An Optimization-Based Sum-of-Squares Approach to Vizing's Conjecture

Elisabeth Gaar,Susan Margulies,Angelika Wiegele,Daniel Krenn

doi:10.1145/3326229.3326239

Elisabeth Gaar, Susan Margulies + Show 2 more

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https://doi.org/10.1145/3326229.3326239

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Abstract

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.

Highlights

Sum-of-squares and its relationship to semidefinite programming is a cutting-edge tool at the forefront of polynomial optimization [5]
Vizing’s conjecture was first proposed in 1968, and relates the sizes of minimum dominating sets in graphs G and H to the size of a minimum dominating set in the Cartesian product graph G□H ; a precise formulation follows as Conjecture 2.1
Our goal is to model Vizing’s conjecture as a semidefinite programming problem

Summary

Introduction

Sum-of-squares and its relationship to semidefinite programming is a cutting-edge tool at the forefront of polynomial optimization [5]. Prior work on polynomial encodings includes colorings [1, 14], stable sets [18, 19], matchings [9], and flows [22] In this project, we combine the modeling strength of systems of polynomial equations with the computational power of semidefinite programming and devise an optimization-based framework for a computational proof of an old, open problem in graph theory, namely Vizing’s conjecture. Algebraic computational results have remained largely untouched In this project, we present an algebraic model of Vizing’s conjecture that equates the validity of the conjecture to the existence of a Positivstellensatz, or a sum-of-squares certificate of nonnegativity modulo a carefully constructed ideal

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