The complexity of the greedoid Tutte polynomial

TL;DR

This paper investigates the computational complexity of evaluating the Tutte polynomial for various classes of greedoids, showing that most evaluations are #P-hard with few exceptions, and connects these results to matroid basis counting.

Contribution

It establishes the #P-hardness of computing the Tutte polynomial for greedoids from rooted graphs, digraphs, and binary matrices at fixed points, with some polynomial-time cases identified.

Findings

Most evaluations are #P-hard except for specific cases.

Evaluation complexity depends on the class of greedoid.

Includes a proof related to matroid basis counting.

Abstract

We consider the Tutte polynomial of three classes of greedoids: those arising from rooted graphs, rooted digraphs and binary matrices. We establish the computational complexity of evaluating each of these polynomials at each fixed rational point (x,y). In each case we show that evaluation is #P-hard except for a small number of exceptional cases when there is a polynomial time algorithm. In the binary case, establishing #P-hardness along one line relies on Vertigan's unpublished result on the complexity of counting bases of a matroid. For completeness, we include an appendix providing a proof if this result.