Recognizing Graph Theoretic Properties with Polynomial Ideals

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure

Exact sampling and counting for fixed-margin matrices

The uniform distribution on matrices with specified row and column sums is often a natural choice of null model when testing for structure in two-way tables (binary or nonnegative integer). Due to the difficulty of sampling from this distribution, many approximate methods have been developed. We will show that by exploiting certain symmetries, exact sampling and counting is in fact possible in many nontrivial real-world cases. We illustrate with real datasets including ecological co-occurrence matrices and contingency tables.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1131 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1104.032