361 research outputs found
How Good Are Popular Matchings?
In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem with Lower Quotas (HRLQ). In this model with two sided preferences, stability is a well accepted notion of optimality. However, in the presence of lower quotas, a stable and feasible matching need not exist. For the HRLQ problem, our goal therefore is to output a good feasible matching assuming that a feasible matching exists. Computing matchings with minimum number of blocking pairs (Min-BP) and minimum number of blocking residents (Min-BR) are known to be NP-Complete. The only approximation algorithms for these problems work under severe restrictions on the preference lists. We present an algorithm which circumvents this restriction and computes a popular matching in the HRLQ instance. We show that on data-sets generated using various generators, our algorithm performs very well in terms of blocking pairs and blocking residents. Yokoi [Yokoi, 2017] recently studied envy-free matchings for the HRLQ problem. We propose a simple modification to Yokoi\u27s algorithm to output a maximal envy-free matching. We observe that popular matchings outperform envy-free matchings on several parameters of practical importance, like size, number of blocking pairs, number of blocking residents.
In the absence of lower quotas, that is, in the Hospital Residents (HR) problem, stable matchings are guaranteed to exist. Even in this case, we show that popularity is a practical alternative to stability. For instance, on synthetic data-sets generated using a particular model, as well as on real world data-sets, a popular matching is on an average 8-10% larger in size, matches more number of residents to their top-choice, and more residents prefer the popular matching as compared to a stable matching. Our comprehensive study reveals the practical appeal of popular matchings for the HR and HRLQ problems. To the best of our knowledge, this is the first study on the empirical evaluation of popular matchings in this setting
Controlled Matching Game for Resource Allocation and User Association in WLANs
In multi-rate IEEE 802.11 WLANs, the traditional user association based on
the strongest received signal and the well known anomaly of the MAC protocol
can lead to overloaded Access Points (APs), and poor or heterogeneous
performance. Our goal is to propose an alternative game-theoretic approach for
association. We model the joint resource allocation and user association as a
matching game with complementarities and peer effects consisting of selfish
players solely interested in their individual throughputs. Using recent
game-theoretic results we first show that various resource sharing protocols
actually fall in the scope of the set of stability-inducing resource allocation
schemes. The game makes an extensive use of the Nash bargaining and some of its
related properties that allow to control the incentives of the players. We show
that the proposed mechanism can greatly improve the efficiency of 802.11 with
heterogeneous nodes and reduce the negative impact of peer effects such as its
MAC anomaly. The mechanism can be implemented as a virtual connectivity
management layer to achieve efficient APs-user associations without
modification of the MAC layer
Popular matchings with two-sided preferences and one-sided ties
We are given a bipartite graph where each vertex has a
preference list ranking its neighbors: in particular, every ranks its
neighbors in a strict order of preference, whereas the preference lists of may contain ties. A matching is popular if there is no matching
such that the number of vertices that prefer to exceeds the number of
vertices that prefer to~. We show that the problem of deciding whether
admits a popular matching or not is NP-hard. This is the case even when
every either has a strict preference list or puts all its neighbors
into a single tie. In contrast, we show that the problem becomes polynomially
solvable in the case when each puts all its neighbors into a single
tie. That is, all neighbors of are tied in 's list and desires to be
matched to any of them. Our main result is an algorithm (where ) for the popular matching problem in this model. Note that this model
is quite different from the model where vertices in have no preferences and
do not care whether they are matched or not.Comment: A shortened version of this paper has appeared at ICALP 201
A structural approach to matching problems with preferences
This thesis is a study of a number of matching problems that seek to match together pairs or groups of agents subject to the preferences of some or all of the agents. We present a number of new algorithmic results for five specific problem domains. Each of these results is derived with the aid of some structural properties implicitly embedded in the problem.
We begin by describing an approximation algorithm for the problem of finding a maximum stable matching for an instance of the stable marriage problem with ties and incomplete lists (MAX-SMTI). Our polynomial time approximation algorithm provides a performance guarantee of 3/2 for the general version of MAX-SMTI, improving
upon the previous best approximation algorithm, which gave a performance guarantee of 5/3.
Next, we study the sex-equal stable marriage problem (SESM). We show that SESM is W[1]-hard, even if the men's and women's preference lists are both of length at most three. This improves upon the previously known hardness results. We contrast this with an exact, low-order exponential time algorithm. This is the first non-trivial exponential time algorithm known for this problem, or indeed for any hard stable matching problem.
