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}
}

Figures from this paper

Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gr\"obner Bases
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)
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
Sums of Squares, Moment Matrices and Optimization Over Polynomials
TLDR
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
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
Complexity of Null-and Positivstellensatz proofs
Quantum entanglement, sum of squares, and the log rank conjecture
TLDR
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).
...
...