Symmetric polynomials and Hall's theorem

Fischer, Klaus G.

doi:10.1016/0012-365x(88)90051-9

Cited by 13 publications

(8 citation statements)

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“…Although the polynomial method is not yet as widely used by combinatorists as, for instance, polyhedral or probabilistic techniques, the literature in this subject continues to grow. Prior work on encoding combinatorial properties includes colourings [2,11,14,17,27,30,31,32], stable sets [11,26,27,39], matchings [15], and flows [2,32,33]. Non-linear encodings of combinatorial problems are often compact.…”

Section: Introductionmentioning

confidence: 99%

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

Loera¹,

Lee²,

Margulies³

et al. 2009

Combinator. Probab. Comp.

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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows.In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.In the second part of this paper, for the purpose of computation, we construct new polynomial encodings 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. We include some applications to graph theory.

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Section: Introductionmentioning

confidence: 99%

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

Loera¹,

Lee²,

Margulies³

et al. 2009

Combinator. Probab. Comp.

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“…An algebraic method to solve combinatorial problems is to encode the problem as a system of polynomial equations and apply the Nullstellensatz or Positivstellensatz. In several areas this and similar approaches have been used to show new results, for example for colorings [1,6,7,9,13,16,18,19], stable sets [6,8,14,16,23], flows [1,19,20] and matchings [10] in graphs. Gaar, Krenn, Margulies and Wiegele [11] used an algebraic method to reformulate Vizing's conjecture as a sum-of-squares (SOS) program.…”

Section: Introductionmentioning

confidence: 99%

Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gröbner Bases

Gaar¹,

Siebenhofer²

2021

Preprint

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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, Krenn, Margulies and Wiegele used the graph class G of all graphs with n G vertices and domination number k G and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of k G , n G , k H , and n H .In this paper, we consider their approach for k G = k H = 1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (n G + n H − 1)/2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with n G + n H − 1 = d for general d, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with k G = k H = 1 and n G + n H ≤ 15.

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“…Systems of polynomial equations and other non-linear models are similarly widely known for their compact and elegant representations of combinatorial problems. Prior work on polynomial encodings includes colorings [1,16], stable sets [20,21], matchings [9], and flows [24]. 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.…”

Section: Introductionmentioning

confidence: 99%

Towards a Computational Proof of Vizing's Conjecture using Semidefinite Programming and Sums-of-Squares

Gaar,

Krenn,

Margulies

et al. 2020

Preprint

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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.An extended abstract containing the ideas of this optimization-based approach for tackling Vizing's conjecture appeared as [10]. This article now also contains the full and complete proofs, new certificates only conjectured in the extended abstract, and further theoretical and computational results for particular cases. This also lead to a restructuring of the whole article, and more examples and many more remarks explaining the implications of the results are provided. The authors gratefully acknowledge the support of Fulbright Austria (via a Visiting Professorship at

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Symmetric polynomials and Hall's theorem

Cited by 13 publications

References 1 publication

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

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

Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gröbner Bases

Towards a Computational Proof of Vizing's Conjecture using Semidefinite Programming and Sums-of-Squares

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