Low Degree Nullstellensatz Certificates for 3-Colorability

Abstract

In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field . In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3colorable in the first iteration of this algorithm when = GF (2). This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most 12 vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when = GF (2).

Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze

“…The bounds we give in Theorems 2 and 4 for our systems of equations are better than the above bounds. Moreover, it has been documented that in practice the degrees of Nullstellensatz certificates of NP-hard problems (e.g., non-3-colorability), tend to be small "in practice" (see, for example, [14,34,36] and the references therein), especially when the polynomial encodings are over finite fields. Note also that when we know the degree of the Nullstellensatz certificate, one can compute explicit coefficients of the Nullstellensatz certificate using a linear algebra system derived by equating the monomials of the identity.…”

“…Note also that when we know the degree of the Nullstellensatz certificate, one can compute explicit coefficients of the Nullstellensatz certificate using a linear algebra system derived by equating the monomials of the identity. This has been exploited in practical computation with great success, see [13,14,34].…”