A lattice point counting generalisation of the Tutte polynomial

Amanda Cameron,Alex Fink

doi:10.46298/dmtcs.6331

Abstract

The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.

Highlights

The Tutte polynomial, originally formulated for graphs, has been generalised to apply to matroids
The corank-nullity polynomial can be defined for polymatroids, but the resulting function gives no accessible information about basis activities, and is not even a Laurent polynomial in the variables x and y
It is possible to salvage some features of this recurrence in restricted cases: this is done by [Oxley and Whittle(1993)] for polymatroids where singletons have rank at most 2, where the corank-nullity polynomial is still universal for a form of deletion-contraction recurrence

Summary

Introduction

The Tutte polynomial, originally formulated for graphs, has been generalised to apply to matroids. The corank-nullity polynomial can be defined for polymatroids, but the resulting function gives no accessible information about basis activities, and is not even a Laurent polynomial in the variables x and y This reflects the difference between matroids and polymatroids that matroids have a well-behaved theory of minors analogous to graph minors: for each ground set element one can define a deletion and contraction, and knowing these two determines the matroid. It is possible to salvage some features of this recurrence in restricted cases: this is done by [Oxley and Whittle(1993)] for polymatroids where singletons have rank at most 2, where the corank-nullity polynomial is still universal for a form of deletion-contraction recurrence Another formula for the Tutte polynomial of a graph or a matroid is defined in terms of activities of bases. We give a geometric interpretation of the coefficients of our polynomial, by way of a particular subdivision of the relevant polytope

Construction

Relation to Tutte

Properties

Coefficients