# An Optimal Monotone Contention Resolution Scheme for Bipartite Matchings via a Polyhedral Viewpoint

## TL;DR

This paper introduces a new polyhedral approach to design optimal monotone contention resolution schemes for bipartite matchings, simplifying analysis and improving bounds in submodular maximization problems.

## Contribution

It presents a novel polyhedral framework for contention resolution schemes, achieving optimal monotone schemes for bipartite matchings and improving correlation gap bounds.

## Key findings

- Developed an optimal monotone contention resolution scheme for bipartite matchings.
- Improved the lower bound on the correlation gap for bipartite matchings.
- Enhanced approximation factors for submodular maximization with matching constraints.

## Abstract

Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints.