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Prajakta Nimbhorkar

Publications and source records attributed to Prajakta Nimbhorkar.

28 records · Page 2Linked to original sources
arXiv2018

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 (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 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.

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arXiv2017

Popular Matching 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 $G = (\mathcal{R} \cup \mathcal{H}, E)$ 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.

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arXiv2017

Dynamic Rank Maximal Matchings

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 U 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., which takes time O(min((r + n, r*sqrt(n))m).

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arXiv2014

Rank Maximal Matchings -- Structure and Algorithms

Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rank-maximal matchings is #P-Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rank-maximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem.

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arXiv2012

Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

Let $G = $ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$ for $G$. More precisely, the algorithm computes a subset $T\subset G$ of size $\tilde{O}(n^2)(1/λ)^{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $λ$-spectral expander (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})^{O(1)}$ for the groups $\Z_d^n$. The earlier known size bound was $O((d+n/\varepsilon^2))^{11/2}$ given by \cite{AMN98}.

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arXiv2010

Popularity at Minimum Cost

We consider an extension of the {\em popular matching} problem in this paper. The input to the popular matching problem is a bipartite graph G = (A U B,E), where A is a set of people, B is a set of items, and each person a belonging to 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 belonging to 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 sqrt{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.

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arXiv2010

Log-space Algorithms for Paths and Matchings in k-trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].

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arXiv2009

A Log-space Algorithm for Canonization of Planar Graphs

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.

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arXiv2008

Longest paths in Planar DAGs in Unambiguous Logspace

We show via two different algorithms that finding the length of the longest path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and also in the complement class co-UL. The result extends to toroidal DAGs as well.

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