The Hospital Residents problem with sizes (HRS) is a generalisation of the well-studied hospital residents (HR) problem. In the HRS problem, an agent a has a size s(a) and the agent occupies s(a) many positions of the hospital h when assigned to h. The notion of stability in this setting is suitably modified, and it is known that deciding whether an HRS instance admits a stable matching is NP-hard under severe restrictions. In this work, we explore a variation of stability, which we term occupancy-based stability. This notion was defined by McDermid and Manlove (J. of Comb. Opt. 2010) but remained unexplored to the best of our knowledge. In our work, we show that every HRS instance admits an occupancy-stable matching. We further show that computing a maximum-size occupancy-stable matching is NP-hard. We complement our hardness result by providing an approximation algorithm with a guarantee strictly better than 3 for the max-size occupancy-stable matching problem. Given that the classical notion of stability adapted for HRS is not guaranteed to exist in general, we show a practical restriction under which a stable matching is guaranteed to exist. We present an efficient algorithm to output a stable matching in the restricted HRS instances. We also provide an alternate NP-hardness proof for the decision version of the stable matching problem for HRS which imposes a severe restriction on the number of neighbours of non-unit sized agents.

We consider the Hospitals/Residents (HR) problem in the presence of ties in preference lists of hospitals. Among the three notions of stability, viz. weak, strong, and super stability, we focus on strong stability. Strong stability is appealing both theoretically and practically; however, its existence is not guaranteed. In this paper, our objective is to optimally augment the quotas of hospitals to ensure that a strongly stable matching exists in the modified instance. Such an augmentation is guaranteed to exist when resident preference lists are strict. We explore two natural optimization criteria: (i) minimizing the total capacity increase across all hospitals (MINSUM) and (ii) minimizing the maximum capacity increase for any hospital (MINMAX). We show that the MINSUM problem admits a polynomial-time algorithm, whereas the MINMAX problem is NP-hard. We prove an analogue of the Rural Hospitals theorem for the MINSUM problem. When each hospital incurs a cost for a unit increase in its quota, the MINSUM problem becomes NP-hard, even for 0/1 costs. In fact, we show that the problem cannot be approximated to any multiplicative factor. We also present a polynomial-time algorithm for optimal MINSUM augmentation when a specified subset of edges is required to be included in the matching.

Matching problems with group-fairness constraints and diversity constraints have numerous applications such as in allocation problems, committee selection, school choice, etc. Moreover, online matching problems have lots of applications in ad allocations and other e-commerce problems like product recommendation in digital marketing. We study two problems involving assigning items to platforms, where items belong to various groups depending on their attributes; the set of items are available offline and the platforms arrive online. In the first problem, we study online matchings with proportional fairness constraints. Here, each platform on arrival should either be assigned a set of items in which the fraction of items from each group is within specified bounds or be assigned no items; the goal is to assign items to platforms in order to maximize the number of items assigned to platforms. In the second problem, we study online matchings with diversity constraints, i.e. for each platform, absolute lower bounds are specified for each group. Each platform on arrival should either be assigned a set of items that satisfy these bounds or be assigned no items; the goal is to maximize the set of platforms that get matched. We study approximation algorithms and hardness results for these problems. The technical core of our proofs is a new connection between these problems and the problem of matchings in hypergraphs. Our experimental evaluation shows the performance of our algorithms on real-world and synthetic datasets exceeds our theoretical guarantees.

Assigning applicants to posts in the presence of the preferences of applicants and quotas associated with posts is extensively investigated. For a post, lower quota guarantees, and upper quota limits the number of applicants assigned to it. Typically, quotas are assumed to be fixed, which need not be the case in practice. We address this by introducing a soft quota setting, in which every post is associated with two values – lower target and upper target which together denote a range for the intended number of applicants in any assignment. Unlike the fixed quota setting, we allow the number of applicants assigned to a post to fall outside the range. This leads to assignments with deviation. Here, we study the problem of computing an assignment that has two orthogonal optimization objectives – minimizing the deviation (maximum or total) w.r.t. soft quotas and ensuring optimality w.r.t. preferences of applicants (rank-maximality or fairness). The order in which these objectives are considered, the different possibilities to optimize deviation combined with the well-studied notions of optimality w.r.t. preferences open up a range of optimization problems of practical importance. We present efficient algorithms based on flow-networks to solve these optimization problems.

We consider the hospital-residents problem where both hospitals and residents can have lower quotas. The input is a bipartite graph G = (ℛ∪ℋ,E), each vertex in ℛ∪ℋ has a strict preference ordering over its neighbors. The sets ℛ and ℋ 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' 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.

