Decision trees, monotone functions, and semimatroids

TL;DR

This paper introduces decision trees for monotone functions on simplicial complexes, explores homology decidability, and connects these concepts to semimatroids, Betti numbers, and the Tutte polynomial.

Contribution

It defines homology decidability for monotone functions and demonstrates its application to semimatroids and related complexes, extending previous notions like semi-nonevasiveness.

Findings

Homology decidability applies to functions related to semimatroids.

Optimal decision trees enable computation of Betti numbers.

Betti numbers relate to evaluations of the Tutte polynomial.

Abstract

We define decision trees for monotone functions on a simplicial complex. We define homology decidability of monotone functions, and show that various monotone functions related to semimatroids are homology decidable. Homology decidability is a generalization of semi-nonevasiveness, a notion due to Jonsson. The motivating example is the complex of bipartite graphs, whose Betti numbers are unknown in general. We show that these monotone functions have optimum decision trees, from which we can compute relative Betti numbers of related pairs of simplicial complexes. Moreover, these relative Betti numbers are coefficients of evaluations of the Tutte polynomial, and every semimatroid collapses onto its broken circuit complex.