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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

For a connected graph G = (V, E) and s, t ∈ V, a non-separating s-t path is a path P between s and t such that the set of vertices of P does not separate G, that is, G - V(P) is connected. An s-t path P is non-disconnecting if G - E(P) is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating s-t path of length at most k is W[1]-hard parameterized by k, while the non-disconnecting counterpart is fixed-parameter tractable (FPT) parameterized by k. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is FPT parameterized by k on planar graphs and on unit disk graphs (where no s, t is given). Further, we give a polynomial-time algorithm on chordal graphs if k is the distance of the shortest path between s and t.

Ankit Abhinav, Susobhan Bandopadhyay, Aritra Banik, Yasuaki Kobayashi, Shunsuke Nagano, Yota Otachi, and Saket Saurabh. Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{abhinav_et_al:LIPIcs.MFCS.2022.6, author = {Abhinav, Ankit and Bandopadhyay, Susobhan and Banik, Aritra and Kobayashi, Yasuaki and Nagano, Shunsuke and Otachi, Yota and Saurabh, Saket}, title = {{Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {6:1--6:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.6}, URN = {urn:nbn:de:0030-drops-168041}, doi = {10.4230/LIPIcs.MFCS.2022.6}, annote = {Keywords: Non-separating path, Parameterized complexity} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.

Aritra Banik, Rajiv Raman, and Saurabh Ray. On Geometric Priority Set Cover Problems. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{banik_et_al:LIPIcs.ISAAC.2021.12, author = {Banik, Aritra and Raman, Rajiv and Ray, Saurabh}, title = {{On Geometric Priority Set Cover Problems}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.12}, URN = {urn:nbn:de:0030-drops-154459}, doi = {10.4230/LIPIcs.ISAAC.2021.12}, annote = {Keywords: Approximation algorithms, geometric set cover, local search, quasi-uniform sampling} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Given n pairs of points, S = {{p_1, q_1}, {p_2, q_2}, ..., {p_n, q_n}}, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.

Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Su Jia, Matthew J. Katz, Tyler Mayer, and Joseph S. B. Mitchell. Network Optimization on Partitioned Pairs of Points. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arkin_et_al:LIPIcs.ISAAC.2017.6, author = {Arkin, Esther M. and Banik, Aritra and Carmi, Paz and Citovsky, Gui and Jia, Su and Katz, Matthew J. and Mayer, Tyler and Mitchell, Joseph S. B.}, title = {{Network Optimization on Partitioned Pairs of Points}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {6:1--6:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.6}, URN = {urn:nbn:de:0030-drops-82700}, doi = {10.4230/LIPIcs.ISAAC.2017.6}, annote = {Keywords: Network Optimization, TSP tour, Matching, Spanning Tree, Pairs, Partition, Algorithms, Complexity} }

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

Frechet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Frechet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes Q={Q_1,Q_2,...,Q_n}, where each class Q_i consists of a pair of points Q_i = {q_i, bar{q_i}}. We call a subset A subset {q_i , bar{q_i}:1 <= i <= n} conflict free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict free subset that optimizes a given function.
Given a line segment l and set Q of a pair of points in R^2, our objective is to find a conflict free subset that minimizes the Frechet distance between l and the point set, where the minimum is taken over all possible conflict free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm which is later derandomized using universal family of sets. We believe that this technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in O(2^k * n * log^2(n)) time, and the derandomized deterministic algorithm runs in O(2^k * k^{O(log(k))} * n * log^2(n)) time, where k, the parameter, is the number of elements in the conflict free subset solution. Finally we present a simple branching algorithm for the problem running in O(2^k * n^{2} *log(n)) time. We also show that the problem is unlikely to have a polynomial sized kernel under standard complexity theoretic assumption.

Aritra Banik, Fahad Panolan, Venkatesh Raman, and Vibha Sahlot. Fréchet Distance Between a Line and Avatar Point Set. 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. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{banik_et_al:LIPIcs.FSTTCS.2016.32, author = {Banik, Aritra and Panolan, Fahad and Raman, Venkatesh and Sahlot, Vibha}, title = {{Fr\'{e}chet Distance Between a Line and Avatar Point Set}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {32:1--32:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.32}, URN = {urn:nbn:de:0030-drops-68676}, doi = {10.4230/LIPIcs.FSTTCS.2016.32}, annote = {Keywords: Frechet Distance, Avatar Problems, Multiple Choice, FPT} }

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

We present two improved algorithms for weighted discrete p-center problem for tree networks with n vertices. One of our proposed algorithms runs in O(n*log(n) + p*log^2(n) * log(n/p)) time. For all values of p, our algorithm thus runs as fast as or faster than the most efficient O(n*log^2(n)) time algorithm obtained by applying Cole's [1987] speed-up technique to the algorithm due to Megiddo and Tamir [1983], which has remained unchallenged for nearly 30 years.
Our other algorithm, which is more practical, runs in O(n*log(n) + p^2*log^2(n/p)) time, and when p=O(sqrt(n)) it is faster than Megiddo and Tamir's O(n*log^2(n) * log(log(n))) time algorithm [1983].

Aritra Banik, Binay Bhattacharya, Sandip Das, Tsunehiko Kameda, and Zhao Song. The p-Center Problem in Tree Networks Revisited. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{banik_et_al:LIPIcs.SWAT.2016.6, author = {Banik, Aritra and Bhattacharya, Binay and Das, Sandip and Kameda, Tsunehiko and Song, Zhao}, title = {{The p-Center Problem in Tree Networks Revisited}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {6:1--6:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.6}, URN = {urn:nbn:de:0030-drops-60296}, doi = {10.4230/LIPIcs.SWAT.2016.6}, annote = {Keywords: Facility location, p-center, parametric search, tree network, sorting network} }

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