Parameterized counting of trees, forests and matroid bases

TL;DR

This paper explores the computational complexity of counting specific combinatorial structures like trees, forests, and matroid bases, revealing hardness results and identifying cases where the problem is fixed parameter tractable.

Contribution

It establishes the W[1]-hardness of counting trees, forests, and matroid bases parameterized by size or rank, and shows fixed parameter tractability for matroids over fixed finite fields.

Findings

Counting trees and forests with k edges is W[1]-hard.

Counting matroid bases is W[1]-hard when parameterized by rank or nullity.

Counting matroid bases over fixed finite fields is fixed parameter tractable.

Abstract

We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with $k$ edges are $\# W[1]$-hard when parameterized by $k$. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for $k$-forests implies $\# W[1]$-hardness of the problem of counting bases of a matroid when parameterized by rank or nullity, even if the matroid is restricted to be representable over a field of characteristic $2$. We complement this result by pointing out that the problem becomes fixed parameter tractable for matroids represented over a fixed finite field.