Locally Stable Marriage with Strict Preferences

We study stable matching problems with locality of information and control. In our model, each agent is a node in a fixed network and strives to be matched to another agent. An agent has a complete preference list over all other agents it can be matched with. Agents can match arbitrarily, and they learn about possible partners dynamically based on their current neighborhood. We consider convergence of dynamics to locally stable matchings -- states that are stable with respect to their imposed information structure in the network. In the two-sided case of stable marriage in which existence is guaranteed, we show that the existence of a path to stability becomes NP-hard to decide. This holds even when the network exists only among one partition of agents. In contrast, if one partition has no network and agents remember a previous match every round, a path to stability is guaranteed and random dynamics converge with probability 1. We characterize this positive result in various ways. For instance, it holds for random memory and for cache memory with the most recent partner, but not for cache memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we present results for centralized computation of locally stable matchings, i.e., computing maximum locally stable matchings in the two-sided case and deciding existence in the roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat

Socially stable matchings in the hospitals / residents problem

In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings. In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem

Matching Dynamics with Constraints

We study uncoordinated matching markets with additional local constraints that capture, e.g., restricted information, visibility, or externalities in markets. Each agent is a node in a fixed matching network and strives to be matched to another agent. Each agent has a complete preference list over all other agents it can be matched with. However, depending on the constraints and the current state of the game, not all possible partners are available for matching at all times. For correlated preferences, we propose and study a general class of hedonic coalition formation games that we call coalition formation games with constraints. This class includes and extends many recently studied variants of stable matching, such as locally stable matching, socially stable matching, or friendship matching. Perhaps surprisingly, we show that all these variants are encompassed in a class of "consistent" instances that always allow a polynomial improvement sequence to a stable state. In addition, we show that for consistent instances there always exists a polynomial sequence to every reachable state. Our characterization is tight in the sense that we provide exponential lower bounds when each of the requirements for consistency is violated. We also analyze matching with uncorrelated preferences, where we obtain a larger variety of results. While socially stable matching always allows a polynomial sequence to a stable state, for other classes different additional assumptions are sufficient to guarantee the same results. For the problem of reaching a given stable state, we show NP-hardness in almost all considered classes of matching games.Comment: Conference Version in WINE 201

MMB & DFT 2014 : Proceedings of the International Workshops ; Modeling, Analysis and Management of Social Networks and their Applications (SOCNET 2014) & Demand Modeling and Quantitative Analysis of Future Generation Energy Networks and Energy-Efficient Systems (FGENET 2014)

At present, a comprehensive set of measurement, modeling, analysis, simulation, and performance evaluation techniques are employed to investigate complex networks. A direct transfer of the developed engineering methodologies to related analysis and design tasks in next-generation energy networks, energy-efficient systems and social networks is enabled by a common mathematical foundation. The International Workshop on "Demand Modeling and Quantitative Analysis of Future Generation Energy Networks and Energy-Efficient Systems" (FGENET 2014) and the International Workshop on "Modeling, Analysis and Management of Social Networks and their Applications" (SOCNET 2014) were held on March 19, 2014, at University of Bamberg in Germany as satellite symposia of the 17th International GI/ITG Conference on "Measurement, Modelling and Evaluation of Computing Systems" and "Dependability and Fault-Tolerance" (MMB & DFT 2014). They dealt with current research issues in next-generation energy networks, smart grid communication architectures, energy-efficient systems, social networks and social media. The Proceedings of MMB & DFT 2014 International Workshops summarizes the contributions of 3 invited talks and 13 reviewed papers and intends to stimulate the readers’ future research in these vital areas of modern information societies.Gegenwärtig wird eine reichhaltige Klasse von Verfahren zur Messung, Modellierung, Analyse, Simulation und Leistungsbewertung komplexer Netze eingesetzt. Die unmittelbare Übertragung entwickelter Ingenieurmethoden auf verwandte Analyse- und Entwurfsaufgaben in Energienetzen der nächsten Generation, energieeffizienten Systemen und sozialen Netzwerken wird durch eine gemeinsame mathematische Basis ermöglicht. Die Internationalen Workshops "Demand Modeling and Quantitative Analysis of Future Generation Energy Net-works and Energy-Efficient Systems" (FGENET 2014) und "Modeling, Analysis and Management of Social Networks and their Applications" (SOCNET 2014) wurden am 19. März 2014 als angegliederte Symposien der 17. Internationalen GI/ITG Konferenz "Measurement, Modelling and Evaluation of Computing Systems" und "Dependability and Fault-Tolerance" (MMB & DFT 2014) an der Otto-Friedrich-Universität Bamberg in Deutschland veranstaltet. Es wurden aktuelle Forschungsfragen in Energienetzen der nächsten Generation, Smart Grid Kommunikationsarchitekturen, energieeffizienten Systemen, sozialen Netzwerken und sozialen Medien diskutiert. Der Tagungsband der Internationalen Workshops MMB & DFT 2014 fasst die Inhalte von 3 eingeladenen Vorträgen und 13 begutachteten Beiträgen zusammen und beabsichtigt, den Lesern Anregungen für ihre eigenen Forschungen auf diesen lebenswichtigen Gebieten moderner Informationsgesellschaften zu vermitteln

