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

Jesus De Loera(deloera***at***math.ucdavis.edu)
Jon Lee(jonlee***at***us.ibm.com)
Susan Margulies(smargulies***at***ucdavis.edu)
Shmuel Onn(onn***at***ie.technion.ac.il)

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.

Keywords: Hilbert's Nullstellensatz, Polynomial method

Category 1: Combinatorial Optimization (Other )

Category 2: Integer Programming (Other )

Category 3: Nonlinear Optimization (Other )

Citation: IBM Resaerch Report RC24276, June 2007

Download: [PDF]

Entry Submitted: 06/13/2007
Entry Accepted: 06/15/2007
Entry Last Modified: 06/13/2007

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