Classified Stable Matching

We introduce the {\sc classified stable matching} problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of ``order types.'' We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality allows us to find various optimal stable matchings using Ellipsoid algorithm. A further ramification of our result is the description of the stable matching polytope for the many-to-many (unclassified) stable matching problem. This answers an open question posed by Sethuraman, Teo and Qian

Envy-free Matchings with Lower Quotas

While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomial time if quotas are paramodular

Stable Roommate Problem with Diversity Preferences

In the multidimensional stable roommate problem, agents have to be allocated to rooms and have preferences over sets of potential roommates. We study the complexity of finding good allocations of agents to rooms under the assumption that agents have diversity preferences [Bredereck et al., 2019]: each agent belongs to one of the two types (e.g., juniors and seniors, artists and engineers), and agents' preferences over rooms depend solely on the fraction of agents of their own type among their potential roommates. We consider various solution concepts for this setting, such as core and exchange stability, Pareto optimality and envy-freeness. On the negative side, we prove that envy-free, core stable or (strongly) exchange stable outcomes may fail to exist and that the associated decision problems are NP-complete. On the positive side, we show that these problems are in FPT with respect to the room size, which is not the case for the general stable roommate problem. Moreover, for the classic setting with rooms of size two, we present a linear-time algorithm that computes an outcome that is core and exchange stable as well as Pareto optimal. Many of our results for the stable roommate problem extend to the stable marriage problem.Comment: accepted to IJCAI'2

An Approximation Algorithm for Maximum Stable Matching with Ties and Constraints

We present a polynomial-time 3/2-approximation algorithm for the problem of finding a maximum-cardinality stable matching in a many-to-many matching model with ties and laminar constraints on both sides. We formulate our problem using a bipartite multigraph whose vertices are called workers and firms, and edges are called contracts. Our algorithm is described as the computation of a stable matching in an auxiliary instance, in which each contract is replaced with three of its copies and all agents have strict preferences on the copied contracts. The construction of this auxiliary instance is symmetric for the two sides, which facilitates a simple symmetric analysis. We use the notion of matroid-kernel for computation in the auxiliary instance and exploit the base-orderability of laminar matroids to show the approximation ratio. In a special case in which each worker is assigned at most one contract and each firm has a strict preference, our algorithm defines a 3/2-approximation mechanism that is strategy-proof for workers

Popular Matchings with Multiple Partners

Our input is a bipartite graph G=(Acup B,E) where each vertex in Acup B has a preference list strictly ranking its neighbors. The vertices in A and in B are called students and courses, respectively. Each student a seeks to be matched to cap(a)geq 1 many courses while each course b seeks cap(b)geq 1 many students to be matched to it. The Gale-Shapley algorithm computes a pairwise-stable matching (one with no blocking edge) in G in linear time. We consider the problem of computing a popular matching in G - a matching M is popular if M cannot lose an election to any matching where vertices cast votes for one matching versus another. Our main contribution is to show that a max-size popular matching in G can be computed by the 2-level Gale-Shapley algorithm in linear time. This is an extension of the classical Gale-Shapley algorithm and we prove its correctness via linear programming

Envy-freeness and Relaxed Stability for Lower-Quotas : A Parameterized Perspective

We consider the problem of assigning agents to resources under the two-sided preference list model with upper and lower-quotas on resources. Krishnaa et al. [17] explore two optimality notions for this setting -- envy-freeness and relaxed stability. They investigate the problem of computing a maximum size envy-free matching (MAXEFM) and a maximum size relaxed stable matching (MAXRSM) that satisfies the lower-quotas. They show that both these optimization problems cannot be approximated within a constant factor unless P = NP. In this work, we investigate parameterized complexity of MAXEFM and MAXRSM. We consider several parameters derived from the instance -- the number of resources with non-zero lower-quota, deficiency of the instance, maximum length of the preference list of a resource with non-zero lower-quota, among others. We show that MAXEFM problem is W [1]-hard for several interesting parameters and MAXRSM problem is para-NP-hard for two natural parameters. We present kernelization results and FPT algorithms on a combination of parameters for both problems.Comment: 14 pages, 2 figures. fullpage used, improved presentation of results, stronger kernelization result for MAXRS