# Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares

@article{Gaar2021TowardsAC, title={Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares}, author={Elisabeth Gaar and Daniel Krenn and Susan Margulies and Angelika Wiegele}, journal={J. Symb. Comput.}, year={2021}, volume={107}, pages={67-105} }

## 2 Citations

Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gr\"obner Bases

- Mathematics
- 2021

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar,…

Conic Linear Optimization for Computer-Assisted Proofs (hybrid meeting)

- Mathematics
- 2022

From a mathematical perspective, optimization is the science of proving inequalities. In this sense, computational optimization is a method for computer-assisted proofs. Conic (linear) optimization…

## References

SHOWING 1-10 OF 30 REFERENCES

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

- Mathematics, Computer ScienceISSAC
- 2019

This paper encodes Vizing's conjecture as an ideal/polynomial pair such that the polynomial is nonnegative if and only if the conjecture is true, and demonstrates how to use semidefinite optimization techniques to computationally obtain numeric sum-of-squares certificates.

Vizing's conjecture: a survey and recent results

- MathematicsJ. Graph Theory
- 2012

Several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs.

Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2009

New polynomial encodings are constructed for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph.

Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz

- Mathematics, Computer Science
- 2007

New polynomial encodings are constructed for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph.

Theta Bodies for Polynomial Ideals

- MathematicsSIAM J. Optim.
- 2010

A hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal is introduced and a geometric description of the first theta body for all ideals is given.

Algebraic characterization of uniquely vertex colorable graphs

- MathematicsJ. Comb. Theory, Ser. B
- 2008

Sums of Squares, Moment Matrices and Optimization Over Polynomials

- Mathematics, Computer Science
- 2009

This work considers the problem of minimizing a polynomial over a semialgebraic set defined byPolynomial equations and inequalities, which is NP-hard in general and reviews the mathematical tools underlying these properties.

Semidefinite Optimization and Convex Algebraic Geometry

- Mathematics
- 2012

This book provides a self-contained, accessible introduction to the mathematical advances and challenges resulting from the use of semidefinite programming in polynomial optimization. This quickly…

Quantum entanglement, sum of squares, and the log rank conjecture

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2017

The algorithm is based on the sum-of-squares hierarchy and its analysis is inspired by Lovett's proof that the communication complexity of every rank-n Boolean matrix is bounded by Õ(√n).