Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
From Jean-Guillaume Dumas' Sparse Integer Matrix Collection,
http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/simc.html
http://arxiv.org/abs/0706.0578
Expressing Combinatorial Optimization Problems by Systems of Polynomial
Equations and the Nullstellensatz
Authors: J.A. De Loera, J. Lee, Susan Margulies, S. Onn
(Submitted on 5 Jun 2007)
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.