
Manipulation Strategies for the Rank Maximal Matching Problem
We consider manipulation strategies for the rankmaximal matching proble...
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AUPCR Maximizing Matchings : Towards a Pragmatic Notion of Optimality for OneSided Preference Matchings
We consider the problem of computing a matching in a bipartite graph in ...
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Matchings and Copeland's Method
Given a graph G = (V,E) where every vertex has weak preferences over its...
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NC Algorithms for Popular Matchings in OneSided Preference Systems and Related Problems
The popular matching problem is of matching a set of applicants to a set...
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Polynomially tractable cases in the popular roommates problem
The input of the popular roommates problem consists of a graph G = (V, E...
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Bipartite EnvyFree Matching
Bipartite EnvyFree Matching (BEFM) is a relaxation of perfect matching....
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Multitasking Capacity: Hardness Results and Improved Constructions
We consider the problem of determining the maximal α∈ (0,1] such that ev...
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Dichotomy Results for Classified RankMaximal Matchings and Popular Matchings
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 belonging to 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 matching in this paper. A classification C_u for a vertex u is a collection of subsets of neighbors of u. Each subset (class) C∈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. We consider two wellstudied notions of optimality namely popularity and rankmaximality. We present an O(E^2)time algorithm for finding a feasible rankmaximal matching, when each classification is a laminar family. We complement this with an NPhardness result when classes are nonlaminar 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) where applicants having a quota of one. Enroute to designing the polynomial time algorithms, we adapt the wellknown Dulmage Mendelsohn decomposition of a bipartite graph w.r.t. a maximum matching to a maximum flow on special flow networks. We believe this generalization is of independent interest.
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