4,780 research outputs found
Rank Maximal Matchings -- Structure and Algorithms
Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P
denotes a set of posts and ranks on the edges denote preferences of the agents
over posts. A matching M in G is rank-maximal if it matches the maximum number
of applicants to their top-rank post, subject to this, the maximum number of
applicants to their second rank post and so on.
In this paper, we develop a switching graph characterization of rank-maximal
matchings, which is a useful tool that encodes all rank-maximal matchings in an
instance. The characterization leads to simple and efficient algorithms for
several interesting problems. In particular, we give an efficient algorithm to
compute the set of rank-maximal pairs in an instance. We show that the problem
of counting the number of rank-maximal matchings is #P-Complete and also give
an FPRAS for the problem. Finally, we consider the problem of deciding whether
a rank-maximal matching is popular among all the rank-maximal matchings in a
given instance, and give an efficient algorithm for the problem
Efficient algorithms for bipartite matching problems with preferences
Matching problems involve a set of participants, where each participant has a capacity and a subset of the participants rank a subset of the others in order of preference (strictly or with ties). Matching problems are motivated in practice by large-scale applications, such as automated matching schemes, which assign participants together based on their preferences over one another.
This thesis focuses on bipartite matching problems in which there are two disjoint sets of participants (such as medical students and hospitals). We present a range of efficient algorithms for finding various types of optimal matchings in the context of these problems. Our optimality criteria involve a diverse range of concepts that are alternatives to classical stability. Examples include so-called popular and Pareto optimal matchings, and also matchings that are optimal with respect to their profile (the number of participants obtaining their first choice, second choice and so on).
The first optimality criterion that we study is the notion of a Pareto optimal matching, a criterion that economists regard as a fundamental property to be satisfied by an optimal matching. We present the first algorithmic results on Pareto optimality for the Capacitated House Allocation problem (CHA), which is a many-to-one variant of the classical House Allocation problem, as well as for the Hospitals-Residents problem (HR), a generalisation of the classical Stable Marriage problem.
For each of these problems, we obtain a characterisation of Pareto optimal matchings, and then use this to obtain a polynomial-time algorithm for finding a maximum Pareto optimal matching.
The next optimality criterion that we study is the notion of a popular matching. We study popular matchings in CHA and present a polynomial-time algorithm for finding a maximum popular matching or reporting that none exists, given any instance of CHA. We extend our findings to the case in CHA where preferences may contain ties (CHAT) by proving the extension of a well-known result in matching theory to the capacitated bipartite graph case, and using this to obtain a polynomial-time algorithm for finding a maximum popular matching, or reporting that none exists.
We next study popular matchings in the Weighted Capacitated House Allocation problem (WCHA), which is a variant of CHA where the agents have weights assigned to them. We identify a structure in the underlying graph of the problem that singles out those edges that cannot belong to a popular matching. We then use this to construct a polynomial-time algorithm for finding a maximum popular matching or reporting that none exists, for the case where preferences are strict.
We then study popular matchings in a variant of the classical Stable Marriage problem with Ties and Incomplete preference lists (SMTI), where preference lists are symmetric. Here, we provide the first characterisation results on popular matchings in the bipartite setting where preferences are two-sided, which can either lead to a polynomial-time algorithm for solving the problem or help establish that it is NP-complete. We also provide the first algorithm for testing if a matching is popular in such a setting.
The remaining optimality criteria that we study involve profile-based optimal matchings. We define three versions of what it means for a matching to be optimal based on its profile, namely so-called greedy maximum, rank-maximal and generous maximum matchings. We study each of these in the context of CHAT and the Hospitals-Residents problem with Ties (HRT).
For each problem model, we give polynomial-time algorithms for finding a greedy maximum, a rank-maximal and a generous maximum matching
Manipulation Strategies for the Rank Maximal Matching Problem
We consider manipulation strategies for the rank-maximal matching problem. In
the rank-maximal matching problem we are given a bipartite graph such that denotes a set of applicants and a set of posts. Each
applicant has a preference list over the set of his neighbours in
, possibly involving ties. Preference lists are represented by ranks on the
edges - an edge has rank , denoted as , if post
belongs to one of 's -th choices. A rank-maximal matching is one in which
the maximum number of applicants is matched to their rank one posts and subject
to this condition, the maximum number of applicants is matched to their rank
two posts, and so on. A rank-maximal matching can be computed in time, where denotes the number of applicants, the
number of edges and the maximum rank of an edge in an optimal solution.
A central authority matches applicants to posts. It does so using one of the
rank-maximal matchings. Since there may be more than one rank- maximal matching
of , we assume that the central authority chooses any one of them randomly.
Let be a manipulative applicant, who knows the preference lists of all
the other applicants and wants to falsify his preference list so that he has a
chance of getting better posts than if he were truthful. In the first problem
addressed in this paper the manipulative applicant wants to ensure that
he is never matched to any post worse than the most preferred among those of
rank greater than one and obtainable when he is truthful. In the second problem
the manipulator wants to construct such a preference list that the worst post
he can become matched to by the central authority is best possible or in other
words, wants to minimize the maximal rank of a post he can become matched
to
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
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
On Randomized Algorithms for Matching in the Online Preemptive Model
We investigate the power of randomized algorithms for the maximum cardinality
matching (MCM) and the maximum weight matching (MWM) problems in the online
preemptive model. In this model, the edges of a graph are revealed one by one
and the algorithm is required to always maintain a valid matching. On seeing an
edge, the algorithm has to either accept or reject the edge. If accepted, then
the adjacent edges are discarded. The complexity of the problem is settled for
deterministic algorithms.
Almost nothing is known for randomized algorithms. A lower bound of
is known for MCM with a trivial upper bound of . An upper bound of
is known for MWM. We initiate a systematic study of the same in this paper with
an aim to isolate and understand the difficulty. We begin with a primal-dual
analysis of the deterministic algorithm due to McGregor. All deterministic
lower bounds are on instances which are trees at every step. For this class of
(unweighted) graphs we present a randomized algorithm which is
-competitive. The analysis is a considerable extension of the
(simple) primal-dual analysis for the deterministic case. The key new technique
is that the distribution of primal charge to dual variables depends on the
"neighborhood" and needs to be done after having seen the entire input. The
assignment is asymmetric: in that edges may assign different charges to the two
end-points. Also the proof depends on a non-trivial structural statement on the
performance of the algorithm on the input tree.
The other main result of this paper is an extension of the deterministic
lower bound of Varadaraja to a natural class of randomized algorithms which
decide whether to accept a new edge or not using independent random choices
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