218,774 research outputs found
Pareto Optimal Allocation under Uncertain Preferences
The assignment problem is one of the most well-studied settings in social
choice, matching, and discrete allocation. We consider the problem with the
additional feature that agents' preferences involve uncertainty. The setting
with uncertainty leads to a number of interesting questions including the
following ones. How to compute an assignment with the highest probability of
being Pareto optimal? What is the complexity of computing the probability that
a given assignment is Pareto optimal? Does there exist an assignment that is
Pareto optimal with probability one? We consider these problems under two
natural uncertainty models: (1) the lottery model in which each agent has an
independent probability distribution over linear orders and (2) the joint
probability model that involves a joint probability distribution over
preference profiles. For both of the models, we present a number of algorithmic
and complexity results.Comment: Preliminary Draft; new results & new author
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
Coalitions and Cliques in the School Choice Problem
The school choice mechanism design problem focuses on assignment mechanisms
matching students to public schools in a given school district. The well-known
Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most
efficient stable mechanism proposed so far as a solution to this problem.
However its inefficiency is well-documented, and recently the Efficiency
Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for
this weakness. In this note we describe two related adjustments to SOSM with
the intention to address the same inefficiency issue. In one we create possibly
artificial coalitions among students where some students modify their
preference profiles in order to improve the outcome for some other students.
Our second approach involves trading cliques among students where those
involved improve their assignments by waiving some of their priorities. The
coalition method yields the EADAM outcome among other Pareto dominations of the
SOSM outcome, while the clique method yields all possible Pareto optimal Pareto
dominations of SOSM. The clique method furthermore incorporates a natural
solution to the problem of breaking possible ties within preference and
priority profiles. We discuss the practical implications and limitations of our
approach in the final section of the article
Convexity and Robustness of Dynamic Traffic Assignment and Freeway Network Control
We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA)
problem to design optimal traffic flow controls for freeway networks as modeled
by the Cell Transmission Model, using variable speed limit, ramp metering, and
routing. We consider two optimal control problems: the DTA problem, where
turning ratios are part of the control inputs, and the Freeway Network Control
(FNC), where turning ratios are instead assigned exogenous parameters. It is
known that relaxation of the supply and demand constraints in the cell-based
formulations of the DTA problem results in a linear program. However, solutions
to the relaxed problem can be infeasible with respect to traffic dynamics.
Previous work has shown that such solutions can be made feasible by proper
choice of ramp metering and variable speed limit control for specific traffic
networks. We extend this procedure to arbitrary networks and provide insight
into the structure and robustness of the proposed optimal controllers. For a
network consisting only of ordinary, merge, and diverge junctions, where the
cells have linear demand functions and affine supply functions with identical
slopes, and the cost is the total traffic volume, we show, using the maximum
principle, that variable speed limits are not needed in order to achieve
optimality in the FNC problem, and ramp metering is sufficient. We also prove
bounds on perturbation of the controlled system trajectory in terms of
perturbations in initial traffic volume and exogenous inflows. These bounds,
which leverage monotonicity properties of the controlled trajectory, are shown
to be in close agreement with numerical simulation results
A combinatorial approximation algorithm for CDMA downlink rate allocation
This paper presents a combinatorial algorithm for downlink rate allocation in Code Division Multiple Access (CDMA) mobile networks. By discretizing the coverage area into small segments, the transmit power requirements are characterized via a matrix representation that separates user and system characteristics. We obtain a closed-form analytical expression for the so-called Perron-Frobenius eigenvalue of that matrix, which provides a quick assessment of the feasibility of the power assignment for a given downlink rate allocation. Based on the Perron-Frobenius eigenvalue, we reduce the downlink rate allocation problem to a set of multiple-choice knapsack problems. The solution of these problems provides an approximation of the optimal downlink rate allocation and cell borders for which the system throughput, expressed in terms of utility functions of the users, is maximized
A multiple-choice knapsack based algorithm for CDMA downlink rate differentiation under uplink coverage restrictions
This paper presents an analytical model for downlink rate allocation in Code Division Multiple Access (CDMA) mobile networks. By discretizing the coverage area into small segments, the transmit power requirements are characterized via a matrix representation that separates user and system characteristics. We obtain a closed-form analytical expression for the so-called Perron-Frobenius eigenvalue of that matrix, which provides a quick assessment of the feasibility of the power assignment for a given downlink rate allocation. Based on the Perron-Frobenius eigenvalue, we reduce the downlink rate allocation problem to a set of multiple-choice knapsack problems. The solution of these problems provides an approximation of the optimal downlink rate allocation and cell borders for which the system throughput, expressed in terms of downlink rates, is maximized. \u
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