# Robust randomized matchings

## TL;DR

This paper introduces a randomized strategy that improves the robustness of matchings in weighted graphs, extending to various independence systems and impacting stochastic optimization problems.

## Contribution

It demonstrates that Alice can achieve a higher robustness ratio using randomized strategies, generalizes the result to broader independence systems, and provides a new LP-based proof of existing bounds.

## Key findings

- Randomized strategies improve robustness ratio to 1/ln(4)
- Extension to general independence systems including matroid intersection and b-matchings
- Improved approximation for maximum priority matching problem

## Abstract

The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the $k$ heaviest edges of $M$ to the maximum weight of a matching of size at most $k$. If $M$ guarantees a payoff of at least $\alpha$ then it is called $\alpha$-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a $1/\sqrt{2}$-robust matching, which is best possible. We show that Alice can improve her payoff to $1/\ln(4)$ by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.