Classified Rank-Maximal Matchings and Popular Matchings – Algorithms and Hardness
19 Jun 2019-pp 244-257
TL;DR: This paper considers the problem of computing an optimal matching in a bipartite graph where elements of A specify preferences over their neighbors in P, possibly involving ties, and each vertex can have capacities and classifications.
Abstract: In this paper, we consider the problem of computing an optimal matching in a bipartite graph \(G=(A\cup P, E)\) where elements of A specify preferences over their neighbors in P, possibly involving ties, and each vertex can have capacities and classifications. A classification \(\mathcal {C}_u\) for a vertex u is a collection of subsets of neighbors of u. Each subset (class) \(C\in \mathcal {C}_u\) has an upper quota denoting the maximum number of vertices from C that can be matched to u. The goal is to find a matching that is optimal amongst all the feasible matchings, which are matchings that respect quotas of all the vertices and classes.
Citations
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TL;DR: In this paper, a set of applicants and posts such that each applicant has a preference list over the posts is given, and a matching $M$ between the applicants and the posted posts is said to be a popu...
Abstract: Assume that we are given a set of applicants and a set of posts such that each applicant has a preference list over the posts. A matching $M$ between the applicants and the posts is said to be popu...
11 citations
01 Jan 2004
TL;DR: An algorithm is given to compute a rank-maximal matching with running time O(min(n + C,C &sqrt;n)m), where n is the number of applicants and posts and m is the total size of the preference lists.
Abstract: Suppose that each member of a set A of applicants ranks a subset of a set P of posts in an order of preference, possibly involving ties. A matching is a set of (applicant, post) pairs such that each applicant and each post appears in at most one pair. A rank-maximal matching is one in which the maximum possible number of applicants are matched to their first choice post, and subject to that condition, the maximum possible number are matched to their second choice post, and so on. This is a relevant concept in any practical matching situation and it was first studied by Irving [2003].We give an algorithm to compute a rank-maximal matching with running time O(min(n + C,C &sqrt;n)m), where C is the maximal rank of an edge used in a rank-maximal matching, n is the number of applicants and posts and m is the total size of the preference lists.
8 citations
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TL;DR: This work addresses group as well as individual fairness constraints in matchings in the context of assigning items to platforms by providing a polynomial-time algorithm that computes a probabilistic individually fair distribution over group fair matchings.
Abstract: We address group as well as individual fairness constraints in matchings in the context of assigning items to platforms. Each item belongs to certain groups and has a preference ordering over platforms. Each platform enforces group fairness by specifying an upper and a lower bound on the number of items that can be matched to it from each group. There could be multiple optimal solutions that satisfy the group fairness constraints. To achieve individual fairness, we introduce ‘probabilistic individual fairness’, where the goal is to compute a distribution over ‘group fair’ matchings such that every item has a reasonable probability of being matched to a platform among its top choices. In the case where each item belongs to exactly one group, we provide a polynomial-time algorithm that computes a probabilistic individually fair distribution over group fair matchings. When an item can belong to multiple groups, and the group fairness constraints are specified as only upper bounds, we rehash the same algorithm to achieve three different polynomial-time approximation algorithms.
2 citations
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TL;DR: This work considers the well-studied many-to-one bipartite matching problem of assigning applicants A to posts P where applicants rank posts in the order of preference and proposes a novel optimality criterion, which is called the “cumulative better signature”.
Abstract: We consider the well-studied many-to-one bipartite matching problem of assigning applicants A to posts P where applicants rank posts in the order of preference. This setting models many important real-world allocation problems like assigning students to courses, applicants to jobs, amongst many others. In such scenarios, it is natural to ask for an allocation that satisfies guarantees of the form “match at least 80% of applicants to one of their top three choices” or “it is unacceptable to leave more than 10% of applicants unassigned”. The well-studied notions of rank-maximality and fairness fail to capture such requirements due to their property of optimizing extreme ends of the signature of a matching. We, therefore, propose a novel optimality criterion, which we call as the “cumulative better signature”. We investigate the computational complexity of the new notion of optimality in the setting where posts have associated fixed quotas. We prove that under the fixed quota setting, the problem turns out to be NP-hard under natural restrictions. We provide randomized algorithms in the fixed quota setting when the number of ranks is constant. We also study the problem under a cost-based quota setting and show that min-cost cumulative better matching can be computed efficiently. Apart from circumventing the hardness, the cost-based quota setting is motivated by real-world applications like course allocation or school choice where the capacities or quotas need not be rigid.
1 citations
21 Aug 2022
TL;DR: In this paper , the authors consider the problem of assigning items to platforms while satisfying group and individual fairness constraints, and provide a polynomial-time algorithm that computes a probabilistic individually fair distribution over group fair matching.
Abstract: We consider the problem of assigning items to platforms while satisfying group and individual fairness constraints. Each item is associated with certain groups and has a preference ordering over platforms. Each platform enforces group fairness by specifying an upper and a lower bound on the number of items that can be matched to it from each group. Although there may be multiple optimal solutions that satisfy the group fairness constraints, we aim to achieve `probabilistic individual fairness' by computing a distribution over `group fair' matchings such that each item has a reasonable probability of being matched to one of its top choices. When each item can belong to multiple groups, the problem of finding a maximum size group-fair matching is NP-hard even when all the group lower bounds are 0, and there are no individual fairness constraints. Given a total of $n$ items, we achieve a $O(\Delta \log n)$ approximation algorithm when an item can belong to at most $\Delta$ groups, and all the group lower bounds are 0. We also provide two approximation algorithms in terms of the total number of groups that have items in the neighborhood of a platform. When each item belongs to a single group, we provide a polynomial-time algorithm that computes a probabilistic individually fair distribution over group fair matching. We further extend our model and algorithms to address the following notions of fairness: `maxmin group fairness', which maximizes the representation of the worst-off groups, and `mindom group fairness', which minimizes the representation of the most dominant groups.
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TL;DR: In this article, the stable marriage problem is extended to the case in which individual preferences are represented by weak orderings instead of linear orderings, which leads to a more natural solution for such problems as the processing of college admissions and the optimal distribution of personnel.
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179 citations