Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

Authors:
J. A. De Loera;J. Lee;P. N. Malkin;S. Margulies
Affiliations:
Univ. of California at Davis, Davis, CA, USA;IBM T.J. Watson Research Center, Yorktown Heights, CA, USA;Univ. of California at Davis, Davis, CA, USA;Univ. of California at Davis, Davis, CA, USA
Venue:
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Year:
2008

Citing 11
Cited 3

On 4-critical planar graphs with high edge density

Discrete Mathematics
Stable sets and polynomials

Proceedings of the first Malta conference on Graphs and combinatorics
Lower bounds for the polynomial calculus and the Gröbner basis algorithm

Computational Complexity
Good Degree Lower Bounds on Nullstellensatz Refutations of the Induction Principle

CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
The division algorithm and the hilbert scheme

The division algorithm and the hilbert scheme
Combinatorial Nullstellensatz

Combinatorics, Probability and Computing
A branch-and-cut algorithm for graph coloring

Discrete Applied Mathematics - Special issue: IV ALIO/EURO workshop on applied combinatorial optimization
Semidefinite representations for finite varieties

Mathematical Programming: Series A and B
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic characterization of uniquely vertex colorable graphs

Journal of Combinatorial Theory Series B
Computer algebra, combinatorics, and complexity: hilbert's nullstellensatz and np-complete problems

Computer algebra, combinatorics, and complexity: hilbert's nullstellensatz and np-complete problems

Computing infeasibility certificates for combinatorial problems through Hilbert's Nullstellensatz

Journal of Symbolic Computation
An algebraic characterization of rainbow connectivity

CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Most Tensor Problems Are NP-Hard

Journal of the ACM (JACM)

Quantified Score

Hi-index	0.00

Visualization

Abstract

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.