Papers by Prajakta Nimbhorkar
We consider the hospital-residents problem where both hospitals and residents can have lower quot... more 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 nonempty, 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.
arXiv: Data Structures and Algorithms, 2020
We consider the problem of matchings under two-sided preferences in the presence of maximum as we... more We consider the problem of matchings under two-sided preferences in the presence of maximum as well as minimum quota requirements for the agents. This setting, studied as the Hospital Residents with Lower Quotas (HRLQ) in literature, models important real world problems like assigning medical interns (residents) to hospitals, and teaching assistants to instructors where a minimum guarantee is essential. When there are no minimum quotas, stability is the de-facto notion of optimality. However, in the presence of minimum quotas, ensuring stability and simultaneously satisfying lower quotas is not an attainable goal in many instances. To address this, a relaxation of stability known as envy-freeness, is proposed in literature. In our work, we thoroughly investigate envy-freeness from a computational view point. Our results show that computing envy-free matchings that match maximum number of agents is computationally hard and also hard to approximate up to a constant factor. Additionall...
In the last lecture, we proved an inapproximability result for the MAX-CUT problem. We also intro... more In the last lecture, we proved an inapproximability result for the MAX-CUT problem. We also introduced the unique label cover problem and Khot’s unique games conjecture. Today, we will see how well unique games can be approximated. In particular, we will see Trevisan’s algorithm for approximating unique games on low-diameter graphs [Tre08]. We will then briefly discuss the algorithm of Arora et al. for approximating unique games on expanders [AKK+08], and a recent sub-exponential time algorithm for unique games on general graphs due to Arora et al [ABS10]. The references for this lecture include the above cited papers and Lecture 8 of the DIMACS tutorial on Limits of approximation [HC09].
Computing and Combinatorics, 2017
In the last lecture, we proved an inapproximability result for the MAX-CUT problem. We also intro... more In the last lecture, we proved an inapproximability result for the MAX-CUT problem. We also introduced the unique label cover problem and Khot’s unique games conjecture. Today, we will see how well unique games can be approximated. In particular, we will see Trevisan’s algorithm for approximating unique games on low-diameter graphs [Tre08]. We will then briefly discuss the algorithm of Arora et al. for approximating unique games on expanders [AKK+08], and a recent sub-exponential time algorithm for unique games on general graphs due to Arora et al [ABS10]. The references for this lecture include the above cited papers and Lecture 8 of the DIMACS tutorial on Limits of approximation [HC09].
We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ)... more 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∪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 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 popula...
We consider an extension of the popular matching problem in this paper. The input to the popular ... more We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph G = (A ∪ B, E), where A is a set of people, B is a set of items, and each person a ∈ A ranks a subset of items in an order of preference, with ties allowed. The popular matching problem seeks to compute a matching M * between people and items such that there is no matching M where more people are happier with M than with M *. Such a matching M * is called a popular matching. However, there are simple instances where no popular matching exists. Here we consider the following natural extension to the above problem: associated with each item b ∈ B is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to "augment" G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of √ n1/2, where n1 is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn1) time, where m is the number of edges.
We prove that the pseudorandom generator introduced by Impagliazzo et al. in [INW94] with proper ... more We prove that the pseudorandom generator introduced by Impagliazzo et al. in [INW94] with proper choice of parameters fools group products of a given finite group G. The seed length is O(log n(|G | O(1) +log 1 δ)), where n is the length of the word and δ is the allowed error. The result implies that the pseudorandom generator with seed length O(log n(2O(w log w) + log 1 δ)) fools read-once permutation branching programs of width w. As an application of the pseudorandom generator one obtains small-bias spaces for products over all finite groups [MZ09].
Planar graph canonization is known to be hard for L this directly follows from L-hardness of tree... more Planar graph canonization is known to be hard for L this directly follows from L-hardness of tree-canonization [Lin92]. We give a log-space algorithm for planar graph canonization. This gives completeness for log-space under AC 0 many-one reductions and improves the previously known upper bound of AC 1 [MR91]. A planar graph can be decomposed into biconnected components. We give a log-space procedure for the decomposition of a biconnected planar graph into a triconnected component tree. The canonization process is based on these decomposition steps. 1
In the k-means problem, we are given a finite set S of points in ℜ m, and integer k ≥ 1, and we w... more In the k-means problem, we are given a finite set S of points in ℜ m, and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [7].
