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arXiv · 0706.0578

Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz

J. A. De Loera·J. Lee·S. Margulies·S. Onn
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Abstract

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. 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. 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- colorability, we found only graphs with Nullstellensatz certificates of degree four.

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J. A. De Loera, J. Lee, S. Margulies, S. Onn. 2007-06-05. Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz. https://arxiv.org/abs/0706.0578

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