Online Matroid Intersection: Beating Half for Random Arrival

TL;DR

This paper introduces a randomized online algorithm for the matroid intersection problem that surpasses the long-standing half-competitiveness barrier, achieving a ratio of 0.5 + delta in expectation for random element arrivals.

Contribution

It presents the first online algorithm with a competitive ratio exceeding half for matroid intersection under random arrivals, and a linear-time offline algorithm with similar improvement.

Findings

Achieves a 0.5 + delta competitive ratio online

First to beat half competitiveness in this setting

Provides a linear-time offline algorithm surpassing half ratio

Abstract

For two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ defined on the same ground set $E$, the online matroid intersection problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniformly random order. At each step, the algorithm must irrevocably decide whether to pick the element, while always maintaining a common independent set. While the natural greedy algorithm---pick an element whenever possible---is half competitive, nothing better was previously known; even for the special case of online bipartite matching in the edge arrival model. We present the first randomized online algorithm that has a $\frac12 + \delta$ competitive ratio in expectation, where $\delta >0$ is a constant. The expectation is over the random order and the coin tosses of the algorithm. As a corollary, we also obtain the first linear time algorithm that beats half competitiveness for offline matroid intersection.