Jesús A. De Loera
David Rolnick
ABSTRACT
We consider a well-known family of polynomial ideals encoding the problem of graph-k-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gröbner bases and Nullstellensatz certificates for the coloring ideals of general graphs. We revisit the fact that computing a Gröbner basis is NP-hard and prove a robust notion of hardness derived from the inapproximability of coloring problems. For chordal graphs, however, we explicitly describe a Gröbner basis for the coloring ideal and provide a polynomial-time algorithm to construct it.
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Index Terms
- Graph-Coloring Ideals: Nullstellensatz Certificates, Gröbner Bases for Chordal Graphs, and Hardness of Gröbner Bases
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