A variant of the Lov\'asz-Theta number based on projection matrices

Francesco Silvestri

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arXiv · 1708.06563

A variant of the Lov\'asz-Theta number based on projection matrices

Francesco Silvestri

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Abstract

We introduce a new model for the chromatic number $\chi(G)$ based on what we call combinatorial projection matrices, which is a special class of doubly stochastic symmetric projection matrices. Relaxing this models yields an SDP whose optimal value is the projection theta number $\hat{\vartheta}(G)$, which is closely related to the Szegedy number $\vartheta^+(G)$, a variant of the Lov\'{a}sz theta number. We characterize that in general, $\hat{\vartheta}(G)\leq \vartheta^+(G)$, with equality if $G$ is vertex-transitive. While this seems to imply that working with binary matrices is a better paradigm than working with binary eigenvalues in this context, our approach is slightly faster than computing the Szegedy number on vertex-transitive graphs.

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Francesco Silvestri. 2017-08-22. A variant of the Lov\'asz-Theta number based on projection matrices. https://arxiv.org/abs/1708.06563

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