N. Gvozdenovic Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and a new, block-diagonal hierarchy is proposed. It has the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. It is applied … Read more
We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and $\chi\left( {\ol G} \right)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\alpha (\ol G)$ and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if … Read more
Recently we investigated in “The operator $\Psi$ for the Chromatic Number of a Graph” hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. In particular, we introduced two hierarchies of lower bounds, the `$\psi$’-hierarchy converging to the fractional chromatic number, and the `$\Psi$’-hierarchy converging to the chromatic number of a graph. … Read more
We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and $\chi(\bar G)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\omega(G)$ and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if $\beta(G)$ is. As an … Read more
Lov\’ asz and Schrijver [1991] have constructed semidefinite relaxations for the stable set polytope of a graph $G=(V,E)$ by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most $\alpha(G)$ steps, where $\alpha(G)$ is the stability number of $G$. Two other hierarchies of semidefinite bounds for the stability number have … Read more