Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz

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-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.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.In the first part of the paper, 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-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.In the second part of this paper, for the purpose of computation, 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-colourable subgraph. We include some applications to graph theory.

Reformulations in Mathematical Programming: Definitions and Systematics

“…We refer the reader to three recent papers [14,33,37] and the references contained therein for a more complete picture of reformulations for BQPs. Reformulations of polynomial [34,57,73,93] and posynomial [75,95] programs also attracted considerable attention. Most work in geometric programming rests on a convex reformulation [40]; a symbolic method to model problems so that the corresponding mathematical program is convex is described in [30].…”

Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

“…It is natural to ask about lower bounds on the degree of the Nullstellensatz certificates. Little is known, but recently it was shown in [6], that for the problem of deciding whether a given graph G has an independent set of a given size, a minimum-degree Nullstellensatz certificate for the non-existence of an independent set of size greater than α(G) (the size of the largest independent set in G) has degree equal to α(G), and it is very dense; specifically, it contains at least one term per independent set in G. For polynomial systems coming from logic there has also been an effort to show degree growth in related polynomial systems (see [3,8] and the references therein). Another question is to provide tighter, more realistic upper bounds for concrete systems of polynomials.…”

“…For example, over the rationals, every odd-wheel has a minimum non-3-colorability certificate of degree four [6]. However, over F2, every odd-wheel has a Nullstellensatz certificate of degree one.…”

Computation with Polynomial Equations and Inequalities Arising in Combinatorial Optimization

“…The techniques we use are a specialization of prior techniques from computational algebra (see [36,20,21,37]). As it turns out this technique is particularly effective when the number of solutions is finite, when K is a finite field, or when the system has nice combinatorial information (see [9]). 2.…”

“…On the negative side, the degrees of the certificates are expected to be high (in the worst case) simply because the NP-hardness of the original combinatorial questions; see e.g. [9]. At the same time, tight exponential upper bounds have been derived (see e.g.…”