Popular matchings

We consider the problem of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a non-empty subset of posts in order of preference, possibly involving ties. We say that a matching M is popular if there is no matching M' such that the number of applicants preferring M' to M exceeds the number of applicants preferring M to M'. In this paper, we give the first polynomial-time algorithms to determine if an instance admits a popular matching, and to find a largest such matching, if one exists. For the special case in which every preference list is strictly ordered (i.e. contains no ties), we give an O(n+m) time algorithm, where n is the total number of applicants and posts, and m is the total length of all the preference lists. For the general case in which preference lists may contain ties, we give an O(√nm) time algorithm, and show that the problem has equivalent time complexity to the maximum-cardinality bipartite matching problem

Weighted Random Popular Matchings

For a set A of n applicants and a set I of m items, we consider a problem of computing a matching of applicants to items, i.e., a function M mapping A to I; here we assume that each applicant $x \in A$ provides a preference list on items in I. We say that an applicant $x \in A$ prefers an item p than an item q if p is located at a higher position than q in its preference list, and we say that x prefers a matching M over a matching M' if x prefers M(x) over M'(x). For a given matching problem A, I, and preference lists, we say that M is more popular than M' if the number of applicants preferring M over M' is larger than that of applicants preferring M' over M, and M is called a popular matching if there is no other matching that is more popular than M. Here we consider the situation that A is partitioned into $A_{1},A_{2},...,A_{k}$, and that each $A_{i}$ is assigned a weight $w_{i}>0$ such that w_{1}>w_{2}>...>w_{k}>0$. For such a matching problem, we say that M is more popular than M' if the total weight of applicants preferring M over M' is larger than that of applicants preferring M' over M, and we call M an k-weighted popular matching if there is no other matching that is more popular than M. In this paper, we analyze the 2-weighted matching problem, and we show that (lower bound) if$m/n^{4/3}=o(1)$, then a random instance of the 2-weighted matching problem with$w_{1} \geq 2w_{2}$has a 2-weighted popular matching with probability o(1); and (upper bound) if$n^{4/3}/m = o(1)$, then a random instance of the 2-weighted matching problem with$w_{1} \geq 2w_{2}$ has a 2-weighted popular matching with probability 1-o(1).Comment: 13 pages, 2 figure

Popular matchings: structure and algorithms

An instance of the popular matching problem (POP-M) consists of a set of applicants and a set of posts. Each applicant has a preference list that strictly ranks a subset of the posts. A matching M of applicants to posts is popular if there is no other matching M' such that more applicants prefer M' to M than prefer M to M'. This paper provides a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a structure called the switching graph, a directed graph computable in linear time from the preference lists. We show that the switching graph can be exploited to yield efficient algorithms for a range of associated problems, including the counting and enumeration of the set of popular matchings and computing popular matchings that satisfy various additional optimality criteria. Our algorithms for computing such optimal popular matchings improve those described in a recent paper by Kavitha and Nasre

Minimal Envy and Popular Matchings

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them

Popular Matchings in Complete Graphs

Our input is a complete graph $G = (V,E)$ on $n$ vertices where each vertex has a strict ranking of all other vertices in $G$. Our 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 $M'$, where each vertex casts a vote for the matching in $\{M,M'\}$ where it gets assigned a better partner. The popular matching problem is to decide whether a popular matching exists or not. The popular matching problem in $G$ is easy to solve for odd $n$. Surprisingly, the problem becomes NP-hard for even $n$, as we show here.Comment: Appeared at FSTTCS 201

Counting Popular Matchings in House Allocation Problems

We study the problem of counting the number of popular matchings in a given instance. A popular matching instance consists of agents A and houses H, where each agent ranks a subset of houses according to their preferences. A matching is an assignment of agents to houses. A matching M is more popular than matching M' if the number of agents that prefer M to M' is more than the number of people that prefer M' to M. A matching M is called popular if there exists no matching more popular than M. McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered. We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem. We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard

Robust Popular Matchings

We study popularity for matchings under preferences. This solution concept captures matchings that do not lose against any other matching in a majority vote by the agents. A popular matching is said to be robust if it is popular among multiple instances. We present a polynomial-time algorithm for deciding whether there exists a robust popular matching if instances only differ with respect to the preferences of a single agent while obtaining NP-completeness if two instances differ only by a downward shift of one alternative by four agents. Moreover, we find a complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable.Comment: Appears in: Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2024