On 4-critical planar graphs with high edge density
Discrete Mathematics
Proceedings of the first Malta conference on Graphs and combinatorics
Exponential lower bounds for the tree-like Hajo´s calculus
Information Processing Letters
Lower bounds for the polynomial calculus and the Gröbner basis algorithm
Computational Complexity
New methods to color the vertices of a graph
Communications of the ACM
Frozen development in graph coloring
Theoretical Computer Science - Phase transitions in combinatorial problems
A New Criterion for Normal Form Algorithms
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Good Degree Lower Bounds on Nullstellensatz Refutations of the Induction Principle
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Systematic Generation of Very Hard Cases for Graph 3-Colorability
TAI '95 Proceedings of the Seventh International Conference on Tools with Artificial Intelligence
The division algorithm and the hilbert scheme
The division algorithm and the hilbert scheme
Polynomials that Vanish on Distinct $n$th Roots of Unity
Combinatorics, Probability and Computing
Combinatorics, Probability and Computing
Semidefinite representations for finite varieties
Mathematical Programming: Series A and B
Constructive generation of very hard 3-colorability instances
Discrete Applied Mathematics
Algebraic characterization of uniquely vertex colorable graphs
Journal of Combinatorial Theory Series B
Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Expressing combinatorial problems by systems of polynomial equations and hilbert's nullstellensatz
Combinatorics, Probability and Computing
Computer algebra, combinatorics, and complexity: hilbert's nullstellensatz and np-complete problems
Computer algebra, combinatorics, and complexity: hilbert's nullstellensatz and np-complete problems
A new look at the easy-hard-easy pattern of combinatorial search difficulty
Journal of Artificial Intelligence Research
A Branch-and-Cut algorithm for graph coloring
Discrete Applied Mathematics - Special issue: IV ALIO/EURO workshop on applied combinatorial optimization
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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 3-colorable, 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 linear-algebra 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.