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How unique is Lovász's theta function?

Abstract : The famous Lovász's ϑ function is computable in polynomial time for every graph, as a semi-definite program (Grötschel, Lovász 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 Lovász'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.
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Contributor : Arnaud Pêcher <>
Submitted on : Monday, December 15, 2014 - 11:34:33 PM
Last modification on : Monday, January 20, 2020 - 12:12:05 PM
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  • HAL Id : hal-01095638, version 1


Arnaud Pêcher, Oriol Serra, Annegret K. Wagler, Xuding Zhu. How unique is Lovász's theta function?. VIII ALIO/EURO Workshop on Applied Combinatorial Optimization, Dec 2014, Montevideo, Uruguay. ⟨hal-01095638⟩



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