Turning our attention to the hospitals / residents problem with couples (HRC), we show that
HRC is NP-complete, even if very severe restrictions are placed on the input. By contrast, we give a linear-time algorithm to find a stable matching with couples (or report that none exists) when stability is defined in terms of the classical Gale-Shapley concept. This result represents the most general polynomial time solvable restriction of HRC that we are aware of.
We then explore the three dimensional stable matching problem (3DSM), in which we seek to find stable matchings
across three sets of agents, rather than two (as in the classical case). We show that under two natural definitions of stability, finding a stable matching
for a 3DSM instance is NP-complete. These hardness results resolve some open questions in the literature.
Finally, we study the popular matching problem (POP-M) in the context of matching a set of applicants to a set of posts. We provide a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a new structure called the switching graph exploited to yield efficient algorithms for a range of associated problems, extending and improving upon the previously best-known results for this problem
Capacity Planning in Stable Matching
We introduce the problem of jointly increasing school capacities and finding
a student-optimal assignment in the expanded market. Due to the impossibility
of efficiently solving the problem with classical methods, we generalize
existent mathematical programming formulations of stability constraints to our
setting, most of which result in integer quadratically-constrained programs. In
addition, we propose a novel mixed-integer linear programming formulation that
is exponentially large on the problem size. We show that its stability
constraints can be separated by exploiting the objective function, leading to
an effective cutting-plane algorithm. We conclude the theoretical analysis of
the problem by discussing some mechanism properties. On the computational side,
we evaluate the performance of our approaches in a detailed study, and we find
that our cutting-plane method outperforms our generalization of existing
mixed-integer approaches. We also propose two heuristics that are effective for
large instances of the problem. Finally, we use the Chilean school choice
system data to demonstrate the impact of capacity planning under stability
conditions. Our results show that each additional seat can benefit multiple
students and that we can effectively target the assignment of previously
unassigned students or improve the assignment of several students through
improvement chains. These insights empower the decision-maker in tuning the
matching algorithm to provide a fair application-oriented solution
Efficient algorithms for optimal matching problems under preferences
In this thesis we consider efficient algorithms for matching problems involving preferences,
i.e., problems where agents may be required to list other agents that they find
acceptable in order of preference. In particular we mainly study the Stable Marriage
problem (SM), the Hospitals / Residents problem (HR) and the Student / Project Allocation
problem (SPA), and some of their variants. In some of these problems the aim
is to find a stable matching which is one that admits no blocking pair. A blocking pair
with respect to a matching is a pair of agents that prefer to be matched to each other
than their assigned partners in the matching if any.
We present an Integer Programming (IP) model for the Hospitals / Residents problem
with Ties (HRT) and use it to find a maximum cardinality stable matching. We also
present results from an empirical evaluation of our model which show it to be scalable
with respect to real-world HRT instance sizes.
Motivated by the observation that not all blocking pairs that exist in theory will lead
to a matching being undermined in practice, we investigate a relaxed stability criterion
called social stability where only pairs of agents with a social relationship have the
ability to undermine a matching. This stability concept is studied in instances of
the Stable Marriage problem with Incomplete lists (smi) and in instances of hr. We
show that, in the smi and hr contexts, socially stable matchings can be of varying
sizes and the problem of finding a maximum socially stable matching (max smiss and
max hrss respectively) is NP-hard though approximable within 3/2. Furthermore we
give polynomial time algorithms for three special cases of the problem arising from
restrictions on the social network graph and the lengths of agentsâ preference lists.
We also consider other optimality criteria with respect to social stability and establish
inapproximability bounds for the problems of finding an egalitarian, minimum regret
and sex equal socially stable matching in the sm context.
We extend our study of social stability by considering other variants and restrictions
of max smiss and max hrss. We present NP-hardness results for max smiss even
under certain restrictions on the degree and structure of the social network graph as
well as the presence of master lists. Other NP-hardness results presented relate to the
problem of determining whether a given man-woman pair belongs to a socially stable
matching and the problem of determining whether a given man (or woman) is part of
at least one socially stable matching. We also consider the Stable Roommates problem
with Incomplete lists under Social Stability (a non-bipartite generalisation of smi under
social stability). We observe that the problem of finding a maximum socially stable
matching in this context is also NP-hard. We present efficient algorithms for three
special cases of the problem arising from restrictions on the social network graph and
the lengths of agentsâ preference lists. These are the cases where (i) there exists a
constant number of acquainted pairs (ii) or a constant number of unacquainted pairs
or (iii) each preference list is of length at most 2.
