To enjoy more social network services, users nowadays are usually involved in multiple online social networks simultaneously. The shared users between different networks are called anchor users, while the remaining unshared users are named as non-anchor users. Connections between accounts of anchor users in different networks are defined as anchor links and networks partially aligned by anchor links can be represented as partially aligned networks. In this paper, we want to predict anchor links between partially aligned social networks, which is formally defined as the partial network alignment problem. The partial network alignment problem is very difficult to solve because of the following two challenges: (1) the lack of general features for anchor links, and (2) the "one-to-one @math " (one to at most one) constraint on anchor links. To address these two challenges, a new method PNA (Partial Network Aligner) is proposed in this paper. PNA (1) extracts a set of explicit anchor adjacency features and latent topological features for anchor links based on the anchor meta path concept and tensor decomposition techniques, and (2) utilizes the generic stable matching to identify the non-anchor users to prune the redundant anchor links attached to them. Extensive experiments conducted on two real-world partially aligned social networks demonstrate that PNA can solve the partial network alignment problem very well and outperform all the other comparison methods with significant advantages.
College admission problem @cite_33 and stable marriage problem @cite_40 have been studied for many years and lots of works have been done in the last century. In recent years, some new papers have come out in these areas. @cite_43 propose to analyze the stability of the equilibrium outcomes in the admission games induced by stable matching rules. Ma @cite_52 analyzes the truncation in stable matching and the small core in nash equilibrium in college admission problems. Flor ' e @cite_23 propose to study the almost stable matching by truncating the Gale-Shapley algorithm.
@cite_33 @cite_52 @cite_43 @cite_40 @cite_23
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Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching. We study strategic issues in the Gale-Shapley stable marriage model. In the first part of the paper, we derive the optimal cheating strategy and show that it is not always possible for a woman to recover her women-optimal stable partner from the men-optimal stable matching mechanism when she can only cheat by permuting her preferences. In fact, we show, using simulation, that the chances that a woman can benefit from cheating are slim. In the second part of the paper, we consider a two-sided matching market found in Singapore. We study the matching mechanism used by the Ministry of Education (MOE) in the placement of primary six students in secondary schools, and discuss why the current method has limited success in accommodating the preferences of the students, and the specific needs of the schools (in terms of the “mix” of admitted students). Using insights from the first part of the paper, we show that stable matching mechanisms are more appropriate in this matching market and explain why the strategic behavior of the students need not be a major concern. (Stable Marriage; Strategic Issues; Gale-Shapley Algorithm; Student Posting Exercise) The original work of Gale and Shapley on an assignment method using the stable marriage criterion has been extended to find all the stable marriage assignments. The algorithm derived for finding all the stable marriage assignments is proved to satisfy all the conditions of the problem. Algorithm 411 applies to this paper. Abstract Two-sided matching markets of the kind known as the “college admissions problem” have been widely thought to be virtually equivalent to the simpler “marriage problem” for which some striking results concerning agents' preferences and incentives have been recently obtained. It is shown here that some of these results do not generalize to the college admissions problem, contrary to a number of assertions in the recent literature. No stable matching procedure exists that makes it a dominant strategy for colleges to reveal their true preferences, and some outcomes may be preferred by all colleges to the college-optimal stable outcome. A stable matching rule is used as the outcome function for the Admission game where colleges behave straightforwardly and the students’ strategies are given by their preferences over the colleges. We show that the college-optimal stable matching rule implements the set of stable matchings via the Nash equilibrium (NE) concept. For any other stable matching rule the strategic behavior of the students may lead to outcomes that are not stable under the true preferences. We then introduce uncertainty about the matching selected and prove that the natural solution concept is that of NE in the strong sense. A general result shows that the random stable matching rule, as well as any stable matching rule, implements the set of stable matchings via NE in the strong sense. Precise answers are given to the strategic questions raised. The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Two-sided matching: a study in game-theoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stableeven if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an egalitarian' and a minimum regret' stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes. The stable marriage problem is a well-known problem of matching men to women so that no man and woman who are not married to each other both prefer each other. Such a problem has a wide variety of practical applications ranging from matching resident doctors to hospitals to matching students to schools. A well-known algorithm to solve this problem is the Gale-Shapley algorithm, which runs in polynomial time. It has been proven that stable marriage procedures can always be manipulated. Whilst the Gale-Shapley algorithm is computationally easy to manipulate, we prove that there exist stable marriage procedures which are NP-hard to manipulate. We also consider the relationship between voting theory and stable marriage procedures, showing that voting rules which are NP-hard to manipulate can be used to define stable marriage procedures which are themselves NP-hard to manipulate. Finally, we consider the issue that stable marriage procedures like Gale-Shapley favour one gender over the other, and we show how to use voting rules to make any stable marriage procedure gender neutral. We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+e)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching. The stable marriage problem, i.e., the problem of assigning the members of two disjoint sets to one another is extended to the case in which individual preferences are represented by weak orderings instead of linear orderings. This leads to a more natural solution for such problems as the processing of college admissions and the optimal distribution of personnel. The paper begins with a formal definition of an assignments function and introduces conditions on it, among them stability. It is shown how the Gale and Shapley algorithm for finding stable assignments can be extended to the more general case considered here. It is shown that if there is an assignment which is preferred by a majority to all other assignments, then this assignment is necessarily stable. Moreover, in a situation where all preference orders are linear orders, all stable assignments are majority assignments. An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within @math , for any e>0, unless P=NP. If the preference lists contain ties, we improve this result to n1−e. Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an sr instance I. It is shown that Stable Matching problems are the same as problems about stable configurations of X-networks. Consequences include easy proofs of old theorems, a new simple algorithm for finding a stable matching, an understanding of the difference between Stable Marriage and Stable Roommates, NP-completeness of Three-party Stable Marriage, CC-completeness of several Stable Matching problems, and a fast parallel reduction from the Stable Marriage problem to the Assignment problem. The stable admissions polytope– the convex hull of the stable assignments of the university admissions problem – is described by a set of linear inequalities. It depends on a new characterization of stability and arguments that exploit and extend a graphical approach that has been fruitful in the analysis of the stable marriage problem. In an instance of the stable marriage problem with ties and incomplete preference lists, stable matchings can have different sizes. It is APX-hard to compute a maximum cardinality stable matching, but there have recently been proposed polynomial-time approximation algorithms, with constant performance guarantees for both the general version of this problem, and for several special cases. Our contribution is to describe a @math -approximation algorithm for the general version of this problem, improving upon the recent @math -approximation algorithm of Kiraly. Interest in such algorithms arises because of the problem's application to centralized matching schemes, the best known of which involve the assignment of graduating medical students to hospitals in various countries. The original stable marriage problem requires all men and women to submit a complete and strictly ordered preference list. This is obviously often unrealistic in practice, and several relaxations have been proposed, including the following two common ones: one is to allow an incomplete list, i.e., a man is permitted to accept only a subset of the women and vice versa. The other is to allow a preference list including ties. Fortunately, it is known that both relaxed problems can still be solved in polynomial time. In this paper, we show that the situation changes substantially if we allow both relaxations (incomplete lists and ties) at the same time: the problem not only becomes NP-hard, but also the optimal cost version has no approximation algorithm achieving the approximation ratio of N1-Ɛ, where N is the instance size, unless P=NP. This book probes the stable marriage problem and its variants as a rich source of problems and ideas that illustrate both the design and analysis of efficient algorithms. It covers the most recent structural and algorithmic work on stable matching problems, simplifies and unifies many earlier proofs, strengthens several earlier results, and presents new results and more efficient algorithms.The authors develop the structure of the set of stable matchings in the stable marriage problem in a more general and algebraic context than has been done previously; they discuss the problem's structure in terms of rings of sets, which allows many of the most useful features to be seen as features of a more general set of problems. The relationship between the structure of the stable marriage problem and the more general stable roommates problem is demonstrated, revealing many commonalities.The results the authors obtain provide an algorithmic response to the practical, and political, problems created by the asymmetry inherent in the Gale Shapley solutions, leading to alternative methods and better compromises than are provided by the Gale Shapley method. And, in contrast to Donald Knuth's earlier work which primarily focused on the application of mathematics to the analysis of algorithms, this book illustrates the productive and almost inseparable relationship between mathematical insight and the design of efficient algorithms.Dan Gusfield is Associate Professor of Computer Science at the University of California, Davis. Robert W. Irving is Senior Lecturer in Computing Science at the University of Glasgow. The Stable Marriage Problem is included in the Foundations of Computing Series, edited by Michael Garey and Albert Meyer. We present the first complete algorithm for the SMTI problem, the stable marriage problem with ties and incomplete lists. We do this in the form of a constraint programming encoding of the problem. With this we are able to carry out the first empirical study of the complete solution of SMTI instances. In the stable marriage problem (SM) [5] we have n men and n women. Each man ranks the women, giving himself a preference list. Similarly each woman ranks the men, giving herself a preference list. The problem is then to marry men and women such that they are stable i.e. such that there is no incentive for individuals to divorce and elope. This problem is polynomial time solvable. However, when preference lists contain ties and are incomplete (SMTI) the problem of determining if there is a stable matching of size n is then NP-complete, as is the optimisation problem of finding the largest or smallest stable matching [6, 10]. In this paper we present constraint programming solutions for the SMTI decision and optimisation problems, a problem generator for random instances of SMTI, and an empirical study of this problem. We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some @d>1 (even if each woman's list is of length at most 4). Couples looking for jobs in the same labor market may cause instabilities. We deter- mine a natural preference domain, the domain of weakly responsive preferences, that guarantees stability. Under a restricted unemployment aversion condition we show that this domain is maximal for the existence of stable matchings. We illustrate how small deviations from (weak) responsiveness, that model the wish of couples to be closer together, cause instability, even when we use a weaker stability notion that excludes myopic blocking. Our remaining results deal with various properties of the set of stable matchings for "responsive couples markets," viz., optimal- ity, filled positions, and manipulation. Journal of Economic Literature Classification Numbers: C78; J41. The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NP-hard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially we can obtain a 2-approximation algorithm. In this article, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2 (1 p L−2) for instances in which only men have ties of length at most L. When both men and women are allowed to have ties but the lengths are limited to two, then we show a ratio of 13 7( 1.1052). Baiou and Balinski characterized the stable admissions polytope using a system of linear inequalities. The structure of feasible solutions to this system of inequalities---fractional stable matchings---is the focus of this paper. The main result associates a geometric structure with each fractional stable matching. This insight appears to be interesting in its own right, and can be viewed as a generalization of the lattice structure (for integral stable matchings) to fractional stable matchings. In addition to obtaining simple proofs of many known results, the geometric structure is used to prove the following two results: First, it is shown that assigning each agent their “median” choice among all stable partners results in a stable matching, which can be viewed as a “fair” compromise; second, sufficient conditions are identified under which stable matchings exist in a problem with externalities, in particular, in the stable matching problem with couples.