Popular Matchings with One-Sided Bias

Let G = (A ? B,E) be a bipartite graph where A consists of agents or main players and B consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so. We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side - so along with overall popularity, we would like to maintain "popularity within the set A". A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings

Popular Matchings with One-Sided Bias

Let $G = (A \cup B,E)$ be a bipartite graph where the set $A$ consists of agents or main players and the set $B$ consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching $M$ is popular if for any matching $N$, the number of vertices that prefer $M$ to $N$ is at least the number that prefer $N$ to $M$. Popular matchings always exist in $G$ since every stable matching is popular. A matching $M$ is $A$-popular if for any matching $N$, the number of agents (i.e., vertices in $A$) that prefer $M$ to $N$ is at least the number of agents that prefer $N$ to $M$. Unlike popular matchings, $A$-popular matchings need not exist in a given instance $G$ and there is a simple linear time algorithm to decide if $G$ admits an $A$-popular matching and compute one, if so. We consider the problem of deciding if $G$ admits a matching that is both popular and $A$-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when $A$ is the more important side -- so along with overall popularity, we would like to maintain ``popularity within the set $A$''. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.Comment: A preliminary version of this paper appeared in Proc. of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 70:1--70:18, 202

Maximum Cardinality Popular Matchings in Strict Two-sided Preference Lists

We consider the problem of computing a maximum cardinality {\em popular} matching in a bipartite graph G = (\A\cup\B, E) where each vertex u \in \A\cup\B ranks its neighbors in a strict order of preference. This is the same as an instance of the {\em 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 $M^*$. \smallskip Popular matchings have been extensively studied in the case of one-sided preference lists, i.e., only vertices of \A have preferences over their neighbors while vertices in \B have no preferences; polynomial time algorithms have been shown here to determine if a given instance admits a popular matching or not and if so, to compute one with maximum cardinality. It has very recently been shown that for two-sided preference lists, the problem of determining if a given instance admits a popular matching or not is NP-complete. However this hardness result assumes that preference lists have {\em ties}. When preference lists are {\em strict}, it is easy to show that popular matchings always exist since stable matchings always exist and they are popular. But the complexity of computing a maximum cardinality popular matching was unknown. In this paper we show an $O(mn)$ algorithm for this problem, where n = |\A| + |\B| and $m = |E|$

Fairly Popular Matchings and Optimality

We consider a matching problem in a bipartite graph G = (A ? B, E) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M is at least the number that prefer N; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity. Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation. We also show the following hardness result: given a matching M, it is NP-complete to decide if there exists a popular matching that defeats M. Interestingly, there exists a set K of at most m popular matchings in G (where |E| = m) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K

Popular Half-Integral Matchings

In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M\u27 to M outnumber those that prefer M to M\u27. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time

Min-Cost Popular Matchings

Let G = (A ? B, E) be a bipartite graph on n vertices where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if there is no matching N such that vertices that prefer N to M outnumber those that prefer M to N. Popular matchings always exist in G since every stable matching is popular. Thus it is easy to find a popular matching in G - however it is NP-hard to compute a min-cost popular matching in G when there is a cost function on the edge set; moreover it is NP-hard to approximate this to any multiplicative factor. An O^*(2?) algorithm to compute a min-cost popular matching in G follows from known results. Here we show: - an algorithm with running time O^*(2^{n/4}) ? O^*(1.19?) to compute a min-cost popular matching; - assume all edge costs are non-negative - then given ? > 0, a randomized algorithm with running time poly(n,1/(?)) to compute a matching M such that cost(M) is at most twice the optimal cost and with high probability, the fraction of all matchings more popular than M is at most 1/2+?

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

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