Popular Matching in Roommates Setting is NP-hard

An input to the Popular Matching problem, in the roommates setting, consists of a graph G 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 M such that there is no matching M where more people are happier with M than with M. 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. ∗University of Bergen, Bergen, Norway. [email protected] †University of Bergen, Bergen, Norway. [email protected] ‡The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected] §Ben-Gurion University, Beersheba, Israel. [email protected] ar X iv :1 80 3. 09 37 0v 1 [ cs .D S] 2 5 M ar 2 01 8