4,216 research outputs found
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 provides a preference list on
items in I. We say that an applicant 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 , and that each
is assigned a weight such that w_{1}>w_{2}>...>w_{k}>0m/n^{4/3}=o(1)w_{1} \geq 2w_{2}n^{4/3}/m = o(1)w_{1} \geq 2w_{2}$ has a 2-weighted popular
matching with probability 1-o(1).Comment: 13 pages, 2 figure
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: 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
Matching under Preferences
Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory.
Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs.
Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process.
Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully
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
Popular Matchings in the Weighted Capacitated House Allocation Problem
We consider the problem of finding a popular matching in the Weighted Capacitated
House Allocation problem (WCHA). An instance of WCHA involves a set of agents
and a set of houses. Each agent has a positive weight indicating his priority, and a
preference list in which a subset of houses are ranked in strict order. Each house has
a capacity that indicates the maximum number of agents who could be matched to
it. A matching M of agents to houses is popular if there is no other matching M′
such that the total weight of the agents who prefer their allocation in M′
to that in
M exceeds the total weight of the agents who prefer their allocation in M to that in
M′
. Here, we give an O(
√
Cn1 + m) algorithm to determine if an instance of WCHA
admits a popular matching, and if so, to find a largest such matching, where C is the
total capacity of the houses, n1 is the number of agents, and m is the total length of
the agents’ preference lists
Popular Matchings in the Capacitated House Allocation Problem
We consider the problem of finding a popular matching in the Capacitated House Allocation problem (CHA). An instance of CHA involves a set of agents and a set of houses. Each agent has a preference list in which a subset of houses are ranked in strict order, and each house may be matched to a number of agents that must not exceed its capacity. A matching M is popular if there is no other matching M′ such that the number of agents who prefer their allocation in M′ to that in M exceeds the number of agents who prefer their allocation in M to that in M′. Here, we give an O(√C+n1m) algorithm to determine if an instance of CHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n1 is the number of agents and m is the total length of the agents’ preference lists. For the case where preference lists may contain ties, we give an O(√Cn1+m) algorithm for the analogous problem
- …