Popular Matchings in the Stable Marriage Problem

Abstract

Abstract. The input is a bipartite graph G = (A ∪ B,E) where each vertex u ∈ A ∪ B ranks its neighbors in a strict order of preference. This is the same as an instance of the stable marriage problem with incomplete lists. A matching M * is said to be popular if there is no matching M such that more vertices are better off in M than in M * . Any stable matching of G is popular, however such a matching is a minimum cardinality popular matching. We consider the problem of computing a maximum cardinality popular matching in G. It has very recently been shown that when preference lists have ties, the problem of determining if a given instance admits a popular matching or not is NP-complete. When preference lists are strict, popular matchings always exist, however the complexity of computing a maximum cardinality popular matching was unknown. In this paper we give a simple characterization of popular matchings when preference lists are strict and a sufficient condition for a maximum cardinality popular matching. We then show an O(mn 0 ) algorithm for computing a maximum cardinality popular matching, where m = |E| and n 0 = min(|A|, |B|).

Popular Matching Games for Correlation-Aware Resource Allocation in the Internet of Things

“…Thus, the device can balance between the achievable rate and the correlation factor between device i and the devices assigned to SBS s. In fact, the second term in (11) enables a device i to report its information to an SBS that cannot get the same information from its assigned devices. Using the utility in (11), the preference relations of the devices can be given by:…”

“…To define the most popular matching [11,12], we introduce the notion of a vote that allows every SBS and device to vote for its preferred matching. For a device i, the vote function…”

“…In particular, we formulate a popular two-sided matching game with externalities between the IoT devices and SBSs [10][11][12]. The devices aim to ensure a given correlation level with the other IoT devices that are assigned to the SBSs while the SBSs are interested in the reliability level with which each of the devices can report the sensed information.…”

Popular matchings

“…He also showed that if all preference lists are strict, then any stable matching is popular; thus a popular matching always exists and can be found in linear time using the well-known deferred acceptance algorithm of Gale and Shapley (1962). Huang and Kavitha (2013) later gave a characterization of popular matchings based on augmenting paths. They also came up with an O(m) algorithm to test whether a given matching is popular.…”

“…As demonstrated by this instance, a strikingly important feature of popular matchings is that they beat stable matchings in size. As a matter of fact, any stable matching is a minimum size popular matching (Huang and Kavitha, 2013). The size of a stable matching in G can be as small as |M max |/2, where M max is a maximum matching in G. Relaxing stability to popularity yields larger matchings and it is easy to show that a largest popular matching has size at least 2 3 • |M max |.…”