Counting Houses of Pareto Optimal Matchings in the House Allocation Problem

Let $A,B$ with $|A| = m$ and $|B| = n\ge m$ be two sets. We assume that every element $a\in A$ has a reference list over all elements from $B$. We call an injective mapping $\tau$ from $A$ to $B$ a matching. A blocking coalition of $\tau$ is a subset $A'$ of $A$ such that there exists a matching $\tau'$ that differs from $\tau$ only on elements of $A'$, and every element of $A'$ improves in $\tau'$, compared to $\tau$ according to its preference list. If there exists no blocking coalition, we call the matching $\tau$ an exchange stable matching (ESM). An element $b\in B$ is reachable if there exists an exchange stable matching using $b$. The set of all reachable elements is denoted by $E^*$. We show $$|E^*| \leq \sum_{i = 1,\ldots, m}{\left\lfloor\frac{m}{i}\right\rfloor} = \Theta(m\log m).$$ This is asymptotically tight. A set $E\subseteq B$ is reachable (respectively exactly reachable) if there exists an exchange stable matching $\tau$ whose image contains $E$ as a subset (respectively equals $E$). We give bounds for the number of exactly reachable sets. We find that our results hold in the more general setting of multi-matchings, when each element $a$ of $A$ is matched with $\ell_a$ elements of $B$ instead of just one. Further, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable elements, i.e., elements of $B$ that are used by all ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise

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

Minimal Envy and Popular Matchings

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them

Popular matchings in the weighted capacitated house allocation problem

We consider the problem of finding a popular matching in the <i>Weighted Capacitated House Allocation</i> problem (WCHA). An instance of WCHA involves a set of agents and a set of houses. Each agent has a positive weight indicating his priority, and a preference list in which a subset of houses are ranked in strict order. Each house has a capacity that indicates the maximum number of agents who could be matched to it. A matching M of agents to houses is popular if there is no other matching M′ such that the total weight of the agents who prefer their allocation in M′ to that in M exceeds the total weight of the agents who prefer their allocation in M to that in M′. Here, we give an [FORMULA] algorithm to determine if an instance of WCHA admits a popular matching, and if so, to find a largest such matching, where C is the total capacity of the houses, n1 is the number of agents, and m is the total length of the agents' preference lists

Set systems related to a house allocation problem

We are given a set of buyers, a set of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping from to , and is strictly better than another house allocation if for every buyer , does not come before in the preference list of . A house allocation is Pareto optimal if there is no strictly better house allocation. Let be the image of i.e., the set of houses sold in the house allocation . We are interested in the largest possible cardinality of the family of sets for Pareto optimal mappings taken over all sets of preference lists of buyers and all sets of houses. This maximum exists since in a Pareto optimal mapping with buyers, each buyer will always be assigned one of their top choices. We improve the earlier upper bound on given by Asinowski et al. (2016), by making a connection between this problem and some problems in extremal set theory

Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.Comment: 31 pages, 6 figure

Minimum Partial-Matching and Hausdorff RMS-Distance under Translation: Combinatorics and Algorithms

We consider the RMS-distance (sum of squared distances between pairs of points) under translation between two point sets in the plane. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum in near-linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. In addition, we show how to compute a local minimum of the partial-matching RMS-distance under translation, in polynomial time

Construir el diálogo científico en la Matemática: la búsqueda del equilibrio entre símbolos y palabras en artículos de investigación sobre Teoría de Juegos

Maestría en Inglés con Orientación en Lingüística AplicadaMost scientific communication is conducted in English, which may be a difficult task and a source of obstacles for researchers whose primary language is not English (Bitchenera & Basturkmen, 2006; Borlogan, 2009; Duff, 2010; Matsuda & Matsuda, 2010). As a matter of concern for language scholars, this situation requires at least two actions: (1) the development of research focused on the problems faced by researchers when writing in a foreign language, and (2) the design and implementation of pedagogical and didactic programmes or services aimed at providing researchers with the tools to enhance their linguistic and rhetorical skills. In both cases, the ultimate objective of these lines of action is to help researchers integrate into and interact with their knowledge communities in an independent, active and successful way. Considering those needs and the emerging interest in English as a lingua franca or as an international language, many scholars have devoted to studying the features of writing and language use across the world and across disciplines (Hyland, 2004; Matsuda & Matsuda, 2010; Mercado, 2010). However, few have explored the case of Mathematics (Lemke, 2002; Morgan, 2008; O’Halloran, 2005; Schleppegrell, 2007), and even fewer have investigated the discourse of scientific research articles (SRA) in this discipline (Graves & Moghadassi, 2013, 2014). In view of this situation, investigation of the discourse of science in the field of Mathematics (Game Theory - GT) as used in the Institute of Applied Mathematics (IMASL), at the National University of San Luis (UNSL), becomes both an answer to local researchers’ needs and an attempt to contribute to current research in writing, evaluative discourse and use of English as an international language for the communication of science. Thus, the main objective of this work is to conduct a comparative description between unpublished GT SRAs written in English by IMASL researchers and published GT SRAs written in English by international authors, in terms of linguistic features used to build authorship and authorial stance. The exploration of the genre is made from the perspective of the system of Appraisal (Hood, 2010; Martin & White, 2005; White, 2000), with the aid of Corpus Linguistics (CL) tools (Cheng, 2012; Meyer, 2002; Tognini-Bonelli, 2001). The results of this research are expected to be useful for the enhancement of knowledge of language professionals devoted to the teaching of writing as well as translation, proofreading, editing and reviewing services. A further goal is to lay the foundations for the production of didactic material which can potentially be incorporated into writing courses or professional writing, translation, reviewing and proofreading training programmes.Fil: Lucero Arrua, Graciela Beatriz. Universidad Nacional de Córdoba. Facultad de Lenguas; Argentina