Group JGD_Margulies

Group Description
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.
Displaying matrices 1 - 20 of 23 in total
Id Name Group Rows Cols Nonzeros Kind Date Download File
2149 cat_ears_2_1 JGD_Margulies 85 85 254 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2150 cat_ears_2_4 JGD_Margulies 1,009 2,689 7,982 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2151 cat_ears_3_1 JGD_Margulies 204 181 542 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2152 cat_ears_3_4 JGD_Margulies 5,226 13,271 39,592 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2153 cat_ears_4_1 JGD_Margulies 377 313 938 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2154 cat_ears_4_4 JGD_Margulies 19,020 44,448 132,888 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2155 flower_4_1 JGD_Margulies 121 129 386 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2156 flower_4_4 JGD_Margulies 1,837 5,529 16,466 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2157 flower_5_1 JGD_Margulies 211 201 602 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2158 flower_5_4 JGD_Margulies 5,226 14,721 43,942 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2159 flower_7_1 JGD_Margulies 463 393 1,178 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2160 flower_7_4 JGD_Margulies 27,693 67,593 202,218 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2161 flower_8_1 JGD_Margulies 625 513 1,538 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
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
2164 kneser_6_2_1 JGD_Margulies 601 676 2,027 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2165 kneser_8_3_1 JGD_Margulies 15,737 15,681 47,042 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2166 wheel_3_1 JGD_Margulies 21 25 74 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2167 wheel_4_1 JGD_Margulies 36 41 122 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market
2168 wheel_5_1 JGD_Margulies 57 61 182 Combinatorial Problem 2008 MATLAB Rutherford Boeing Matrix Market