... They have done a lot for me and I will always be grateful to them. A special thanks to my sis... more ... They have done a lot for me and I will always be grateful to them. A special thanks to my sister, Prachi, who has been a great companion right from my childhood. She has provided an invaluable support during my stay at TIFR. ...
ArXiv, 2018
In this paper, we consider the problem of computing an optimal matching in a bipartite graph wher... more 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 $\mathcal{C}_u$ 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...
We study computation by formulas over (min,+). We consider the computation of max{x_1,...,x_n} ov... more We study computation by formulas over (min,+). We consider the computation of max{x_1,...,x_n} over N as a difference of (min,+) formulas, and show that size n + n \log n is sufficient and necessary. Our proof also shows that any (min,+) formula computing the minimum of all sums of n-1 out of n variables must have n \log n leaves; this too is tight. Our proofs use a complexity measure for (min,+) functions based on minterm-like behaviour and on the entropy of an associated graph.
In this paper, we consider the problem of computing an optimal matching in a bipartite graph \(G=... more 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.
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In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem... more 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 [20] recently studied envy-free matchings for the ...
We consider the problem of assigning applicants to posts when each applicant has a strict prefere... more We consider the problem of assigning applicants to posts when each applicant has a strict preference ordering over a subset of posts, and each post has all its neighbors in a single tie. That is, a post is indifferent amongst all its neighbours. Each post has a capacity denoting the maximum number of applicants that can be assigned to it. An assignment M, referred to as a matching, is said to be popular, if there is no other assignment \(M'\) such that the number of votes \(M'\) gets compared to M is more than the number of votes M gets compared to \(M'\). Here votes are cast by applicants and posts for comparing M and \(M'\). An applicant a votes for M over \(M'\) if a gets a more preferred partner in M than in \(M'\). A post p votes for M over \(M'\) if p gets more applicants assigned to it in M than in \(M'\). The number of votes a post p casts gives rise to two models. Let M(p) denote the set of applicants p gets in M. If \(|M(p)|>|M'(p)|\)...
ArXiv, 2021
We consider the problem of assigning items to platforms in the presence of group fairness constra... more 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 realworld 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 s...
We consider the problem of assigning residents to hospitals when hospitals have upper and lower q... more We consider the problem of assigning residents to hospitals when hospitals have upper and lower quotas. Apart from this, both residents and hospitals have a preference list which is a strict ordering on a subset of the other side. Stability is a well-known notion of optimality in this setting. Every Hospital-Residents (HR) instance without lower quotas admits at least one stable matching. When hospitals have lower quotas (HRLQ), there exist instances for which no matching that is simultaneously stable and feasible exists. We investigate envy-freeness which is a relaxation of stability for such instances. Yokoi (ISAAC 2017) gave a characterization for HRLQ instances that admit a feasible and envy-free matching. Yokoi's algorithm gives a minimum size feasible envy-free matching, if there exists one. We investigate the complexity of computing a maximum size envy-free matching in an HRLQ instance (MAXEFM problem) which is equivalent to computing an envy-free matching with minimum nu...
In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem... more 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 (MinBP) and minimum number of blocking residents (MinBR) 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 (ISAAC 2017) recently studied envy-free matchings fo...
We consider the problem of matching applicants to posts where applicants have preferences over po... more We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A ∪ P, E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applicants to their rank 1 posts, subject to this the maximum number of applicants to their rank 2 posts, and so on. We consider this problem in a dynamic setting, where vertices and edges can be added and deleted at any point. Let n and m be the number of vertices and edges in an instance G, and r be the maximum rank used by any rank-maximal matching in G. We give a simple O(r(m + n))-time algorithm to update an existing rank-maximal matching under each of these changes. When r = o(n), this is faster than recomputing a rank-maximal matching completely using a known algorithm like that of Irving et al....
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Papers by Prajakta Nimbhorkar