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

Popular Matchings with Lower Quotas

We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph G = (R U H, E) where R and H denote sets of residents and hospitals, respectively. Every vertex has a preference list that imposes a strict ordering on its neighbors. In addition, each hospital has an associated upper-quota and a lower-quota. A matching M in G is an assignment of residents to hospitals, and M is said to be feasible if every resident is assigned to at most one hospital and a hospital is assigned at least its lower-quota many residents and at most its upper-quota many residents. Stability is a de-facto notion of optimality in a model where both sets of vertices have preferences. A matching is stable if no unassigned pair has an incentive to deviate from it. It is well-known that an instance of the HRLQ problem need not admit a feasible stable matching. In this paper, we consider the notion of popularity for the HRLQ problem. A matching M is popular if no other matching M\u27 gets more votes than M when vertices vote between M and M\u27. When there are no lower quotas, there always exists a stable matching and it is known that every stable matching is popular. We show that in an HRLQ instance, although a feasible stable matching need not exist, there is always a matching that is popular in the set of feasible matchings. We give an efficient algorithm to compute a maximum cardinality matching that is popular amongst all the feasible matchings in an HRLQ instance

Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas

We consider the hospital-residents problem where both hospitals and residents can have lower quotas. The input is a bipartite graph G = (???,E), each vertex in ??? has a strict preference ordering over its neighbors. The sets ? and ? denote the sets of residents and hospitals respectively. Each hospital has an upper and a lower quota denoting the maximum and minimum number of residents that can be assigned to it. Residents have upper quota equal to one, however, there may be a requirement that some residents must not be left unassigned in the output matching. We call this as the residents\u27 lower quota. We show that whenever the set of matchings satisfying all the lower and upper quotas is non-empty, there always exists a matching that is popular among the matchings in this set. We give a polynomial-time algorithm to compute such a matching

How Good Are Popular Matchings?

In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem with Lower Quotas (HRLQ). In this model with two sided preferences, stability is a well accepted notion of optimality. However, in the presence of lower quotas, a stable and feasible matching need not exist. For the HRLQ problem, our goal therefore is to output a good feasible matching assuming that a feasible matching exists. Computing matchings with minimum number of blocking pairs (Min-BP) and minimum number of blocking residents (Min-BR) are known to be NP-Complete. The only approximation algorithms for these problems work under severe restrictions on the preference lists. We present an algorithm which circumvents this restriction and computes a popular matching in the HRLQ instance. We show that on data-sets generated using various generators, our algorithm performs very well in terms of blocking pairs and blocking residents. Yokoi [Yokoi, 2017] recently studied envy-free matchings for the HRLQ problem. We propose a simple modification to Yokoi\u27s algorithm to output a maximal envy-free matching. We observe that popular matchings outperform envy-free matchings on several parameters of practical importance, like size, number of blocking pairs, number of blocking residents. In the absence of lower quotas, that is, in the Hospital Residents (HR) problem, stable matchings are guaranteed to exist. Even in this case, we show that popularity is a practical alternative to stability. For instance, on synthetic data-sets generated using a particular model, as well as on real world data-sets, a popular matching is on an average 8-10% larger in size, matches more number of residents to their top-choice, and more residents prefer the popular matching as compared to a stable matching. Our comprehensive study reveals the practical appeal of popular matchings for the HR and HRLQ problems. To the best of our knowledge, this is the first study on the empirical evaluation of popular matchings in this setting

Matching with Couples: a Multidisciplinary Survey

This survey deals with two-sided matching markets where one set of agents (workers/residents) has to be matched with another set of agents (firms/hospitals). We first give a short overview of a selection of classical results. Then, we review recent contributions to a complex and representative case of matching with complementarities, namely matching markets with couples. We discuss contributions from computer scientists, economists, and game theorists.matching; couples; stability; computational complexity; incentive compatibility; restricted domains; large markets

Investigating the temporal dynamics of inter-organizational exchange: patient transfers among Italian hospitals

Previous research on interaction behavior among organizations (resource exchange, collaboration, communication) has typically aggregated records of those behaviors over time to constitute a ‘network’ of organizational relationships. We instead directly study structural-temporal patterns in organizational exchange, focusing on the dynamics of reciprocation. Applying this lens to a community of Italian hospitals during the period 2003-2007, we observe two mechanisms of interorganizational reciprocation: organizational embedding and resource dependence. We flesh out these two mechanisms by showing how they operate in distinct time frames: Dependence operates on contemporaneous exchange structures, whereas embedding develops through longer-term historical patterns. We also show how these processes operate differently in competitive and noncompetitive contexts, operationalized in terms of market differentiation and geographic space. In noncompetitive contexts, we observe both logics of reciprocation, dependence in the short term and embedding over the long term, developing into patterns of generalized exchange in this population. In competitive contexts, we observe neither form of reciprocation and instead observe the microfoundations of status hierarchies in exchange

Understanding Entry-Level Health Care Employment in Chicago

Restructuring within the health care industry over the past ten years has sought to improve the quality and delivery of health care services while reducing employers' costs. However, coupled with the tight local and regional labor market, these organizational changes have introduced new challenges for employers trying to find workers who are adequately prepared to deliver quality health care services.Earlier this year, the Chicago Jobs Council received funding from the Richard H. Driehaus Foundation and BP Amoco to conduct research in the health care industry. This report summarizes the most recent industry and occupational data on Chicago's health care industry and provides a snapshot of the entry-level recruitment and retention needs of Chicago's health care employers, job training programs, and low-income job seekers. Based on our findings, the report also makes recommendations to employers, training providers, public agencies and workforce boards

Popular Edges with Critical Nodes

In the popular edge problem, the input is a bipartite graph G = (A ? B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A? B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M\u27 such that the number of vertices that prefer M\u27 to M is more than the number of vertices that prefer M to M\u27. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018]. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes

Treatment of Choice or A Last Resort? A Review of Residential Mental Health Placements For Children and Adolescents

Residential treatment is often regarded as a treatment of ‘last resort’ and, increasingly, residential treatment programs are being asked to address the needs of very troubled children and adolescents. This paper is an effort to summarize what is currently known about the effects of residential treatment for children and adolescents. The review is organized into two sections: studies of the effectiveness of group home residential treatment and studies of the effectiveness of residential treatment delivered in residential treatment centres. In both areas, we attempt to identify trends within treatment, as well as patterns found in the literature that characterize post residential treatment adaptation. We also discuss several additional factors that appear to share a relationship with residential treatment outcomes crossing both short-term and long-term trends. We conclude our review with suggestions for future directions in residential treatment for children and adolescents