Popular Matchings and Limits to Tractability
Abstract
We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if admits a popular matching that is neither stable nor dominant. We also show a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem. In non-bipartite graphs, we show a strong negative result: it is NP-hard to decide whether a popular matching exists or not, and the same result holds if we replace popular with dominant. On the positive side, we show the following results in any graph: - we identify a subclass of dominant matchings called strongly dominant matchings and show a linear time algorithm to decide if a strongly dominant matching exists or not; - we show an efficient algorithm to compute a popular matching of minimum cost in a graph with edge costs and bounded treewidth.
Cite
@article{arxiv.1805.11473,
title = {Popular Matchings and Limits to Tractability},
author = {Yuri Faenza and Telikepalli Kavitha and Vladlena Powers and Xingyu Zhang},
journal= {arXiv preprint arXiv:1805.11473},
year = {2018}
}
Comments
arXiv admin note: text overlap with arXiv:1804.00141