Algorithmica

, Volume 61, Issue 3, pp 738–757

# Bounded Unpopularity Matchings

• Chien-Chung Huang
• Telikepalli Kavitha
• Dimitrios Michail
• Meghana Nasre
Article

DOI: 10.1007/s00453-010-9434-9

Huang, CC., Kavitha, T., Michail, D. et al. Algorithmica (2011) 61: 738. doi:10.1007/s00453-010-9434-9

## Abstract

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M′ such that more people prefer M′ to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by Abraham et al. (SIAM J. Comput. 37(4):1030–1045, 2007). If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity—unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching M with the minimum value of u(M) or g(M) is NP-hard, and that if G does not admit a popular matching, then we have u(M)≥2 for all matchings M in G.

Here we show that a matching M that achieves u(M)=2 can be computed in $$O(m\sqrt{n})$$ time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H=H2,H3,…,Hk such that if Hk admits a matching that matches all people, then we can compute in $$O(km\sqrt{n})$$ time a matching M such that u(M)≤k−1 and $$g(M)\le n(1-\frac{2}{k})$$. Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.

### Keywords

Matching with preferences Popularity Approximation algorithms

## Authors and Affiliations

• Chien-Chung Huang
• 1
• Telikepalli Kavitha
• 2
• Dimitrios Michail
• 3
• Meghana Nasre
• 2
1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
2. 2.Indian Institute of ScienceBangaloreIndia
3. 3.Department of Informatics and TelematicsHarokopion University of AthensAthensGreece