We also present algorithmic results for finding matchings in the spa context that are
optimal with respect to profile, which is the vector whose ith component is the number
of students assigned to their ith-choice project. We present an efficient algorithm for
finding a greedy maximum matching in the spa context â this is a maximum matching
whose profile is lexicographically maximum. We then show how to adapt this algorithm
to find a generous maximum matching â this is a matching whose reverse profile is
lexicographically minimum. We demonstrate how this approach can allow additional
constraints, such as lecturer lower quotas, to be handled flexibly. We also present
results of empirical evaluations carried out on both real world and randomly generated
datasets. These results demonstrate the scalability of our algorithms as well as some
interesting properties of these profile-based optimality criteria.
Practical applications of spa motivate the investigation of certain special cases of the
problem. For instance, it is often desired that the workload on lecturers is evenly distributed
(i.e. load balanced). We enforce this by either adding lower quota constraints
on the lecturers (which leads to the potential for infeasible problem instances) or adding
a load balancing optimisation criterion. We present efficient algorithms in both cases.
Another consideration is the fact that certain projects may require a minimum number
of students to become viable. This can be handled by enforcing lower quota constraints
on the projects (which also leads to the possibility of infeasible problem instances). A
technique of handling this infeasibility is the idea of closing projects that do not meet
their lower quotas (i.e. leaving such project completely unassigned). We show that the
problem of finding a maximum matching subject to project lower quotas where projects
can be closed is NP-hard even under severe restrictions on preference lists lengths and
project upper and lower quotas. To offset this hardness, we present polynomial time
heuristics that find large feasible matchings in practice. We also present ip models
for the spa variants discussed and show results obtained from an empirical evaluation
carried out on both real and randomly generated datasets. These results show that
our algorithms and heuristics are scalable and provide good matchings with respect to
profile-based optimalit
Towards a fair distribution mechanism for asylum
It has been suggested that the distribution of refugees over host countries can be made more fair or efficient if policy makers take into account not only numbers of refugees to be distributed but also the goodness of the matches between refugees and their possible host countries. There are different ways to design distribution mechanisms that incorporate this practice, which opens up a space for normative considerations. In particular, if the mechanism takes countriesâ or refugeesâ preferences into account, there may be trade-offs between satisfying their preferences and the number of refugees distributed. This article argues that, in such cases, it is not a reasonable policy to satisfy preferences. Moreover, conditions are given which, if satisfied, prevent the trade-off from occurring. Finally, it is argued that countries should not express preferences over refugees, but rather that priorities for refugees should be imposed, and that fairness beats efficiency in the context of distributing asylum. The framework of matching theory is used to make the arguments precise, but the results are general and relevant for other distribution mechanisms such as the relocations currently in effect in the European Unio
Investigation of Matching Problems using Constraint Programming and Optimisation Methods
This thesis focuses on matching under ordinal preferences, i.e. problems where agents may be required to list other agents that they find acceptable in order of preference. In particular, we focus on two main cases: the popular matching and the kidney exchange problem. These problems are important in practice and in this thesis we develop novel algorithms and techniques to solve them as combinatorial optimisation problems. The first part of the thesis focuses on one-sided matching on a bipartite graph, specifically the popular matching. When the participants express their preferences in an ordinal order, one might want to guarantee that no two applicants are inclined to form a coalition in order to maximise their welfare, thus finding a stable matching is needed. Popularity is a concept that offers an attractive trade- off between these two notions. In particular, we examine the popular matching in the context of constraint programming using global constraints. We discuss the possibility to find a popular matching even for the instances that does not admit one.
The second part of the thesis focuses on non-bipartite graphs, i.e. the kidney exchange problem. Kidney transplant is the most effective treatment to cure end-stage renal disease, affecting one in every thousand European citizen. Motivated by the observation that the kidney exchange is inherently a stochastic online problem, first, we give a stochastic online method, which provides an expected value estimation that is correct within the limit of sampling errors. Second, we show that by taking into consideration a probabilistic model of future arrivals and drop-offs, we can get reduce sampling scenarios, and we can even construct a sampling-free probabilistic model, called the Abstract Exchange Graph (AEG). A final contribution of this thesis is related to finding robust solutions when uncertainty occurs. Uncertainty is inherent to most real world problems
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