An optimization-based sum-of-squares approach to Vizing's conjecture

  • Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. In particular, we encode the conjecture as an ideal/polynomial pair such that the polynomial is nonnegative if and only if the conjecture is true. We demonstrate how to use semidefinite optimization techniques to computationally obtain numeric sum-of-squares certificates, and then show how to transform these numeric certificates into symbolic certificates approving nonnegativity of our polynomial. After outlining the theoretical structure of this computer-based proof of Vizing's conjecture, we present computational and theoretical results. In particular, we present exact low-degree sparse sum-of-squares certificates for particular families of graphs.

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Metadaten
Author:Elisabeth GaarGND, Angelika Wiegele, Daniel Krenn, Susan Margulies
URN:urn:nbn:de:bvb:384-opus4-1122752
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/112275
ISBN:978-1-4503-6084-5OPAC
Parent Title (English):ISSAC '19: Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation, Beijing, China, July 15-18, 2019
Publisher:ACM
Place of publication:New York, NY
Editor:James Davenport, Dongming Wang, Manuel Kauers, Russell Bradford
Type:Conference Proceeding
Language:English
Year of first Publication:2019
Publishing Institution:Universität Augsburg
Release Date:2024/04/04
First Page:155
Last Page:162
DOI:https://doi.org/10.1145/3326229.3326239
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Diskrete Mathematik, Optimierung und Operations Research
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):CC-BY 4.0: Creative Commons: Namensnennung (mit Print on Demand)

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