How unique is Lovász's theta function?

Abstract

The famous Lovasz's ϑ function is computable in polynomial time for every graph, as a semi-definite program (Grotschel, Lovasz and Schrijver, 1981). The chromatic number and the clique number of every perfect graph G are computable in polynomial time. Despite numerous efforts since the last three decades, stimulated by the Strong Perfect Graph Theo-rem (Chudnovsky, Robertson, Seymour and Thomas, 2006), no combinatorial proof of this result is known. In this work, we try to understand why the key properties of Lovasz's ϑ function make it so unique. We introduce an infinite set of convex functions, which includes the clique number ω and ϑ . This set includes a sequence of linear programs which are monotone increasing and converging to ϑ . We provide some evidences that ϑ is the unique function in this setting allowing to compute the chromatic number of perfect graphs in polynomial time.

Locations

• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF