12 research outputs found
Pareto optimality in many-to-many matching problems
Consider a many-to-many matching market that involves two finite disjoint sets, a set A of applicants and a set C of courses. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having a limited budget available. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal can be accomplished in 0(1 A 12 I C 12) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study some structural questions related to POM. We show that, unlike in the one-to-one case, finding a maximum cardinality POM is NP-hard for many-to-many markets. (C) 2014 Elsevier B.V. All rights reserved
Counting Houses of Pareto Optimal Matchings in the House Allocation Problem
Let with and be two sets. We assume that every
element has a reference list over all elements from . We call an
injective mapping from to a matching. A blocking coalition of
is a subset of such that there exists a matching that
differs from only on elements of , and every element of
improves in , compared to according to its preference list. If
there exists no blocking coalition, we call the matching an exchange
stable matching (ESM). An element is reachable if there exists an
exchange stable matching using . The set of all reachable elements is
denoted by . We show This is
asymptotically tight. A set is reachable (respectively exactly
reachable) if there exists an exchange stable matching whose image
contains as a subset (respectively equals ). We give bounds for the
number of exactly reachable sets. We find that our results hold in the more
general setting of multi-matchings, when each element of is matched
with elements of instead of just one. Further, we give complexity
results and algorithms for corresponding algorithmic questions. Finally, we
characterize unavoidable elements, i.e., elements of that are used by all
ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise
Pareto Optimal Matchings in Many-to-Many Markets with Ties
We consider Pareto-optimal matchings (POMs) in a many-to-many market of
applicants and courses where applicants have preferences, which may include
ties, over individual courses and lexicographic preferences over sets of
courses. Since this is the most general setting examined so far in the
literature, our work unifies and generalizes several known results.
Specifically, we characterize POMs and introduce the \emph{Generalized Serial
Dictatorship Mechanism with Ties (GSDT)} that effectively handles ties via
properties of network flows. We show that GSDT can generate all POMs using
different priority orderings over the applicants, but it satisfies truthfulness
only for certain such orderings. This shortcoming is not specific to our
mechanism; we show that any mechanism generating all POMs in our setting is
prone to strategic manipulation. This is in contrast to the one-to-one case
(with or without ties), for which truthful mechanisms generating all POMs do
exist
Pareto optimal matchings in many-to-many markets with ties
We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants
and courses where applicants have preferences, which may include ties, over
individual courses and lexicographic preferences over sets of courses. Since this is the
most general setting examined so far in the literature, our work unifies and generalizes
several known results. Specifically, we characterize POMs and introduce the Generalized
Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties
via properties of network flows. We show that GSDT can generate all POMs using
different priority orderings over the applicants, but it satisfies truthfulness only for
certain such orderings. This shortcoming is not specific to our mechanism; we show
that any mechanism generating all POMs in our setting is prone to strategic manipulation.
This is in contrast to the one-to-one case (with or without ties), for which
truthful mechanisms generating all POMs do exist
Size versus truthfulness in the house allocation problem
We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants
Pareto Optimal Matchings of Students to Courses in the Presence of Prerequisites
We consider the problem of allocating applicants to courses, where each applicant
has a subset of acceptable courses that she ranks in strict order of preference. Each
applicant and course has a capacity, indicating the maximum number of courses and
applicants they can be assigned to, respectively. We thus essentially have a many-tomany
bipartite matching problem with one-sided preferences, which has applications
to the assignment of students to optional courses at a university.
We consider additive preferences and lexicographic preferences as two means of extending
preferences over individual courses to preferences over bundles of courses.
We additionally focus on the case that courses have prerequisite constraints: we will
mainly treat these constraints as compulsory, but we also allow alternative prerequisites.
We further study the case where courses may be corequisites.
For these extensions to the basic problem, we present the following algorithmic results,
which are mainly concerned with the computation of Pareto optimal matchings
(POMs). Firstly, we consider compulsory prerequisites. For additive preferences, we
show that the problem of finding a POM is NP-hard. On the other hand, in the
case of lexicographic preferences we give a polynomial-time algorithm for finding a
POM, based on the well-known sequential mechanism. However we show that the
problem of deciding whether a given matching is Pareto optimal is co-NP-complete.
We further prove that finding a maximum cardinality (Pareto optimal) matching is
NP-hard. Under alternative prerequisites, we show that finding a POM is NP-hard
for either additive or lexicographic preferences. Finally we consider corequisites. We
prove that, as in the case of compulsory prerequisites, finding a POM is NP-hard
for additive preferences, though solvable in polynomial time for lexicographic preferences.
In the latter case, the problem of finding a maximum cardinality POM is
NP-hard and very difficult to approximate
Solving the Maximum Popular Matching Problem with Matroid Constraints
We consider the problem of finding a maximum popular matching in a
many-to-many matching setting with two-sided preferences and matroid
constraints. This problem was proposed by Kamiyama (2020) and solved in the
special case where matroids are base orderable. Utilizing a newly shown matroid
exchange property, we show that the problem is tractable for arbitrary
matroids. We further investigate a different notion of popularity, where the
agents vote with respect to lexicographic preferences, and show that both
existence and verification problems become NP-hard, even in the -matching
case.Comment: 16 pages, 2 figure
Size versus truthfulness in the House Allocation problem
We study the House Allocation problem (also known as the Assignment problem),
i.e., the problem of allocating a set of objects among a set of agents, where
each agent has ordinal preferences (possibly involving ties) over a subset of
the objects. We focus on truthful mechanisms without monetary transfers for
finding large Pareto optimal matchings. It is straightforward to show that no
deterministic truthful mechanism can approximate a maximum cardinality Pareto
optimal matching with ratio better than 2. We thus consider randomised
mechanisms. We give a natural and explicit extension of the classical Random
Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation
problem where preference lists can include ties. We thus obtain a universally
truthful randomised mechanism for finding a Pareto optimal matching and show
that it achieves an approximation ratio of . The same bound
holds even when agents have priorities (weights) and our goal is to find a
maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On
the other hand we give a lower bound of on the approximation
ratio of any universally truthful Pareto optimal mechanism in settings with
strict preferences. In the case that the mechanism must additionally be
non-bossy with an additional technical assumption, we show by utilising a
result of Bade that an improved lower bound of holds. This
lower bound is tight since RSDM for strict preference lists is non-bossy. We
moreover interpret our problem in terms of the classical secretary problem and
prove that our mechanism provides the best randomised strategy of the
administrator who interviews the applicants.Comment: To appear in Algorithmica (preliminary version appeared in the
Proceedings of EC 2014
Serial Rules in a Multi-Unit Shapley-Scarf Market
We study generalized Shapley-Scarf exchange markets where each agent is endowed with multiple units of an indivisible and agent-specific good and monetary compensations are not possible. An outcome is given by a circulation which consists of a balanced exchange of goods. We focus on circulation rules that only require as input ordinal preference rankings of individual goods, and agents are assumed to have responsive preferences over bundles of goods. We study the properties of serial dictatorship rules which allow agents to choose either a single good or an entire bundle sequentially, according to a fixed ordering of the agents. We also introduce and explore extensions of these serial dictatorship rules that ensure individual rationality. The paper analyzes the normative and incentive properties of these four families of serial dictatorships and also shows that the individually rational extensions can be implemented with efficient graph algorithms.PBiró gratefully acknowledges the financial support by the Hungarian Academy of Sciences, Momentum Grant No. LP2021-2, and by the Hungarian
Scientific Research Fund, OTKA, Grant No. K143858.
F. Klijn gratefully acknowledges financial support from AGAUR–Generalitat de Catalunya (2017-SGR-1359) and the Spanish Agencia Estatal de
Investigación (AEI) through grants ECO2017-88130-P and PID2020-114251GB-I00 (funded by MCIN/ AEI /10.13039/501100011033) and the Severo Ochoa
Programme for Centres of Excellence in R&D (Barcelona School of Economics, SEV-2015-0563 and CEX2019-000915-S).
S. Pápai gratefully acknowledges financial support from an FRQSC grant titled “Formation des coalitions et des réseaux dans les situations économiques
et sociales avec des externalités” (SE-144698