Two-edge-connected (not necessarily spanning) subgraphs and polyhedra

TL;DR

This paper studies the polytope of 2-edge-connected subgraphs in a graph, providing linear inequalities for optimization, characterizing facet-defining inequalities, and testing their practical effectiveness.

Contribution

It introduces a new ILP formulation for maximum weight 2-edge-connected subgraphs and characterizes the facet structure of the associated polytope.

Findings

Linear inequalities describe the lattice points of the polytope.

Characterization of when inequalities define facets.

Practical evaluation of inequalities on various graph classes.

Abstract

Given a graph $G$, we study the $2$-edge-connected subgraph polytope $\mathrm{TECSP}(G)$, which is given by the convex hull of the incidence vectors of all $2$-edge-connected subgraphs of $G$. We describe the lattice points of this polytope by linear inequalities which provides an ILP-algorithm for finding a $2$-edge-connected subgraph of maximum weight. Furthermore, we characterize when these inequalities define facets of $\mathrm{TECSP}(G)$. We also consider further types of supporting hyperplanes of $\mathrm{TECSP}(G)$ and study when they are facet-defining. Finally, we investigate the efficiency of our considered inequalities practically on some classes of graphs.