Efficient algorithms for optimal matching problems under preferences

In this thesis we consider efficient algorithms for matching problems involving preferences, i.e., problems where agents may be required to list other agents that they find acceptable in order of preference. In particular we mainly study the Stable Marriage problem (SM), the Hospitals / Residents problem (HR) and the Student / Project Allocation problem (SPA), and some of their variants. In some of these problems the aim is to find a stable matching which is one that admits no blocking pair. A blocking pair with respect to a matching is a pair of agents that prefer to be matched to each other than their assigned partners in the matching if any. We present an Integer Programming (IP) model for the Hospitals / Residents problem with Ties (HRT) and use it to find a maximum cardinality stable matching. We also present results from an empirical evaluation of our model which show it to be scalable with respect to real-world HRT instance sizes. Motivated by the observation that not all blocking pairs that exist in theory will lead to a matching being undermined in practice, we investigate a relaxed stability criterion called social stability where only pairs of agents with a social relationship have the ability to undermine a matching. This stability concept is studied in instances of the Stable Marriage problem with Incomplete lists (smi) and in instances of hr. We show that, in the smi and hr contexts, socially stable matchings can be of varying sizes and the problem of finding a maximum socially stable matching (max smiss and max hrss respectively) is NP-hard though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem arising from restrictions on the social network graph and the lengths of agents’ preference lists. We also consider other optimality criteria with respect to social stability and establish inapproximability bounds for the problems of finding an egalitarian, minimum regret and sex equal socially stable matching in the sm context. We extend our study of social stability by considering other variants and restrictions of max smiss and max hrss. We present NP-hardness results for max smiss even under certain restrictions on the degree and structure of the social network graph as well as the presence of master lists. Other NP-hardness results presented relate to the problem of determining whether a given man-woman pair belongs to a socially stable matching and the problem of determining whether a given man (or woman) is part of at least one socially stable matching. We also consider the Stable Roommates problem with Incomplete lists under Social Stability (a non-bipartite generalisation of smi under social stability). We observe that the problem of finding a maximum socially stable matching in this context is also NP-hard. We present efficient algorithms for three special cases of the problem arising from restrictions on the social network graph and the lengths of agents’ preference lists. These are the cases where (i) there exists a constant number of acquainted pairs (ii) or a constant number of unacquainted pairs or (iii) each preference list is of length at most 2. We also present algorithmic results for finding matchings in the spa context that are optimal with respect to profile, which is the vector whose ith component is the number of students assigned to their ith-choice project. We present an efficient algorithm for finding a greedy maximum matching in the spa context — this is a maximum matching whose profile is lexicographically maximum. We then show how to adapt this algorithm to find a generous maximum matching — this is a matching whose reverse profile is lexicographically minimum. We demonstrate how this approach can allow additional constraints, such as lecturer lower quotas, to be handled flexibly. We also present results of empirical evaluations carried out on both real world and randomly generated datasets. These results demonstrate the scalability of our algorithms as well as some interesting properties of these profile-based optimality criteria. Practical applications of spa motivate the investigation of certain special cases of the problem. For instance, it is often desired that the workload on lecturers is evenly distributed (i.e. load balanced). We enforce this by either adding lower quota constraints on the lecturers (which leads to the potential for infeasible problem instances) or adding a load balancing optimisation criterion. We present efficient algorithms in both cases. Another consideration is the fact that certain projects may require a minimum number of students to become viable. This can be handled by enforcing lower quota constraints on the projects (which also leads to the possibility of infeasible problem instances). A technique of handling this infeasibility is the idea of closing projects that do not meet their lower quotas (i.e. leaving such project completely unassigned). We show that the problem of finding a maximum matching subject to project lower quotas where projects can be closed is NP-hard even under severe restrictions on preference lists lengths and project upper and lower quotas. To offset this hardness, we present polynomial time heuristics that find large feasible matchings in practice. We also present ip models for the spa variants discussed and show results obtained from an empirical evaluation carried out on both real and randomly generated datasets. These results show that our algorithms and heuristics are scalable and provide good matchings with respect to profile-based optimalit

Locally Stable Marriage with Strict Preferences

Abstract. We study two-sided matching markets with locality of in-formation and control. Each male (female) agent has an arbitrary strict preference list over all female (male) agents. In addition, each agent is a node in a fixed network. Agents learn about possible partners dynam-ically based on their current network neighborhood. We consider con-vergence of dynamics to locally stable matchings that are stable with respect to their imposed information structure in the network. While ex-istence of such states is guaranteed, we show that reachability becomes NP-hard to decide. This holds even when the network exists only among one side. In contrast, if only one side has no network and agents re-member a previous match every round, reachability is guaranteed and random dynamics converge with probability 1. We characterize this pos-itive result in various ways. For instance, it holds for random memory and for memory with the most recent partner, but not for memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we conclude with results on approximating maximum locally stable matchings.