Quasi-Popular Matchings, Optimality, and Extended Formulations

Let [Formula: see text] be an instance of the stable marriage problem in which every vertex ranks its neighbors in a strict order of preference. A matching [Formula: see text] in [Formula: see text] is popular if [Formula: see text] does not lose a head-to-head election against any matching. Popular matchings generalize stable matchings. Unfortunately, when there are edge costs, to find or even approximate up to any factor a popular matching of minimum cost is NP-hard. Let [Formula: see text] be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most [Formula: see text] by paying the price of mildly relaxing popularity. Our main positive results are two bicriteria algorithms that find in polynomial time a “quasi-popular” matching of cost at most [Formula: see text]. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching, which could be much smaller than [Formula: see text]. Key to the other algorithm is a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

Popular Matching in Roommates Setting Is NP-hard

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

Popularity, Mixed Matchings, and Self-Duality

Our input instance is a bipartite graph G where each vertex has a preference list ranking its neighbors in a strict order of preference. A matching M is popular if there is no matching N such that the number of vertices that prefer N to M outnumber those that prefer M to N. Each edge is associated with a utility and we consider the problem of matching vertices in a popular and utility-optimal manner. It is known that it is NP-hard to compute a max-utility popular matching. So we consider mixed matchings: a mixed matching is a probability distribution or a lottery over matchings. Our main result is that the popular fractional matching polytope PG is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching which is the average of two integral matchings. So in order to implement a max-utility popular mixed matching in G, we need just a single random bit. We analyze the popular fractional matching polytope whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality. The self-duality of this LP plays a crucial role in our proof. Our result implies that a max-utility popular half-integral matching in G and also in the roommates problem (where the input graph need not be bipartite) can be computed in polynomial time.

Popular Matchings in Complete Graphs

Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching $$M'$$ M ′ : here each vertex casts a vote for the matching in $$\{M,M'\}$$ { M , M ′ } in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes $$\texttt {NP}$$ NP -complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and $$\texttt {NP}$$ NP -complete when n has the other parity.

Preprint of "The Impact of Mate Value in First and Subsequent Real-Life Romantic Encounters"

We provide a first systematic investigation of the most prominent hypotheses about the impact of mate value on interpersonal attraction in real-life early-stage romantic encounters. Using Response Surface Analysis, we simultaneously examined how (a) people’s perception of their own mate value, (b) their perception of a potential partner’s mate value, and (c) the interplay between the two mate values impact initial romantic attraction and selection as well as subsequent interpersonal outcomes after selection. Data came from the “Date me for Science” speed-dating study (n = 398), in which participants who mutually selected each other at the speed-dating event were followed up with 3 assessments in the 6 weeks after the event to assess subsequent outcomes. Participants’ romantic attraction, likelihood of selecting, and subsequent interpersonal outcomes with a dating partner almost exclusively depended on their perception of their dating partner’s mate value: the higher, the better. There was no evidence for the popular matching hypothesis, which states that people feel attracted to and select dating partners whom they perceive to have a mate value similar to their own. Implications of these findings for theory and research on the impact of mate value on romantic attraction and selection are discussed.