SWAT 2008: Algorithm Theory – SWAT 2008 pp 127-137

# Bounded Unpopularity Matchings

• Chien-Chung Huang
• Telikepalli Kavitha
• Dimitrios Michail
• Meghana Nasre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5124)

## 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 in [2]. 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.

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© Springer-Verlag Berlin Heidelberg 2008

## Authors and Affiliations

• Chien-Chung Huang
• 1
• Telikepalli Kavitha
• 2
• Dimitrios Michail
• 3
• Meghana Nasre
• 2
1. 1.Dartmouth CollegeUSA
2. 2.Indian Institute of ScienceIndia
3. 3.INRIA Sophia Antipolis - MéditerranéeFrance