# Max-size popular matchings and extensions

## TL;DR

This paper investigates the computational complexity of max-size and max-weight popular matchings in roommates and bipartite graphs, revealing NP-hardness results and providing efficient algorithms for special cases.

## Contribution

It proves NP-hardness of max-size popular matchings in roommates and max-weight popular matchings in bipartite graphs, and introduces a linear-time algorithm for strongly dominant matchings.

## Key findings

- Max-size popular matching is NP-hard in roommates graphs.
- Max-weight popular matching is NP-hard even with simple weights.
- A linear-time algorithm exists for strongly dominant matchings in roommates graphs.

## Abstract

We consider the max-size popular matching problem in a roommates instance G = (V,E) with strict preference lists. A matching M is popular if there is no matching M' in G such that the vertices that prefer M' to M outnumber those that prefer M to M'. We show it is NP-hard to compute a max-size popular matching in G. This is in contrast to the tractability of this problem in bipartite graphs where a max-size popular matching can be computed in linear time. We define a subclass of max-size popular matchings called strongly dominant matchings and show a linear time algorithm to solve the strongly dominant matching problem in a roommates instance. We consider a generalization of the max-size popular matching problem in bipartite graphs: this is the max-weight popular matching problem where there is also an edge weight function w and we seek a popular matching of largest weight. We show this is an NP-hard problem and this is so even when w(e) is either 1 or 2 for every edge e. We also show an algorithm with running time O*(2^{n/4}) to find a max-weight popular matching matching in G = (A U B,E)$ on n vertices.