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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

In the Stable Marriage problem, when the preference lists are complete, all agents of the smaller side can be matched. However, this need not be true when preference lists are incomplete. In most real-life situations, where agents participate in the matching market voluntarily and submit their preferences, it is natural to assume that each agent wants to be matched to someone in his/her preference list as opposed to being unmatched. In light of the Rural Hospital Theorem, we have to relax the "no blocking pair" condition for stable matchings in order to match more agents. In this paper, we study the question of matching more agents with fewest possible blocking edges. In particular, the goal is to find a matching whose size exceeds that of a stable matching in the graph by at least t and has at most k blocking edges. We study this question in the realm of parameterized complexity with respect to several natural parameters, k,t,d, where d is the maximum length of a preference list. Unfortunately, the problem remains intractable even for the combined parameter k+t+d. Thus, we extend our study to the local search variant of this problem, in which we search for a matching that not only fulfills each of the above conditions but is "closest", in terms of its symmetric difference to the given stable matching, and obtain an FPT algorithm.

Sushmita Gupta, Pallavi Jain, Sanjukta Roy, Saket Saurabh, and Meirav Zehavi. On the (Parameterized) Complexity of Almost Stable Marriage. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2020.24, author = {Gupta, Sushmita and Jain, Pallavi and Roy, Sanjukta and Saurabh, Saket and Zehavi, Meirav}, title = {{On the (Parameterized) Complexity of Almost Stable Marriage}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {24:1--24:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.24}, URN = {urn:nbn:de:0030-drops-132655}, doi = {10.4230/LIPIcs.FSTTCS.2020.24}, annote = {Keywords: Stable Matching, Parameterized Complexity, Local Search} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time O*(((1+\sqrt{5})/2)^k) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a "divide-and-conquer-like" approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures "divided-and-conquered" pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O*(2^k) and polynomial space. Here, we show how a result by Björklund et al. [JCSS, 2017] can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.

Sushmita Gupta, Sanjukta Roy, Saket Saurabh, and Meirav Zehavi. Parameterized Algorithms and Kernels for Rainbow Matching. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 71:1-71:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gupta_et_al:LIPIcs.MFCS.2017.71, author = {Gupta, Sushmita and Roy, Sanjukta and Saurabh, Saket and Zehavi, Meirav}, title = {{Parameterized Algorithms and Kernels for Rainbow Matching}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {71:1--71:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.71}, URN = {urn:nbn:de:0030-drops-81245}, doi = {10.4230/LIPIcs.MFCS.2017.71}, annote = {Keywords: Rainbow Matching, Parameterized Algorithm, Bounded Search Trees, Divide-and-Conquer, 3-Set Packing, 3-Dimensional Matching} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

In this paper we study the "independent" version of the classic Feedback Vertex Set problem in the realm of parameterized algorithms and moderately exponential time algorithms. More precisely, we study the Independent Feedback Vertex Set problem, where we are given an undirected graph G on n vertices and a positive integer k, and the objective is to check if there is an independent feedback vertex set of size at most k. A set S subseteq V(G) is called an independent feedback vertex set (ifvs) if S is an independent set and G\S is a forest. In this paper we design two deterministic exact algorithms for Independent Feedback Vertex Set with running times O*(4.1481^k) and O*(1.5981^n). In fact, the algorithm with O*(1.5981^n) running time finds the smallest sized ifvs, if an ifvs exists. Both the algorithms are based on interesting measures and improve the best known algorithms for the problem in their respective domains. In particular, the algorithm with running time O*(4.1481^k) is an improvement over the previous algorithm that ran in time O*(5^k). On the other hand, the algorithm with running time O*(1.5981^n) is the first moderately exponential time algorithm that improves over the naive algorithm that enumerates all the subsets of V(G). Additionally, we show that the number of minimal ifvses in any graph on n vertices is upper bounded by 1.7485^n.

Akanksha Agrawal, Sushmita Gupta, Saket Saurabh, and Roohani Sharma. Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2016.2, author = {Agrawal, Akanksha and Gupta, Sushmita and Saurabh, Saket and Sharma, Roohani}, title = {{Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {2:1--2:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.2}, URN = {urn:nbn:de:0030-drops-69400}, doi = {10.4230/LIPIcs.IPEC.2016.2}, annote = {Keywords: independent feedback vertex set, fixed parameter tractable, exact algorithm, enumeration} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

In this paper we consider a problem that arises from a strategic issue in the stable matching model (with complete preference lists) from the viewpoint of exact-exponential time algorithms. Specifically, we study the Stable Extension of Partial Matching (SEOPM) problem, where the input consists of the complete preference lists of men, and a partial matching. The objective is to find (if one exists) a set of preference lists of women, such that the men-optimal Gale Shapley algorithm outputs a perfect matching that contains the given partial matching. Kobayashi and Matsui [Algorithmica, 2010] proved this problem is NP-complete. In this article, we give an exact-exponential algorithm for SEOPM running in time 2^{O(n)}, where n denotes the number of men/women. We complement our algorithmic finding by showing that unless Exponential Time Hypothesis (ETH) fails, our algorithm is asymptotically optimal. That is, unless ETH fails, there is no algorithm for SEOPM running in time 2^{o(n)}. Our algorithm is a non-trivial combination of a parameterized algorithm for Subgraph Isomorphism, a relationship between stable matching and finding an out-branching in an appropriate graph and enumerating non-isomorphic out-branchings.

Sushmita Gupta and Sanjukta Roy. Stable Matching Games: Manipulation via Subgraph Isomorphism. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2016.29, author = {Gupta, Sushmita and Roy, Sanjukta}, title = {{Stable Matching Games: Manipulation via Subgraph Isomorphism}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.29}, URN = {urn:nbn:de:0030-drops-68642}, doi = {10.4230/LIPIcs.FSTTCS.2016.29}, annote = {Keywords: stable matching, Gale-Shapley algorithm, suitor graph, subgraph iso- morphism, exact-exponential time algorithms} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

The stable marriage problem (SMP) can be seen as a typical game, where each player wants to obtain the best possible partner by manipulating his/her preference list. Thus the set Q of preference lists submitted to the matching agency may differ from P, the set of true preference lists. In this paper, we study the stability of the stated lists in Q. If Q is not Nash equilibrium, i.e., if a player can obtain a strictly better partner (with respect to the preference order in P) by only changing his/her list, then in the view of standard game theory, Q is vulnerable. In the case of SMP, however, we need to consider another factor, namely that all valid matchings should not include any "blocking pairs" with respect to P. Thus, if the above manipulation of a player introduces blocking pairs, it would prevent this manipulation. Consequently, we say Q is totally stable if either Q is a Nash equilibrium or if any attempt at manipulation by a single player causes blocking pairs with respect to P. We study the complexity of testing the total stability of a stated strategy. It is known that this question is answered in polynomial time if the instance (P,Q) always satisfies P=Q. We extend this polynomially solvable class to the general one, where P and Q may be arbitrarily different.

Sushmita Gupta, Kazuo Iwama, and Shuichi Miyazaki. Total Stability in Stable Matching Games. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 23:1-23:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gupta_et_al:LIPIcs.SWAT.2016.23, author = {Gupta, Sushmita and Iwama, Kazuo and Miyazaki, Shuichi}, title = {{Total Stability in Stable Matching Games}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {23:1--23:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.23}, URN = {urn:nbn:de:0030-drops-60450}, doi = {10.4230/LIPIcs.SWAT.2016.23}, annote = {Keywords: stable matching, Gale-Shapley algorithm, manipulation, stability, Nash equilibrium} }

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