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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

In this paper, we give the first constant factor approximation algorithm for capacitated knapsack median problem (CKnM) for hard uniform capacities, violating the budget by a factor of 1+epsilon and capacities by a 2+epsilon factor. To the best of our knowledge, no constant factor approximation is known for the problem even with capacity/budget/both violations. Even for the uncapacitated variant of the problem, the natural LP is known to have an unbounded integrality gap even after adding the covering inequalities to strengthen the LP. Our techniques for CKnM provide two types of results for the capacitated k-facility location problem. We present an O(1/epsilon^2) factor approximation for the problem, violating capacities by (2+epsilon). Another result is an O(1/epsilon) factor approximation, violating the capacities by a factor of at most (1 + epsilon) using at most 2k facilities for a fixed epsilon>0. As a by-product, a constant factor approximation algorithm for capacitated facility location problem with uniform capacities is presented, violating the capacities by (1 + epsilon) factor. Though constant factor results are known for the problem without violating the capacities, the result is interesting as it is obtained by rounding the solution to the natural LP, which is known to have an unbounded integrality gap without violating the capacities. Thus, we achieve the best possible from the natural LP for the problem. The result shows that the natural LP is not too bad.

Sapna Grover, Neelima Gupta, Samir Khuller, and Aditya Pancholi. Constant Factor Approximation Algorithm for Uniform Hard Capacitated Knapsack Median Problem. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{grover_et_al:LIPIcs.FSTTCS.2018.23, author = {Grover, Sapna and Gupta, Neelima and Khuller, Samir and Pancholi, Aditya}, title = {{Constant Factor Approximation Algorithm for Uniform Hard Capacitated Knapsack Median Problem}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {23:1--23:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.23}, URN = {urn:nbn:de:0030-drops-99224}, doi = {10.4230/LIPIcs.FSTTCS.2018.23}, annote = {Keywords: Capacitated Knapsack Median, Capacitated k -Facility Location} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

In this paper, we propose and analyze a local search algorithm for the Universal facility location problem. Our algorithm improves the approximation ratio of this problem from 5.83, given by Angel et al., to 5. A second major contribution of the paper is that it gets rid of the expensive multi operation that was a mainstay of all previous local search algorithms for capacitated facility location and universal facility location problem. The only operations that we require to prove the 5-approximation are add, open, and close. A multi operation is basically a combination of the open and close operations. The 5-approximation algorithm for the capacitated facility location problem, given by Bansal et al., also uses the multi operation. However, on careful observation, it turned out that add, open, and close operations are sufficient to prove a 5-factor for the problem. This resulted into an improved algorithm for the universal facility location problem, with an improved factor.

Manisha Bansal, Naveen Garg, and Neelima Gupta. A 5-Approximation for Universal Facility Location. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 24:1-24:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bansal_et_al:LIPIcs.FSTTCS.2018.24, author = {Bansal, Manisha and Garg, Naveen and Gupta, Neelima}, title = {{A 5-Approximation for Universal Facility Location}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {24:1--24:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.24}, URN = {urn:nbn:de:0030-drops-99239}, doi = {10.4230/LIPIcs.FSTTCS.2018.24}, annote = {Keywords: Facility location, Approximation Algorithms, Local Search} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

The replica placement problem has been well studied on trees. In this paper, we study this problem on directed acyclic graphs. The replica placement problem on general DAGs generalizes the set cover problem. We present a constant factor approximation algorithm for the special case of DAGs having bounded degree and bounded tree-width (BDBT-DAGs). We also present a constant factor approximation algorithm for DAGs composed of local BDBT-DAGs connected in a tree like manner (TBDBT-DAGs). The latter class of DAGs generalizes trees as well; we improve upon the previously best known approximation ratio for the problem on trees. Our algorithms are based on the LP rounding technique; the core component of our algorithm exploits the structural properties of tree-decompositions to massage the LP solution into an integral solution.

Sonika Arora, Venkatesan T. Chakaravarthy, Kanika Gupta, Neelima Gupta, and Yogish Sabharwal. Replica Placement on Directed Acyclic Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 213-225, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{arora_et_al:LIPIcs.FSTTCS.2014.213, author = {Arora, Sonika and Chakaravarthy, Venkatesan T. and Gupta, Kanika and Gupta, Neelima and Sabharwal, Yogish}, title = {{Replica Placement on Directed Acyclic Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {213--225}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.213}, URN = {urn:nbn:de:0030-drops-48449}, doi = {10.4230/LIPIcs.FSTTCS.2014.213}, annote = {Keywords: Approximation Algorithms, LP Rounding} }

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

In this paper, we study the replica placement problem on trees and present a constant factor approximation algorithm (with an additional additive constant factor). This improves the best known previous algorithm having an approximation ratio dependent on the maximum degree of the tree. Our techniques also extend to the partial cover version. Our algorithms are based on the LP rounding technique. The core component of our algorithm exploits a connection between the natural LP solutions of the replica placement problem and the capacitated vertex cover problem.

Sonika Arora, Venkatesan T. Chakaravarthy, Neelima Gupta, Koyel Mukherjee, and Yogish Sabharwal. Replica Placement via Capacitated Vertex Cover. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 263-274, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{arora_et_al:LIPIcs.FSTTCS.2013.263, author = {Arora, Sonika and Chakaravarthy, Venkatesan T. and Gupta, Neelima and Mukherjee, Koyel and Sabharwal, Yogish}, title = {{Replica Placement via Capacitated Vertex Cover}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {263--274}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.263}, URN = {urn:nbn:de:0030-drops-43784}, doi = {10.4230/LIPIcs.FSTTCS.2013.263}, annote = {Keywords: Approximation Algorithms, LP Rounding} }

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