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...

Popular Matchings with Ties and Matroid Constraints

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.

Rank-Maximal Matchings

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.

Classified Rank-Maximal Matchings and Popular Matchings – Algorithms and Hardness

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.

Bipartite Matchings with Group Fairness and Individual Fairness Constraints

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.

Optimal Matchings with One-Sided Preferences: Fixed and Cost-Based Quotas

In this paper, we consider the problem of computing an optimal matching in a bipartite graph where elements of one side of the bipartition specify preferences over the other side, and one or both sides can have capacities and classifications. The input instance is a bipartite graph G=(A U P,E), where A is a set of applicants, P is a set of posts, and each applicant ranks its neighbors in an order of preference, possibly involving ties. Moreover, each vertex v in A U P has a quota q(v) denoting the maximum number of partners it can have in any allocation of applicants to posts - referred to as a {\em matching} in this paper. A classification for a vertex u is a collection of subsets of neighbors of u. Each subset (class) has an upper quota denoting the maximum number of vertices from the class 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. 
We consider two well-studied notions of optimality namely popularity and rank-maximality. The notion of rank-maximality involves finding a matching in $G$ with maximum number of rank-$1$ edges, subject to that, maximum number of rank-2 edges and so on. We present an O(|E|^2)-time algorithm for finding a feasible rank-maximal matching, when each classification is a laminar family. We complement this with an NP-hardness result when classes are non-laminar even under strict preference lists, and even when only posts have classifications, and each applicant has a quota of one. We show an analogous dichotomy result for computing a popular matching amongst feasible matchings (if one exists) in a bipartite graph with posts having capacities and classifications and applicants having a quota of one.

Nada Pulath

Papers

Classified Rank-Maximal Matchings and Popular Matchings – Algorithms and Hardness

Dichotomy Results for Classified Rank-Maximal Matchings and Popular Matchings