Popular Matching in Roommates Setting is NP-hard
Sushmita Gupta, Pranabendu Misra, Saket Saurabh, Meirav Zehavi

TL;DR
This paper proves that determining the existence of a popular matching in the roommates setting is NP-complete, resolving a long-standing open question in computational social choice.
Contribution
It establishes the NP-hardness of the popular matching problem in the roommates setting, settling an open problem from the literature.
Findings
Popular matching problem in roommates setting is NP-complete.
Resolves a decade-long open question.
Implications for computational complexity in matching theory.
Abstract
An input to the Popular Matching problem, in the roommates setting, consists of a graph and each vertex ranks its neighbors in strict order, known as its preference. In the Popular Matching problem the objective is to test whether there exists a matching such that there is no matching where more people are happier with than with . In this paper we settle the computational complexity of the Popular Matching problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly, explicitly asked over the last decade.
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