Group JGD_Margulies

Group Description
Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
From Jean-Guillaume Dumas' Sparse Integer Matrix Collection,

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.
Displaying collection matrices 21 - 23 of 23 in total
Id Name Group Rows Cols Nonzeros Kind Date Download File
2162 flower_8_4 JGD_Margulies 55,081 125,361 375,266 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2163 kneser_10_4_1 JGD_Margulies 349,651 330,751 992,252 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2169 wheel_601 JGD_Margulies 902,103 723,605 2,170,814 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market