Near-linear time samplers for matroid independent sets with applications

TL;DR

This paper introduces a near-linear time algorithm for sampling independent sets in matroids, enabling faster graph connectivity sampling and network reliability estimation, and extends to the random cluster model for q ≤ 1.

Contribution

It presents the first near-linear time sampler for matroid independent sets, improving efficiency in graph and network reliability computations and generalizing to the random cluster model.

Findings

Achieves on(n) time complexity for matroid sampling

Enables on(|E|) time sampling of connected spanning subgraphs

Extends to the random cluster model with q  1

Abstract

We give a $\widetilde{O}(n)$ time almost uniform sampler for independent sets of a matroid, whose ground set has $n$ elements and is given by an independence oracle. As a consequence, one can sample connected spanning subgraphs of a given graph $G=(V,E)$ in $\widetilde{O}(|E|)$ time. This leads to improved running time on estimating all-terminal network reliability. Furthermore, we generalise this near-linear time sampler to the random cluster model with $q\le 1$.