%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/JGD_Margulies/flower_4_4
% name: JGD_Margulies/flower_4_4
% [Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis]
% id: 2156
% date: 2008
% author: S. Margulies
% ed: J.-G. Dumas
% fields: name title A id date author ed kind notes
% kind: combinatorial problem
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% notes:
% 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.                            
%                                                                         
% Filename in JGD collection: Margulies/flower_4_4.sms                    
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