45,285 research outputs found
Matching Theory for Future Wireless Networks: Fundamentals and Applications
The emergence of novel wireless networking paradigms such as small cell and
cognitive radio networks has forever transformed the way in which wireless
systems are operated. In particular, the need for self-organizing solutions to
manage the scarce spectral resources has become a prevalent theme in many
emerging wireless systems. In this paper, the first comprehensive tutorial on
the use of matching theory, a Nobelprize winning framework, for resource
management in wireless networks is developed. To cater for the unique features
of emerging wireless networks, a novel, wireless-oriented classification of
matching theory is proposed. Then, the key solution concepts and algorithmic
implementations of this framework are exposed. Then, the developed concepts are
applied in three important wireless networking areas in order to demonstrate
the usefulness of this analytical tool. Results show how matching theory can
effectively improve the performance of resource allocation in all three
applications discussed
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
Matching under Preferences
Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory.
Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs.
Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process.
Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully
Distributed Channel Assignment in Cognitive Radio Networks: Stable Matching and Walrasian Equilibrium
We consider a set of secondary transmitter-receiver pairs in a cognitive
radio setting. Based on channel sensing and access performances, we consider
the problem of assigning channels orthogonally to secondary users through
distributed coordination and cooperation algorithms. Two economic models are
applied for this purpose: matching markets and competitive markets. In the
matching market model, secondary users and channels build two agent sets. We
implement a stable matching algorithm in which each secondary user, based on
his achievable rate, proposes to the coordinator to be matched with desirable
channels. The coordinator accepts or rejects the proposals based on the channel
preferences which depend on interference from the secondary user. The
coordination algorithm is of low complexity and can adapt to network dynamics.
In the competitive market model, channels are associated with prices and
secondary users are endowed with monetary budget. Each secondary user, based on
his utility function and current channel prices, demands a set of channels. A
Walrasian equilibrium maximizes the sum utility and equates the channel demand
to their supply. We prove the existence of Walrasian equilibrium and propose a
cooperative mechanism to reach it. The performance and complexity of the
proposed solutions are illustrated by numerical simulations.Comment: submitted to IEEE Transactions on Wireless Communicaitons, 13 pages,
10 figures, 4 table
Stable marriage with general preferences
We propose a generalization of the classical stable marriage problem. In our
model, the preferences on one side of the partition are given in terms of
arbitrary binary relations, which need not be transitive nor acyclic. This
generalization is practically well-motivated, and as we show, encompasses the
well studied hard variant of stable marriage where preferences are allowed to
have ties and to be incomplete. As a result, we prove that deciding the
existence of a stable matching in our model is NP-complete. Complementing this
negative result we present a polynomial-time algorithm for the above decision
problem in a significant class of instances where the preferences are
asymmetric. We also present a linear programming formulation whose feasibility
fully characterizes the existence of stable matchings in this special case.
Finally, we use our model to study a long standing open problem regarding the
existence of cyclic 3D stable matchings. In particular, we prove that the
problem of deciding whether a fixed 2D perfect matching can be extended to a 3D
stable matching is NP-complete, showing this way that a natural attempt to
resolve the existence (or not) of 3D stable matchings is bound to fail.Comment: This is an extended version of a paper to appear at the The 7th
International Symposium on Algorithmic Game Theory (SAGT 2014
Consistency and Monotonicity in One-Sided Assignment Problems
One-sided assignment problems combine important features of two well-known matching models. First, as in roommate problems, any two agents can be matched and second, as in two-sided assignment problems, the payoffs of a matching can be divided between the agents. We take a similar approach to one-sided assignment problems as Sasaki (1995) for two-sided assignment problems and we analyze various desirable properties of solutions including consistency and weak pairwise-monotonicity. We show that for the class of solvable one-sided assignment problems (i.e., the subset of one-sided assignment problems with a non-empty core), if a subsolution of the core satisfies [indifference with respect to dummy agents, continuity, and consistency] or [Pareto indifference and consistency], then it coincides with the core (Theorems 1 and 2). However, we also prove that on the class of all one-sided assignment problems (solvable or not), no solution satisfies consistency and coincides with the core whenever the core is non-empty (Theorem 3). Finally, we comment on the difficulty in obtaining further positive results for the class of solvable one-sided assignment problems in line with Sasaki's (1995) characterizations of the core for two-sided assignment problems.(One-sided) assignment problems, consistency, core, matching.
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