The matroid secretary problem for minor-closed classes and random matroids

TL;DR

This paper establishes constant-competitive algorithms for the matroid secretary problem within minor-closed classes of matroids representable over prime fields, leveraging advanced matroid structure theory, and analyzes typical matroids' competitiveness.

Contribution

It introduces a constant-competitive algorithm for minor-closed classes of matroids and shows that most matroids are nearly optimally solvable with a simple random basis approach.

Findings

Existence of constant-competitive algorithms for matroids in minor-closed classes.

Most matroids are nearly 2-competitive with a simple random basis algorithm.

Assuming the paving conjecture, almost all matroids admit a near-optimal algorithm.

Abstract

We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.