We consider the problem of assigning items to platforms in the presence of group fairness constraints. In the input, each item belongs to certain categories, called classes in this paper. Each platform specifies the group fairness constraints through an upper bound on the number of items it can serve from each class. Additionally, each platform also has an upper bound on the total number of items it can serve. The goal is to assign items to platforms so as to maximize the number of items assigned while satisfying the upper bounds of each class. This problem models several important real-world problems like ad-auctions, scheduling, resource allocations, school choice etc. We show that if the classes are arbitrary, then the problem is NP-hard and has a strong inapproximability. We consider the problem in both online and offline settings under natural restrictions on the classes. Under these restrictions, the problem continues to remain NP-hard but admits approximation algorithms with small approximation factors. We also implement some of the algorithms. Our experiments show that the algorithms work well in practice both in terms of efficiency and the number of items that get assigned to some platform.

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's 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.

We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph $G = (\mathcal{R} \cup \mathcal{H}, E)$ where $\mathcal{R}$ and $\mathcal{H}$ 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 $h$ has an associated upper-quota $q^+(h)$ and lower-quota $q^-(h)$. 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 $h$ is assigned at least $q^-(h)$ and at most $q^+(h)$ 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 $M'$ gets more votes than $M$ when vertices vote between $M$ and $M'$. 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.

We consider the strategic issues of the popular matchings problem. Let $G = (\mathcal{A} \cup \mathcal{P}, E)$ be a bipartite graph, where $\mathcal{A}$ denotes a set of agents, $\mathcal{P}$ denotes a set of posts, and the edges in $E$ are ranked. Each agent ranks a subset of posts in an order of preference, possibly involving ties. A matching $M$ is popular if there exists no matching $M'$ such that the number of agents that prefer $M'$ to $M$ exceeds the number of agents that prefer $M$ to $M'$. Consider a centralized market where agents submit their preferences and a central authority matches agents to posts according to the notion of popularity. Since a popular matching need not be unique, we assume that the central authority chooses an arbitrary popular matching. Let $a_1$ be the sole manipulative agent who is aware of the true preference lists of all other agents. The goal of $a_1$ is to falsify her preference list to get better always, that is, in the falsified instance (i) every popular matching matches $a_1$ to a post that is at least as good as the most preferred post that she gets when she was truthful, and (ii) some popular matching matches $a_1$ to a post better than the most preferred post $p$ that she gets when she was truthful, assuming that $p$ is not one of $a_1$'s (true) most preferred posts. We show that the optimal cheating strategy for a manipulative agent to get better always can be computed in $O(m+n)$ time when preference lists are all strict and in $O(\sqrt{n}m)$ time when preference lists are allowed to contain ties. Here $n = |\mathcal{A}| + |\mathcal{P}|$ and $m = |E|$. To compute the cheating strategies, we develop a switching graph characterization of the popular matchings problem involving ties. The switching graph characterization was studied for the case of strict lists by McDermid and Irving [J. Comb. Optim., 22 (2011), pp. 339--358] and was open for the case of ties. We show an $O(\sqrt{n}m)$ time algorithm to compute the set of popular pairs using the switching graph. These results are of independent interest and answer a part of the open questions posed by McDermid and Irving.

We consider the cheating strategies for the popular matchings problem. Let G = (\A \cup \p, E) be a bipartite graph where \A denotes a set of agents, p denotes a set of posts and the edges in E are ranked. Each agent ranks a subset of posts in an order of preference, possibly involving ties. A matching M is popular if there exists no matching M' such that the number of agents that prefer M' to M exceeds the number of agents that prefer M to M'. Consider a centralized market where agents submit their preferences and a central authority matches agents to posts according to the notion of popularity. Since a popular matching need not be unique, we assume that the central authority chooses an arbitrary popular matching. Let a_1 be the sole manipulative agent who is aware of the true preference lists of all other agents. The goal of a_1 is to falsify her preference list to get better always, that is, to improve the set of posts she gets matched to in the falsified instance. We show that the optimal cheating strategy for a single agent to get better always can be computed in O(m+n) time when preference lists are all strict and in O(\sqrt{n}m) time when preference lists are allowed to contain ties. Here n = |\A| + |\p| and m = |E|. To compute the cheating strategies, we develop a switching graph characterization of the popular matchings problem involving ties. The switching graph characterization was studied for the case of strict lists by McDermid and Irving (J. Comb. Optim. 2011) and was open for the case of ties. We show an O(\sqrt{n}m) time algorithm to compute the set of popular pairs using the switching graph. These results are of independent interest and answer a part of the open questions posed by McDermid and Irving.

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such that G is (strongly) rainbow connected. In this paper we study the rainbow connectivity problem and the strong rainbow connectivity problem from a computational point of view. Our main results can be summarised as below: 1) For every fixed k >= 3, it is NP-Complete to decide whether src(G) <= k even when the graph G is bipartite. 2) For every fixed odd k >= 3, it is NP-Complete to decide whether rc(G) <= k. This resolves one of the open problems posed by Chakraborty et al. (J. Comb. Opt., 2011) where they prove the hardness for the even case. 3) The following problem is fixed parameter tractable: Given a graph G, determine the maximum number of pairs of vertices that can be rainbow connected using two colors. 4) For a directed graph G, it is NP-Complete to decide whether rc(G) <= 2.