Popular Edges with Critical Nodes

In the popular edge problem, the input is a bipartite graph \(G = (A \cup B,E)\) where \(A\) and \(B\) denote a set of men and a set of women respectively, and each vertex in \(A\cup B\) has a strict preference ordering over its neighbours. A matching \(M\) in \(G\) is said to be {\em popular} if there is no other matching \(M'\) such that the number of vertices that prefer \(M'\) to \(M\) is more than the number of vertices that prefer \(M\) to \(M'\). The goal is to determine, whether a given edge \(e\) belongs to some popular matching in \(G\). A polynomial-time algorithm for this problem appears in \cite{CK18}. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from \cite{Kavitha14,Kavitha21,NNRS21,NN17} that, when \(G\) admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.