Fair and large stable matchings in the stable marriage and student-project allocation problems

In this thesis, we present new algorithmic and complexity results for specific matching problems involving preferences. In particular we study the Stable Marriage problem (SM) and the Student-Project Allocation problem (SPA) and their variants. A matching in these scenarios is an allocation of men to women (SM) or students to projects (SPA). Primarily we are interested in finding matchings that are stable. A stable matching is a matching that admits no blocking pair, which is a pair of agents (not already allocated together) who would rather deviate from the given matching and become assigned to each other. In addition to stability, other objectives may be applied. We focus on finding either fair or large stable matchings in SM and SPA. In the Stable Marriage problem with Incomplete lists (SMI), the rank of a matched man or woman, with respect to a matching, is the position of their assigned partner on their preference list. The degree of the set of men in a matching is the rank of a worst-off man, over all matched men. A similar definition exists for the set of women. The cost of the set of men in a matching is sum of ranks of all matched men. Again a similar definition exists for the set of women. We introduce the following degree-based definitions of fairness in SMI. A stable matching is regret-equal if it minimises the difference in degree between the set of men and the set of women, over all stable matchings. Additionally, a stable matching is min-regret sum if it minimises the sum of the degree of men and the degree of women, over all stable matchings. We present polynomial-time algorithms to find these types of fair stable matchings, given an instance of SMI, and perform experiments to both test the performance of two algorithms to find a regret-equal stable matching, and to compare properties of several other types of fair stable matchings over both degree- and cost-based measures. Also in SMI, we investigate fairness in the form of profile-based stable matchings. The profile of a matching is a vector of integers, where the ith vector element indicates the number of agents assigned to their ith-choice partner. A stable matching is rank-maximal if its profile is lexicographically maximum taken over all stable matchings. A stable matching is generous if its reverse profile is lexicographically minimum taken over all stable matchings. A polynomial-time algorithm exists to find a rank-maximal stable matching, using weights that are exponential in the number of men or women [32]. We adapt this algorithm to work with polynomially-bounded weight vectors, and show using randomly-generated instances that our approach is significantly less costly in terms of space. Further experiments are carried out to compare these profile-based optimal matchings over several cost- and profile-based fairness properties. We additionally show that in the Stable Roommates problem, each of the problems of finding a rank-maximal stable matching or a generous stable matching is NP-hard. In the Student-Project Allocation problem with lecturer preferences over Students including Ties (SPA-ST) we study the problem of finding large stable matchings. A 3/2-approximation algorithm exists to find a maximum-sized stable matching in HRT, and we extend this to the SPA-ST case, developing a linear-time 3/2-approximation algorithm for the problem of finding a maximum-sized stable matching in SPA-ST. We test the performance of our approximation algorithm using the implementation of a new Integer Programming (IP) model that finds a maximum-sized stable matching, and show that in practice, our approximation algorithm produces stable matchings of size far closer to optimal than the 3/2 bound. Additionally, we give an example to show that this bound is tight. Finally, we look at fairness in the context of the Student-Project Allocation problem with lecturer preferences over Students including Ties and Lecturer targets (SPA-STL), an extension to SPA-ST in which each lecturer has an associated target, indicating their preferred number of allocations (or the number of allocations preferred by the matching scheme administrator). We first investigate load balancing without the presence of stability. A load-max-balanced matching is a matching in which the maximum difference between a lecturer’s target and their number of allocations is minimised. A load-sum-balanced matching is a matching in which the sum of differences between all lecturers’ targets and their number of allocations is minimised. Finally, a load-balanced matching is a matching that is both load-max-balanced and load-sum-balanced. We provide new polynomial-time algorithms to find matchings of these types. Additionally we show that in the presence of stability, each of the problems of finding a stable matching that is load-max-balanced, load-sum-balanced or load-balanced is NP-hard. Finally, we present new IP models for finding such types of optimal stable matching

Disconnected Skeleton: Shape at its Absolute Scale

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.Comment: The work excluding {\S}V and {\S}VI has first appeared in 2005 ICCV: Aslan, C., Tari, S.: An Axis-Based Representation for Recognition. In ICCV(2005) 1339- 1346.; Aslan, C., : Disconnected Skeletons for Shape Recognition. Masters thesis, Department of Computer Engineering, Middle East Technical University, May 200

Stable Marriage Problem Based Adaptation for Clone Detection and Service Selection

