Popularity at Minimum Cost

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We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph G = (A ∪ B, E), where A is a set of people, B is a set of items, and each person a ∈ A ranks a subset of items in an order of preference, with ties allowed. The popular matching problem seeks to compute a matching M∗ between people and items such that there is no matching M where more people are happier with M than with M∗ . Such a matching M∗ 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 ∈ 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 th...

Telikepalli Kavitha, Meghana Nasre, Prajakta Nimbh

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