the conjecture $\alpha\text{2EC} = \frac{6}{5}$ is true for any cost functions optimized by some $x^{*}\in J$.