Current software engineering topics such as clone detection and service selection need to improve the capability of detection process and selection process. The clone detection is the process of finding duplicated code through the system for several purposes such as removal of repeated portions as maintenance part of legacy system. Service selection is the process of finding the appropriate web service which meets the consumer’s request. Both problems can be converted into a matching problem. Matching process forms an essential part of software engineering activities. In this research, a well-known mathematical algorithm Stable Marriage Problem (SMP) and its variations are investigated to fulfil the purposes of matching processes in software engineering area. We aim to provide a competitive matching algorithm that can help to detect cloned software accurately and ensure high scalability, precision and recall. We also aim to apply matching algorithm on incoming request and service profile to deal with the web service as a clever independent object so that we can allow the services to accept or decline requests (equal opportunity) rather than the current state of service selection (search-based), in which service lacks of interacting as an independent candidate. In order to meet the above aims, the traditional SMP algorithm has been extended to achieve the cardinality of many-to-many. This adaptation is achieved by defining the selective strategy which is the main engine of the new adaptations. Two adaptations, Dual-Proposed and Dual-Multi-Allocation, have been proposed to both service selection and clone detection process. The proposed approach (SMP-based) shows very competitive results compare to existing software clone approaches, especially in identifying type 3 (copy with further modifications such update, add and delete statements) of cloned software. It performs the detection process with a relatively high precision and recall compare to the CloneDR tool and shows good scalability on a middle sized program. For service selection, the proposed approach has several advantages such as service protection and service quality. The services gain equal opportunity against the incoming requests. Therefore, the intelligent service interaction is achieved, and both stability and satisfaction of the candidates are ensured. This dissertation contributes to several contributions firstly, the new extended SMP algorithm by introducing selective strategy to accommodate many-to-many matching problems, to improve overall features. Secondly, a new SMP-based clone detection approach to detect cloned software accurately and ensures high precision and recall. Ultimately, a new SMPbased service selection approach allows equal opportunity between services and requests. This led to improve service protection and service quality. Case studies are carried out for experiments with the proposed approach, which show that the new adaptations can be applied effectively to clone detection and service selection processes with several features (e.g. accuracy). It can be concluded that the match based approach is feasible and promising in software engineering domain.Royal Embassy of Saudi Arabi

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

Computing With Distributed Information

The age of computing with massive data sets is highlighting new computational challenges. Nowadays, a typical server may not be able to store an entire data set, and thus data is often partitioned and stored on multiple servers in a distributed manner. A natural way of computing with such distributed data is to use distributed algorithms: these are algorithms where the participating parties (i.e., the servers holding portions of the data) collaboratively compute a function over the entire data set by sending (preferably small-size) messages to each other, where the computation performed at each participating party only relies on the data possessed by it and the messages received by it. We study distributed algorithms focused on two key themes: convergence time and data summarization. Convergence time measures how quickly a distributed algorithm settles on a globally stable solution, and data summarization is the approach of creating a compact summary of the input data while retaining key information. The latter often leads to more efficient computation and communication. The main focus of this dissertation is on design and analysis of distributed algorithms for important problems in diverse application domains centering on the themes of convergence time and data summarization. Some of the problems we study include convergence time of double oral auction and interdomain routing, summarizing graphs for large-scale matching problems, and summarizing data for query processing

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

Filling position incentives in matching markets

One of the main problems in the hospital-doctor matching is the maldistribution of doctor assignments across hospitals. Namely, many hospitals in rural areas are matched with far fewer doctors than what they need. The so called "Rural Hospital Theorem" (Roth (1984)) reveals that it is unavoidable under stable assignments. On the other hand, the counterpart of the problem in the school choice context|low enrollments at schools| has important consequences for schools as well. In the current study, we approach the problem from a different point of view and investigate whether hospitals can increase their filled positions by misreporting their preferences under well-known Boston, Top Trading Cycles, and stable rules. It turns out that while it is impossible under Boston and stable mechanisms, Top Trading Cycles rule is manipulable in that sense

A J-Spectral Factorization Approach to ℋ∞ Control

Necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with ℋ∞ control, are derived using J-spectral factorization theory. The existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled J-spectral factorization problems, with the second factor providing a parametrization of all solutions to the model matching problem. The existence of the J-spectral factors is then shown to be equivalent to the existence of nonnegative definite, stabilizing solutions to two indefinite algebraic Riccati equations, allowing a state-space formula for a linear fractional representation of all controllers to be given. A virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework, and no additional auxiliary Riccati equations are required

Local search for stable marriage problems with ties and incomplete lists

The stable marriage problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. We consider a useful variation of the stable marriage problem, where the men and women express their preferences using a preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these preference lists. In this setting, we study the problem of finding a stable matching that marries as many people as possible. Stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. This problem is NP-hard. We tackle this problem using local search, exploiting properties of the problem to reduce the size of the neighborhood and to make local moves efficiently. Experimental results show that this approach is able to solve large problems, quickly returning stable matchings of large and often optimal size.Comment: 12 pages, Proc. PRICAI 2010 (11th Pacific Rim International Conference on Artificial Intelligence), Byoung-Tak Zhang and Mehmet A. Orgun eds., Springer LNA