Paper Info
Title
Popularity at minimum cost
Abstract
We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph $G = (\mathcal{A}\cup\mathcal{B},E)$ , where $\mathcal{A}$ is a set of people, $\mathcal{B}$ is a set of items, and each person $a \in\mathcal{A}$ ranks a subset of items in order of preference, with ties allowed. The popular matching problem seeks to compute a matching M 驴 between people and items such that there is no matching M where more people are happier with M than with M 驴. Such a matching M 驴 is called a popular matching. However, there are simple instances where no popular matching exists.Here we consider the following natural extension to the above problem: associated with each item $b \in\mathcal{B}$ is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to "augment" G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of $\sqrt{n_{1}}/2$ , where n 1 is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn 1) time, where m is the number of edges.
Year
DOI
Venue
2014
10.1007/s10878-012-9537-0
Clinical Orthopaedics and Related Research
Keywords
Field
DocType
Bipartite graphs,Matchings,One-sided preference lists,NP-hardness
Discrete mathematics,Combinatorics,Computer science,Popularity,Bipartite graph
Journal
Volume
Issue
ISSN
27
3
1382-6905
Citations 
PageRank 
References 
3
0.45
12
Authors
3
Name
Order
Citations
PageRank
Telikepalli Kavitha153448.32
Meghana Nasre29812.80
Prajakta Nimbhorkar317015.04