Applied Hilbert's Nullstellensatz for Combinatorial Problems

Abstract

Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \textit{stable set} of a given size, finding the \textit{chromatic number} of a graph or more generally, determining the feasibility of an \textit{Integer Programming problem} are classical examples of this. In this thesis we study a powerful tool from Algebraic Geometry, called \textit{Hilbert's Nullstellensatz}. It characterizes the \textit{infeasibility} of a system of polynomial equations by the \textit{feasibility} of a possibly very large system of \textit{linear equations}. The solutions to this linear system provide \textit{certificates} for the infeasibility of the polynomial system, called \textit{Nullstellensatz Certificates}.

In this thesis we focus on the study of Nullstellensatz Certificates for the existence of \textit{proper colorings} of graphs. We use basic ideas from \textit{duality theory} to determine various properties of the Nullstellensatz Certificates. We give new proofs to several known results in the current literature and present some new results that shed some light on the relationship between the sparsity of a graph and the \textit{size} of the Nullstellensatz Certificates for \textit{$k$-colorability}.