607 research outputs found
Popular matchings in the weighted capacitated house allocation problem
We consider the problem of finding a popular matching in the <i>Weighted Capacitated House Allocation</i> 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 Mā² such that the total weight of the agents who prefer their allocation in Mā² to that in M exceeds the total weight of the agents who prefer their allocation in M to that in Mā². Here, we give an [FORMULA] 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
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
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
Truthful Assignment without Money
We study the design of truthful mechanisms that do not use payments for the
generalized assignment problem (GAP) and its variants. An instance of the GAP
consists of a bipartite graph with jobs on one side and machines on the other.
Machines have capacities and edges have values and sizes; the goal is to
construct a welfare maximizing feasible assignment. In our model of private
valuations, motivated by impossibility results, the value and sizes on all
job-machine pairs are public information; however, whether an edge exists or
not in the bipartite graph is a job's private information.
We study several variants of the GAP starting with matching. For the
unweighted version, we give an optimal strategyproof mechanism; for maximum
weight bipartite matching, however, we show give a 2-approximate strategyproof
mechanism and show by a matching lowerbound that this is optimal. Next we study
knapsack-like problems, which are APX-hard. For these problems, we develop a
general LP-based technique that extends the ideas of Lavi and Swamy to reduce
designing a truthful mechanism without money to designing such a mechanism for
the fractional version of the problem, at a loss of a factor equal to the
integrality gap in the approximation ratio. We use this technique to obtain
strategyproof mechanisms with constant approximation ratios for these problems.
We then design an O(log n)-approximate strategyproof mechanism for the GAP by
reducing, with logarithmic loss in the approximation, to our solution for the
value-invariant GAP. Our technique may be of independent interest for designing
truthful mechanisms without money for other LP-based problems.Comment: Extended abstract appears in the 11th ACM Conference on Electronic
Commerce (EC), 201
Social status in economic theory: a review.
Social distinction or status is an important motivation of human behaviour. This paper provides a selective survey of recent advances in the economic analysis of the origins and consequences of social status. First, a selection of empirical research from a variety of scientific disciplines is discussed to underpin the further theoretical analysis. I then consider the origins and determinants of tastes for status, discuss the endogenous derivation of such a preferences for relative standing and assess the different formalisations these preferences. Subsequently, the consequences of preferences for status are studied for a variety of problems and settings. The last section discusses a number of implications of status concerns for normative economics and public policy.
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 countriesā or refugeesā 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
Distributed Caching in Small Cell Networks
The dense deployment of small cells in indoor and outdoor areas contributes mainly in increasing the capacity of cellular networks. On the other hand, the high number of deployed base stations coupled with the increasing growth of data traffic have prompted the apparition of base stations fi tted with storage capacity to avoid network saturation. The storage devices are used as caching units to overcome the limited backhaul capacity in small cells networks (SCNs). Extending the concept of storage to SCNs, gives rise to many new challenges related to the specific characteristics of these networks such as the heterogeneity of the base stations. Formulating the caching problem while taking into account all these specific characteristics with the aim to satisfy the users expectations result in combinatorial optimization problems. However, classical optimization tools do not ensure the optimality of the provided solutions or often the proposed algorithms have an exponential complexity. While most of the existing works are based on the classical optimization tools, in this thesis, we explore another approach to provide a practical solution for the caching problem. In particular, we focus on matching theory which is a game theoretic approach that provides mathematical tools to formulate, analyze and understand scenarios between sets of players. We model the caching problem as a one-to-one matching game between a set of files and a set of base stations and then, we propose an iterative extension of the deferred acceptance algorithm that needs a stable and optimal matching between the two sets. The experimental results show that the proposed algorithm reduces the backhaul load by 10-15 % compared to a random caching algorithm
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