Matroid Polytopes: Algorithms, Theory, and Applications

TL;DR

This dissertation explores properties, algebraic combinatorics, and algorithmic solutions related to matroid polytopes, including polynomial-time Ehrhart polynomial computation, conjectures on their structure, and practical heuristics for optimization problems.

Contribution

It introduces polynomial-time algorithms for Ehrhart polynomials, provides evidence for conjectures on matroid polytope properties, and develops heuristics for large-scale matroid optimization.

Findings

Ehrhart polynomials are computable in polynomial time for fixed rank.

Support for conjectures on the h^*-vector and Ehrhart coefficients.

New heuristics effectively solve large matroid optimization problems.

Abstract

This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. Second, we discuss theoretical results regarding the algebraic combinatorics of matroid polytopes. We discuss two conjectures about the h^*-vector and coefficients of Ehrhart polynomials of matroid polytopes and provide theoretical and computational evidence for their validity. We also explore a variant of White's conjecture which states that every matroid polytope has a regular unimodular triangulation. We provide extensive computational evidence supporting this new conjecture and propose a combinatorial condition on simplices sufficient for unimodularity. Finally, motivated by recent work on algorithmic theory for non-linear and multicriteria matroid optimization, we have developed algorithms and heuristics aimed at practical solutions of large instances of these difficult problems. Our methods primarily use the local adjacency structure inherent in matroid polytopes to pivot to feasible solutions which may or may not be optimal. We also present a modified breadth-first-search heuristic that uses adjacency to enumerate a subset of feasible solutions. We present other heuristics, and provide computational evidence supporting these new techniques. We implemented all of our algorithms in the software package MOCHA (Matroids Optimization Combinatorial Heuristics and Algorithms).