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/kneser_8_3_1.sms