Application of the Lovász-Schrijver Lift-and-Project Operator to Compact Stable Set Integer Programs

TL;DR

The paper addresses tightening the Lovász theta bound $\theta(G)$ for the stability number $\alpha(G)$ by applying the Lovász–Schrijver lift-and-project operator $N_+(\cdot)$ to tailored LP relaxations. It constructs two starting LP families, a clique-based relaxation $QSTAB(G,\mathcal{C})$ and a nodal relaxation $\operatorname{NOD}(G,r)$ with $r\in\{\Gamma,\theta,\alpha\}$, and obtains SDP relaxations via $M_+(\cdot)$, comparing them to $N_+(FRAC(G))$ and the base $\theta(G)$; a cutting-plane approach manages exponential constraint sets. The results show all proposed relaxations dominate $\theta(G)$, with $\mu(G,\mathcal{C})$ and $\nu(G,\alpha)$ delivering substantial gap reductions; clique-based methods excel on sparse graphs while nodal-based methods excel on dense graphs, and the polynomial-time bound $\nu(G,\theta)$ provides meaningful gains as well. The findings demonstrate that practical SDP relaxations based on Lovász–Schrijver lifting can surpass $\theta(G)$ at a reasonable computational cost, suggesting extensions to related problems such as graph coloring.

Abstract

The Lovász theta function $θ(G)$ 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, $θ(G)$ 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 Lovász 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 $θ(G)$. Our computational experience, including larger graphs than those previously documented, shows that upper bounds stronger than $θ(G)$ can be accessed by a reasonable additional effort using the clique-based formulation on sparse graphs and the nodal-based one on dense graphs.