# Position-Based Matching with Multi-Modal Preferences

pp 1373-1381

TL;DR: In this article , the authors introduce three position-based matching models, which minimize the "dissatisfaction score", which measures matchings from different perspectives, and present diverse complexity results for these three models, among others, polynomial-time solvability for the first model.

Abstract: Many models have been proposed for computing a one-to-one matching between two equal-sized sets/sides of agents, each assigned with one preference list of the agents in the opposite side. The most prominent one might be the Stable Matching model. Re-cently, the Stable Matching model has been extended to the multimodal setting [6, 13, 29], where each agent has more than one preference lists, each representing a criterion based on which the agents of the opposite side are evaluated. We use a layer to denote the set of preference lists of agents, which are based on the same criterion. Thus, the single modal matching problem has only one layer. This setting finds applications in many real-world scenarios. However, it turns out that stable matchings might not exist with multi-modal preferences and the determination is NP-hard and W-hard with respect to several natural parameters. Here, we introduce three position-based matching models, which minimize the “dissatisfaction score”. We define four dissatisfaction scores, which measure matchings from different perspectives. The first model minimizes the total respective dissatisfaction score over all layers, while the second minimizes the maximum of the respective score over all layers. The third model seeks for a matching 𝑀 which is Layer Pareto-optimal, meaning that there does not exist a matching 𝑀 ′ , which is at least as good as 𝑀 with respect to the respective dissatisfaction score in all layers, but is strictly better in at least one layer. We present diverse complexity results for these three models, among others, polynomial-time solvability for the first model. We also investigate the generalization which given an upper bound on the dissatisfaction score, computes a matching involving subsets of agents and a subset of layers. Hereby, we mainly focus on the parameterized complexity with respect to parameters such as the size of agent subsets, or the size of the layer subset and achieve fixed-parameter tractability as well as intractability results.

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16 May 2022

TL;DR: In this paper , the complexity of stable matching problems with multilayer approval preferences was studied and eleven stability notions derived from three well-established stability notions for stable matchings with ties and four adaptions proposed by Chen et al.

Abstract: We study stable matching problems where agents have multilayer preferences: There are ℓ layers each consisting of one preference relation for each agent. Recently, Chen et al. [EC ’18] studied such problems with strict preferences, establishing four multilayer adaptions of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptions proposed by Chen et al. For each stability notion, we show that the problem of ﬁnding a stable matching is either polynomial-time solvable or NP-hard. Furthermore, we examine the inﬂuence of the number of layers and the desired “degree of stability” on the problems’ complexity. Somewhat surprisingly, we discover that assuming approval preferences instead of strict preferences does not consider-ably simplify the situation (and sometimes even makes polynomial-time solvable problems NP-hard).

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TL;DR: In this paper , the authors considered a new model of election control that by assigning different rules to the votes from different layers, makes the special candidate p being the winner of the election (a rule can be assigned to different layers).

Abstract: We study the election control problem with multi-votes, where each voter can present a single vote according different views (or layers, we use"layer"to represent"view"). For example, according to the attributes of candidates, such as: education, hobby or the relationship of candidates, a voter may present different preferences for the same candidate set. Here, we consider a new model of election control that by assigning different rules to the votes from different layers, makes the special candidate p being the winner of the election (a rule can be assigned to different layers). Assuming a set of candidates C among a special candidate"p", a set of voters V, and t layers, each voter gives t votes over all candidates, one for each layer, a set of voting rules R, the task is to find an assignment of rules to each layer that p is acceptable for voters (possible winner of the election). Three models are considered (denoted as sum-model, max-model, and min-model) to measure the satisfaction of each voter. In this paper, we analyze the computational complexity of finding such a rule assignment, including classical complexity and parameterized complexity. It is interesting to find out that 1) it is NP-hard even if there are only two voters in the sum-model, or there are only two rules in sum-model and max-model; 2) it is intractable with the number of layers as parameter for all of three models; 3) even the satisfaction of each vote is set as dichotomous, 1 or 0, it remains hard to find out an acceptable rule assignment. Furthermore, we also get some other intractable and tractable results.

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TL;DR: This paper has always been one of my favorite children, combining as it does elements of the duality of linear programming and combinatorial tools from graph theory, and it may be of some interest to tell the story of its origin this article.

Abstract: This paper has always been one of my favorite “children,” combining as it does elements of the duality of linear programming and combinatorial tools from graph theory. It may be of some interest to tell the story of its origin.

11,096 citations

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RAND Corporation

^{1}TL;DR: In this article, the authors studied the relationship between college admission and the stability of marriage in the United States, and found that college admission is correlated with the number of stable marriages.

Abstract: (2013). College Admissions and the Stability of Marriage. The American Mathematical Monthly: Vol. 120, No. 5, pp. 386-391.

5,655 citations

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TL;DR: In this paper, algorithms for the solution of the general assignment and transportation problems are presen, and the algorithm is generalized to one for the transportation problem.

Abstract: In this paper we presen algorithms for the solution of the general assignment and transportation problems. In Section 1, a statement of the algorithm for the assignment problem appears, along with a proof for the correctness of the algorithm. The remarks which constitute the proof are incorporated parenthetically into the statement of the algorithm. Following this appears a discussion of certain theoretical aspects of the problem. In Section 2, the algorithm is generalized to one for the transportation problem. The algorithm of that section is stated as concisely as possible, with theoretical remarks omitted.

3,918 citations

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TL;DR: Random serial dictatorship and the core from random endowments in house allocation problems as mentioned in this paper were used to solve the problem of house allocation in a house allocation problem in the 1990s.

Abstract: Random serial dictatorship and the core from random endowments in house allocation problems

604 citations

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TL;DR: The Boston Public Schools (BPS) system for assigning students to schools is described in this paper, where the authors describe some of the difficulties with the current assignment mechanism and some elements of the design and evaluation of possible replacement mechanisms.

Abstract: After the publication of “School Choice: A Mechanism Design Approach” by Abdulkadiroglu and Sonmez (2003), a Boston Globe reporter contacted us about the Boston Public Schools (BPS) system for assigning students to schools. The Globe article highlighted the difficulties that Boston’s system may give parents in strategizing about applying to schools. Briefly, Boston tries to give students their firstchoice school. But a student who fails to get her first choice may find her later choices filled by students who chose them first. So there is a risk in ranking a school first if there is a chance of not being admitted; other schools that would have been possible had they been listed first may also be filled. Valerie Edwards, then Strategic Planning Manager at BPS, and her colleague Carleton Jones invited us to a meeting in October 2003. BPS agreed to a study of their assignment system and provided us with micro-level data sets on choices and characteristics of students in the grades at which school choices are made (K, 1, 6, and 9), and school characteristics. Based on the pending results of this study, the Superintendent has asked for our advice on the design of a new assignment mechanism. This paper describes some of the difficulties with the current mechanism and some elements of the design and evaluation of possible replacement mechanisms. School choice in Boston has been partly shaped by desegregation. In 1974, Judge W. Arthur Garrity ordered busing for racial balance. In 1987, the U.S. Court of Appeals freed BPS to adopt a new, choice-based assignment plan. In 1999 BPS eliminated racial preferences in assignment and adopted the current mechanism.

580 citations