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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In this paper, we introduce and study the Non-Uniform k-Center (NUkC) problem. Given a finite metric space (X, d) and a collection of balls of radii {r_1 >= ... >= r_k}, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation alpha, such that the union of balls of radius alpha*r_i around the i-th center covers all the points in X. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds, or as a wireless router placement problem with routers of different powers/ranges.
The NUkC problem generalizes the classic k-center problem when all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k-center with outliers (kCwO for short) problem when there are k balls of radius 1 and l balls of radius 0. There are 2-approximation and 3-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years.
We first observe that no O(1)-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an (O(1), O(1))-bi-criteria approximation result: we give an O(1)-approximation to the optimal dilation, however, we may open Theta(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community. We show NUkC is as hard as the firefighter problem.
While we don't know if the converse is true, we are able to adapt ideas from recent works [Chalermsook/Chuzhoy, SODA 2010; Asjiashvili/Baggio/Zenklusen, arXiv 2016] in non-trivial ways to obtain our constant factor bi-criteria approximation.

Deeparnab Chakrabarty, Prachi Goyal, and Ravishankar Krishnaswamy. The Non-Uniform k-Center Problem. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chakrabarty_et_al:LIPIcs.ICALP.2016.67, author = {Chakrabarty, Deeparnab and Goyal, Prachi and Krishnaswamy, Ravishankar}, title = {{The Non-Uniform k-Center Problem}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {67:1--67:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.67}, URN = {urn:nbn:de:0030-drops-62178}, doi = {10.4230/LIPIcs.ICALP.2016.67}, annote = {Keywords: Clustering, k-Center, Approximation Algorithms, Firefighter Problem} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on n vertices and a positive integer parameter k, find if there exist k edges(arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the
vertices have even degrees--where the resulting graph is Eulerian,the problem is called Undirected Eulerian Edge Deletion. The corresponding
problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its
out-degree is called Directed Eulerian Edge Deletion. Cygan et al.[Algorithmica, 2014] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time
2^O(k log k)n^O(1). They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time
2^O(k)n^O(1). It was also posed as an open problem at the School on Parameterized Algorithms and Complexity 2014, Bedlewo, Poland.
In this paper we answer their question in the affirmative: using the technique of computing representative families of co-graphic matroids we design algorithms which solve these problems in time 2^O(k)n^O(1). The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity.

Prachi Goyal, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Finding Even Subgraphs Even Faster. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 434-447, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2015.434, author = {Goyal, Prachi and Misra, Pranabendu and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{Finding Even Subgraphs Even Faster}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {434--447}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.434}, URN = {urn:nbn:de:0030-drops-56336}, doi = {10.4230/LIPIcs.FSTTCS.2015.434}, annote = {Keywords: Eulerian Edge Deletion, FPT, Representative Family.} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Given a universe U := U_1 + .... + U_r and a r-uniform family F
which is a subset of U_1 x .... x U_r, the r-dimensional matching
problem asks if F admits a collection of k mutually disjoint sets. The
special case when r=3 is the classic 3-Dimensional Matching problem.
Recently, several improvements have been suggested for these (and closely related) problems in the setting of randomized parameterized algorithms. Also, many approaches have evolved for deterministic parameterized algorithms. For instance, for the 3-Dimensional Matching problem, a combination of color coding and iterative expansion yields a running time of O^*(2.80^{(3k)}), and for the r-dimensional matching problem, a recently developed derandomization for known algebraic techniques leads to a running time of O^*(5.44^{(r-1)k}).
In this work, we employ techniques based on dynamic programming and
representative families, leading to a deterministic algorithm with running time O^*(2.85^{(r-1)k}) for the r-Dimensional Matching problem. Further, we incorporate the principles of iterative expansion used in the literature [TALG 2012] to obtain a better algorithm for 3D-matching, with a running time of O^*(2.003^{(3k)}). Apart from the significantly improved running times, we believe that these algorithms demonstrate an interesting application of representative families in conjunction with more traditional techniques.

Prachi Goyal, Neeldhara Misra, and Fahad Panolan. Faster Deterministic Algorithms for r-Dimensional Matching Using Representative Sets. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 237-248, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2013.237, author = {Goyal, Prachi and Misra, Neeldhara and Panolan, Fahad}, title = {{Faster Deterministic Algorithms for r-Dimensional Matching Using Representative Sets}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {237--248}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.237}, URN = {urn:nbn:de:0030-drops-43761}, doi = {10.4230/LIPIcs.FSTTCS.2013.237}, annote = {Keywords: 3-Dimensional Matching, Fixed-Parameter Algorithms, Iterative Expansion} }

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