Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz (original) (raw)
Related papers
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-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, 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-colorable subgraph.
Computing infeasibility certificates for combinatorial problems through Hilbert’s Nullstellensatz
Journal of Symbolic Computation, 2011
Systems of polynomial equations with coefficients over a field K can be used to concisely model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over the algebraic closure of the field K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics, and based on fast large-scale linearalgebra computations over K. We also describe several mathematical ideas for optimizing our algorithm, such as using alternative forms of the Nullstellensatz for computation, adding carefully constructed polynomials to our system, branching and exploiting symmetry. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph instances with almost two thousand nodes and tens of thousands of edges. colorability of graphs can be modeled via a system of polynomial equations [2]. More generally, one can easily prove the following lemma:
Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility
2008
Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.
Recognizing graph theoretic properties with polynomial ideals
the electronic journal of …, 2010
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
An application of the combinatorial Nullstellensatz to a graph labelling problem
Journal of Graph Theory, 2010
An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,…,m}, such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K2, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n=pk vertices, where p is an odd prime and k is a positive integer that admits a Cp-factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50 (2005), 263–272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1–2) (1999), 7–29]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 70–82, 2010.
2015
In this paper we observe the problem of counting graph colorings using polynomials. Several reformulations of The Four Color Conjecture are considered (among them algebraic, probabilistic and arithmetic). In the last section Tutte polynomials are mentioned. 1.
New Polynomial Classes for #2SAT Established Via Graph-Topological Structure
Engineering Letters, 2007
We address the problem of designing efficient procedures for counting models of Boolean formulas and, in this task, we establish new classes of instances where #2SAT is solved in polynomial time. Those instances are recognized by the topological structure of the underlying graph of the instances. We show that, if the depth-search over the constrained graph of a formula generates a tree where the set of fundamental cycles are disjointed (there are not common edges between any pair of fundamental cycles), then #2SAT is tractable. This class of instances do not set restrictions on the number of occurrences of a variable in a Boolean formula. Our proposal can be applied to impact directly in the reduction of the complexity time of the algorithms for other counting problems.
Ideals , varieties , stability , colorings and combinatorial designs ∗
2008
A combinatorial design is equivalent to a stable set in a suitably chosen Johnson graph, whose vertices correspond to all k-sets that could be blocks of the design. In order to find maximum stable sets of a graph G, two ideals are associated with G, one constructed from the Motzkin-Strauss formula and one reported by Lovász in connection with the stability polytope. These ideals are shown to coincide and form the stability ideal of G. Graph stability ideals belong to a class of 0-1 ideals. These ideals are shown to be radical, and therefore have a strong structure. Stability ideals of Johnson graphs provide an algebraic characterization that can be used to generate Steiner triple systems. Two different ideals for the generation of Steiner triple systems, and a third for Kirkman triple systems, are developed. The last of these combines stability and colorings. 2010 Mathematics Subject Classification: 05B07,13P10.
Combinatorial hardness proofs for polynomial evaluation
Lecture Notes in Computer Science, 1998
We exhibit a new method for showing lower bounds for the time complexity of polynomial evaluation procedures given by straightline programs. Time, denoted by L, is measured in terms of nonscalar arithmetic operations. The time complexity function considered here is L 2 . As main difference with the previously known methods to study this problem, our general complexity method is purely combinatorial and does not need number theory or powerful tools from algebraic geometry. Using this method we are able to exhibit new families of polynomials "hard to compute" (this means that the time complexity function L 2 increases linearly in the degree, for some universal constant c > 0). We are also able to present in a uniform and easy way, almost all known specific families of univariate polynomials which are known to be hard to compute. Our method can also be applied to classical questions of transcendency in number theory and geometry. A list of (old and new) formal power series is given whose transcendency can be proved easily by our method.
The Complexity of some Problems Related to Graph 3-colorability
Discrete Applied Mathematics, 1998
It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given graph has a stable set whose deletion results in (3) the complement of a bipartite graph, (4) a split graph or (5) a threshold graph. 0 1998 Elsevier Science B.V. All rights reserved.