Popular matchings with weighted voters
Klaus Heeger, \'Agnes Cseh

TL;DR
This paper studies a weighted generalization of the popular matching problem, proving NP-hardness in general but providing a polynomial-time algorithm for instances where one side's vertices have weight greater than 3 and the other's weight is 1.
Contribution
It introduces a weighted variant of the popular matching problem and offers a polynomial-time algorithm for specific weight configurations, contrasting with NP-hardness in the general case.
Findings
NP-hardness of finding popular matchings in the weighted case
Polynomial-time algorithm for instances with one side's vertices having weight > 3
Existence or non-existence proof for popular matchings in special cases
Abstract
In the Popular Matching problem, we are given a bipartite graph and for each vertex , strict preferences over the neighbors of . Given two matchings and , matching is more popular than if the number of vertices preferring to is larger than the number of vertices preferring to . A matching is called popular if there is no matching that is more popular than . We consider a natural generalization of Popular Matching where every vertex has a weight. Then, we call a matching more popular than matching if the weight of vertices preferring to is larger than the weight of vertices preferring to . For this case, we show that it is NP-hard to find a popular matching. Our main result its a polynomial-time algorithm that delivers a popular matching or a proof for it non-existence in…
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Taxonomy
TopicsGame Theory and Voting Systems · Complexity and Algorithms in Graphs · Auction Theory and Applications
