\(\)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 … Read more
\(\) We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-Schrijver SDP operator \( \text{LS}_+\), with a particular focus on a search for relatively small graphs with high \( \text{LS}_+\)-rank (the least number of iterations of the \( \text{LS}_+\) operator on the fractional stable set polytope to compute … Read more
De Klerk and Pasechnik (2002) introduced the bounds $\vartheta^{(r)}(G)$ ($r\in \mathbb{N}$) for the stability number $\alpha(G)$ of a graph $G$ and conjectured exactness at order $\alpha(G)-1$: $\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)$. These bounds rely on the conic approximations $\mathcal{K}_n^{(r)}$ by Parrilo (2000) for the copositive cone $\text{COP}_n$. A difficulty in the convergence analysis of $\vartheta^{(r)}$ is the bad behaviour … Read more
We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp.875–892], who conjectured convergence to $\alpha(G)$ in $r=\alpha(G) -1$ steps. Even the weaker conjecture claiming finite convergence is still open. … Read more
The stable set problem is a fundamental combinatorial optimisation problem, that is known to be very difficult in both theory and practice. Some of the solution algorithms in the literature are based on 0-1 linear programming formulations. We examine an entire family of such formulations, based on so-called clique and nodal inequalities. As well as … Read more
The Lovasz theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovasz and the other to Groetschel et al. The former SDP is often thought to be preferable … Read more
We study the Lov\'{a}sz-Schrijver lift-and-project operator ($\LS_+$) based on the cone of symmetric, positive semidefinite matrices, applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the $\LS_+$-operator generates the stable set polytope in one step has been open since 1990. We call these graphs … Read more
We propose a novel solution approach for polynomial programming problems with equality constraints. By means of a generic transformation, we show that solution schemes for the (typically simpler) problem without equalities can be used to address the problem with equalities. In particular, we propose new solution schemes for mixed binary programs, pure 0-1 quadratic programs, … Read more
Recently, Aguilera et al.\ exposed a beautiful relationship between antiblocker duality and the lift-and-project operator proposed by Balas et al. We present a very short proof of their result that the \BCC-rank of the clique polytope is invariant under complementation. The proof of Aguilera et al. relies on their main technical result, which describes a … Read more
We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lov\’asz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures’ performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations … Read more