Application of the Lov\'asz-Schrijver Lift-and-Project Operator to Compact Stable Set Integer Programs

TL;DR

This paper explores applying Lovász-Schrijver lift-and-project operators to improve upper bounds on the stability number of graphs, achieving stronger bounds efficiently for certain graph classes.

Contribution

It introduces novel SDP formulations based on clique and nodal inequalities that match or surpass the Lovász theta bound with less computational effort.

Findings

Stronger upper bounds than theta(G) are achievable with new formulations.

Clique-based formulation works well on sparse graphs.

Nodal-based formulation is effective on dense graphs.

Abstract

The Lov\'asz theta function $\theta(G)$ provides a very good upper bound on the stability number of a graph $G$. It can be computed in polynomial time by solving a semidefinite program (SDP), which also turns out to be fairly tractable in practice. Consequently, $\theta(G)$ achieves a hard-to-beat trade-off between computational effort and strength of the bound. Indeed, several attempts to improve the theta bound are documented, mainly based on playing around the application of the $N_+(\cdot)$ lifting operator of Lov\'asz and Schrijver to the classical formulation of the maximum stable set problem. Experience shows that solving such SDP-s often struggles against practical intractability and requires highly specialized methods. We investigate the application of such an operator to two different linear formulations based on clique and nodal inequalities, respectively. Fewer inequalities describe these two and yet guarantee that the resulting SDP bound is at least as strong as $\theta(G)$. Our computational experience, including larger graphs than those previously documented, shows that upper bounds stronger than $\theta(G)$ can be accessed by a reasonable additional effort using the clique-based formulation on sparse graphs and the nodal-based one on dense graphs.