We analyze the (parameterized) computational complexity of "fair" variants of bipartite many-to-one matching, where each vertex from the "left" side is matched to exactly one vertex and each vertex...
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We analyze the (parameterized) computational complexity of "fair" variants of bipartite many-to-one matching, where each vertex from the "left" side is matched to exactly one vertex and each vertex from the "right" side may be matched to multiple vertices. We want to find a "fair" matching, in which each vertex from the right side is matched to a "fair" set of vertices. Assuming that each vertex from the left side has one color modeling its "attribute", we study two fairness criteria. For instance, in one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is, for instance, relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively. We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. We establish the correctness of our integer programs, based on Frank's separation theorem [Frank, Discrete Math. 1982]. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.
Stochastic majorization is a tool that has been used in many areas of probability and statistics (such as multivariate statistical analysis, queueing theory and reliability theory) in order to obtain useful bounds and...
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Stochastic majorization is a tool that has been used in many areas of probability and statistics (such as multivariate statistical analysis, queueing theory and reliability theory) in order to obtain useful bounds and inequalities. In this paper we study the relations among several notions of stochastic majorization and stochastic convexity and obtain sufficient (and sometimes necessary) conditions which imply some of these notions. Extensions and generalizations of several results in the literature are obtained. Some examples and applications regarding stochastic comparisons of order statistics are also presented in order to illustrate the results of the paper.
Many scenarios where agents with preferences compete for resources can be cast as maximum matching problems on bipartite graphs. Our focus is on resource allocation problems where agents may have preferences that make...
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ISBN:
(纸本)9781450392136
Many scenarios where agents with preferences compete for resources can be cast as maximum matching problems on bipartite graphs. Our focus is on resource allocation problems where agents may have preferences that make them incompatible with some resources. We assume that a Principal chooses a maximum matching randomly so that each agent is matched to a resource with some probability. Agents would like to improve their chances of being matched by modifying their preferences within certain limits. The Principal's goal is to advise an unsatisfied agent to relax its restrictions so that the total cost of relaxation is within a budget (chosen by the agent) and the increase in the probability of being assigned a resource is maximized. We develop efficient algorithms for some variants of this budget-constrained maximization problem and establish hardness results for other variants. For the latter variants, we also develop algorithms with performance guarantees. We experimentally evaluate our methods on synthetic datasets as well as on two novel real-world datasets: a vacation activities dataset and a classrooms dataset.
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