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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

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.

Meghana Nasre, Prajakta Nimbhorkar, Keshav Ranjan, and Ankita Sarkar. Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{nasre_et_al:LIPIcs.FSTTCS.2021.30, author = {Nasre, Meghana and Nimbhorkar, Prajakta and Ranjan, Keshav and Sarkar, Ankita}, title = {{Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {30:1--30:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.30}, URN = {urn:nbn:de:0030-drops-155419}, doi = {10.4230/LIPIcs.FSTTCS.2021.30}, annote = {Keywords: Matching, Popularity, Lower quota, Preferences} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

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.

Krishnapriya A M, Meghana Nasre, Prajakta Nimbhorkar, and Amit Rawat. How Good Are Popular Matchings?. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{am_et_al:LIPIcs.SEA.2018.9, author = {A M, Krishnapriya and Nasre, Meghana and Nimbhorkar, Prajakta and Rawat, Amit}, title = {{How Good Are Popular Matchings?}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.9}, URN = {urn:nbn:de:0030-drops-89440}, doi = {10.4230/LIPIcs.SEA.2018.9}, annote = {Keywords: bipartite graphs, hospital residents, lower-quotas, popular matchings} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

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' 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.

Meghana Nasre and Prajakta Nimbhorkar. Popular Matchings with Lower Quotas. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{nasre_et_al:LIPIcs.FSTTCS.2017.44, author = {Nasre, Meghana and Nimbhorkar, Prajakta}, title = {{Popular Matchings with Lower Quotas}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.44}, URN = {urn:nbn:de:0030-drops-83937}, doi = {10.4230/LIPIcs.FSTTCS.2017.44}, annote = {Keywords: bipartite matchings, preferences, hospital residents, lower-quota, popular matchings} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

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.

Meghana Nasre. Popular Matchings: Structure and Cheating Strategies. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 412-423, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{nasre:LIPIcs.STACS.2013.412, author = {Nasre, Meghana}, title = {{Popular Matchings: Structure and Cheating Strategies}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {412--423}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.412}, URN = {urn:nbn:de:0030-drops-39520}, doi = {10.4230/LIPIcs.STACS.2013.412}, annote = {Keywords: bipartite matchings, preferences, cheating strategies} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

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.

Prabhanjan Ananth, Meghana Nasre, and Kanthi K. Sarpatwar. Rainbow Connectivity: Hardness and Tractability. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 241-251, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{ananth_et_al:LIPIcs.FSTTCS.2011.241, author = {Ananth, Prabhanjan and Nasre, Meghana and Sarpatwar, Kanthi K.}, title = {{Rainbow Connectivity: Hardness and Tractability}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {241--251}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.241}, URN = {urn:nbn:de:0030-drops-33535}, doi = {10.4230/LIPIcs.FSTTCS.2011.241}, annote = {Keywords: Computational Complexity, Rainbow Connectivity, Graph Theory, Fixed Parameter Tractable Algorithms} }