Susan Margulies

On the complexity of Hilbert refutations for partition

Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilberts Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant of the partition matrix is a polynomial that factors into an iteration over all possible partitions of W.

Graph-Coloring Ideals: Nullstellensatz Certificates, Gröbner Bases for Chordal Graphs, and Hardness of Gröbner Bases

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.

The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming

In 2007, W. H. Cunningham and J. Geelen describe an algorithm for solving

Branch-decomposition heuristics for linear matroids

\max\{c^Tx\COLON Ax = b,\,x \geq 0,\,x \in \Bbb{Z}^n\}

Weak orientability of matroids and polynomial equations

, where

Systems of Polynomials for Detecting Orientable Matroids

A \in \Bbb{Z}_{\geq 0}^{m \times n}

Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz

,

A Note on Total and Paired Domination of Cartesian Product Graphs

b \in \Bbb{Z}^m

Integer domination of Cartesian product graphs

, and

Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals

c \in \Bbb{Z}^n