An input to the P OPULAR M ATCHING problem, in roommates setting (as opposed marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors strict order, known as preference. In problem objective is test whether there exists matching * such that no more vertices prefer their matched status (in terms preferences) over *. this article, we settle computa...

A generally-accepted minimum requirement for an algorithm to be considered ‘efficient’ is that its running time is polynomial: O(nc) for some constant c, where n is the size of the input.1 Researchers recognized early on that not all problems can be solved this quickly, but we had a hard time figuring out exactly which ones could and which ones couldn’t. there are several so-called NP-hard prob...