72,029 research outputs found
Approximately Stable, School Optimal, and Student-Truthful Many-to-One Matchings (via Differential Privacy)
We present a mechanism for computing asymptotically stable school optimal
matchings, while guaranteeing that it is an asymptotic dominant strategy for
every student to report their true preferences to the mechanism. Our main tool
in this endeavor is differential privacy: we give an algorithm that coordinates
a stable matching using differentially private signals, which lead to our
truthfulness guarantee. This is the first setting in which it is known how to
achieve nontrivial truthfulness guarantees for students when computing school
optimal matchings, assuming worst- case preferences (for schools and students)
in large markets
An Algorithm for Strong Stability in the Student-Project Allocation Problem With Ties
We study a variant of the Student-Project Allocation problem with lecturer preferences over Students where ties are allowed in the preference lists of students and lecturers (spa-st). We investigate the concept of strong stability in this context. Informally, a matching is strongly stable if there is no student and lecturer l such that if they decide to form a private arrangement outside of the matching via one of l’s proposed projects, then neither party would be worse off and at least one of them would strictly improve. We describe the first polynomial-time algorithm to find a strongly stable matching or report that no such matching exists, given an instance of spa-st. Our algorithm runs in O(m2) time, where m is the total length of the students’ preference lists
Average Sensitivity of Graph Algorithms
In modern applications of graphs algorithms, where the graphs of interest are
large and dynamic, it is unrealistic to assume that an input representation
contains the full information of a graph being studied. Hence, it is desirable
to use algorithms that, even when only a (large) subgraph is available, output
solutions that are close to the solutions output when the whole graph is
available. We formalize this idea by introducing the notion of average
sensitivity of graph algorithms, which is the average earth mover's distance
between the output distributions of an algorithm on a graph and its subgraph
obtained by removing an edge, where the average is over the edges removed and
the distance between two outputs is the Hamming distance.
In this work, we initiate a systematic study of average sensitivity. After
deriving basic properties of average sensitivity such as composition, we
provide efficient approximation algorithms with low average sensitivities for
concrete graph problems, including the minimum spanning forest problem, the
global minimum cut problem, the minimum - cut problem, and the maximum
matching problem. In addition, we prove that the average sensitivity of our
global minimum cut algorithm is almost optimal, by showing a nearly matching
lower bound. We also show that every algorithm for the 2-coloring problem has
average sensitivity linear in the number of vertices. One of the main ideas
involved in designing our algorithms with low average sensitivity is the
following fact; if the presence of a vertex or an edge in the solution output
by an algorithm can be decided locally, then the algorithm has a low average
sensitivity, allowing us to reuse the analyses of known sublinear-time
algorithms and local computation algorithms (LCAs). Using this connection, we
show that every LCA for 2-coloring has linear query complexity, thereby
answering an open question.Comment: 39 pages, 1 figur
Mechanism Design without Money via Stable Matching
Mechanism design without money has a rich history in social choice
literature. Due to the strong impossibility theorem by Gibbard and
Satterthwaite, exploring domains in which there exist dominant strategy
mechanisms is one of the central questions in the field. We propose a general
framework, called the generalized packing problem (\gpp), to study the
mechanism design questions without payment. The \gpp\ possesses a rich
structure and comprises a number of well-studied models as special cases,
including, e.g., matroid, matching, knapsack, independent set, and the
generalized assignment problem.
We adopt the agenda of approximate mechanism design where the objective is to
design a truthful (or strategyproof) mechanism without money that can be
implemented in polynomial time and yields a good approximation to the socially
optimal solution. We study several special cases of \gpp, and give constant
approximation mechanisms for matroid, matching, knapsack, and the generalized
assignment problem. Our result for generalized assignment problem solves an
open problem proposed in \cite{DG10}.
Our main technical contribution is in exploitation of the approaches from
stable matching, which is a fundamental solution concept in the context of
matching marketplaces, in application to mechanism design. Stable matching,
while conceptually simple, provides a set of powerful tools to manage and
analyze self-interested behaviors of participating agents. Our mechanism uses a
stable matching algorithm as a critical component and adopts other approaches
like random sampling and online mechanisms. Our work also enriches the stable
matching theory with a new knapsack constrained matching model
Student-project allocation with preferences over projects
We study the problem of allocating students to projects, where both students and lecturers have preferences over projects, and both projects and lecturers have capacities. In this context we seek a stable matching of students to projects, which respects these preference and capacity constraints. Here, the stability definition generalises the corresponding notion in the context of the classical Hospitals/Residents problem. We show that stable matchings can have different sizes, which motivates max-spa-p, the problem of finding maximum cardinality stable matching. We prove that max-spa-p is NP-hard and not approximable within δ, for some δ>1, unless P=NP. On the other hand, we give an approximation algorithm with a performance guarantee of 2 for max-spa-p
Contagious Synchronization and Endogenous Network Formation in Financial Networks
When banks choose similar investment strategies the financial system becomes
vulnerable to common shocks. We model a simple financial system in which banks
decide about their investment strategy based on a private belief about the
state of the world and a social belief formed from observing the actions of
peers. Observing a larger group of peers conveys more information and thus
leads to a stronger social belief. Extending the standard model of Bayesian
updating in social networks, we show that the probability that banks
synchronize their investment strategy on a state non-matching action critically
depends on the weighting between private and social belief. This effect is
alleviated when banks choose their peers endogenously in a network formation
process, internalizing the externalities arising from social learning.Comment: 41 pages, 10 figures, Journal of Banking & Finance 201
A constraint programming approach to the hospitals/residents problem
An instance I of the Hospitals/Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a <i>stable matching</i>, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. We provide additional motivation for our models by indicating how side constraints can be added easily in order to solve hard variants of HR
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