Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares. (November 2021)

• Record Type: Journal Article
• Authors: Gaar, Elisabeth; Krenn, Daniel; Margulies, Susan; Wiegele, Angelika
• Is Part Of: Journal of symbolic computation. Volume 107(2021)
• Page Start: 67
• Page End: 105
• Publication Date: 2021-11
• Subjects: Vizing's conjecture -- Algebraic model -- Gröbner basis -- Sum-of-squares problems -- Semidefinite programming; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.01.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial pair such that the polynomial is non-negative on the variety associated with the ideal if and only if the conjecture is true for this graph class. Using semidefinite programming we obtain numeric sum-of-squares certificates, which we then manage to transform into symbolic certificates confirming non-negativity of our polynomials. Specifically, we obtain exact low-degree sparse sum-of-squares certificates for particular classes of graphs. The obtained certificates allow generalizations for larger graph classes. Besides computational verification of these more general certificates, we also present theoretical proofs as well as conjectures and questions for further investigations.