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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Given an edge weighted graph and a forest F, the 2-edge connectivity augmentation problem is to pick a minimum weighted set of edges, E', such that every connected component of E' ∪ F is 2-edge connected. Williamson et al. gave a 2-approximation algorithm (WGMV) for this problem using the primal-dual schema. We show that when edge weights are integral, the WGMV procedure can be modified to obtain a half-integral dual. The 2-edge connectivity augmentation problem has an interesting connection to routing flow in graphs where the union of supply and demand is planar. The half-integrality of the dual leads to a tight 2-approximate max-half-integral-flow min-multicut theorem.

Naveen Garg and Nikhil Kumar. Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{garg_et_al:LIPIcs.ESA.2020.55, author = {Garg, Naveen and Kumar, Nikhil}, title = {{Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {55:1--55:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.55}, URN = {urn:nbn:de:0030-drops-129214}, doi = {10.4230/LIPIcs.ESA.2020.55}, annote = {Keywords: Combinatorial Optimization, Multicommodity Flow, Network Design} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the online problem of scheduling jobs on identical machines, where jobs have precedence constraints. We are interested in the demanding setting where the jobs sizes are not known up-front, but are revealed only upon completion (the non-clairvoyant setting). Such precedence-constrained scheduling problems routinely arise in map-reduce and large-scale optimization. For minimizing the total weighted completion time, we give a constant-competitive algorithm. And for total weighted flow-time, we give an O(1/epsilon^2)-competitive algorithm under (1+epsilon)-speed augmentation and a natural "no-surprises" assumption on release dates of jobs (which we show is necessary in this context).
Our algorithm proceeds by assigning virtual rates to all waiting jobs, including the ones which are dependent on other uncompleted jobs. We then use these virtual rates to decide on the actual rates of minimal jobs (i.e., jobs which do not have dependencies and hence are eligible to run). Interestingly, the virtual rates are obtained by allocating time in a fair manner, using a Eisenberg-Gale-type convex program (which we can solve optimally using a primal-dual scheme). The optimality condition of this convex program allows us to show dual-fitting proofs more easily, without having to guess and hand-craft the duals. This idea of using fair virtual rates may have broader applicability in scheduling problems.

Naveen Garg, Anupam Gupta, Amit Kumar, and Sahil Singla. Non-Clairvoyant Precedence Constrained Scheduling. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{garg_et_al:LIPIcs.ICALP.2019.63, author = {Garg, Naveen and Gupta, Anupam and Kumar, Amit and Singla, Sahil}, title = {{Non-Clairvoyant Precedence Constrained Scheduling}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {63:1--63:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.63}, URN = {urn:nbn:de:0030-drops-106394}, doi = {10.4230/LIPIcs.ICALP.2019.63}, annote = {Keywords: Online algorithms, Scheduling, Primal-Dual analysis, Nash welfare} }

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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 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon.

Bhaskar Ray Chaudhury, Yun Kuen Cheung, Jugal Garg, Naveen Garg, Martin Hoefer, and Kurt Mehlhorn. On Fair Division for Indivisible Items. 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. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chaudhury_et_al:LIPIcs.FSTTCS.2018.25, author = {Chaudhury, Bhaskar Ray and Cheung, Yun Kuen and Garg, Jugal and Garg, Naveen and Hoefer, Martin and Mehlhorn, Kurt}, title = {{On Fair Division for Indivisible Items}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {25:1--25:17}, 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.2018.25}, URN = {urn:nbn:de:0030-drops-99242}, doi = {10.4230/LIPIcs.FSTTCS.2018.25}, annote = {Keywords: Fair Division, Indivisible Goods, Envy-Free} }

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Complete Volume

**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

LIPIcs, Volume 40, APPROX/RANDOM'15, Complete Volume

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Proceedings{garg_et_al:LIPIcs.APPROX-RANDOM.2015, title = {{LIPIcs, Volume 40, APPROX/RANDOM'15, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015}, URN = {urn:nbn:de:0030-drops-54012}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015}, annote = {Keywords: Data Structures, Coding and Information Theory, Theory of Computation, Computation by Abstract Devices, Modes of Computation, Complexity Measures and Problem Complexity, Numerical Algorithms and Problems, Nonnumerical Algorithms and Problems, Approximation, Numerical Linear Algorithms and Problems} }

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Front Matter

**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. i-xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{garg_et_al:LIPIcs.APPROX-RANDOM.2015.i, author = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, title = {{Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {i--xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.i}, URN = {urn:nbn:de:0030-drops-53474}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.i}, annote = {Keywords: Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We study the Unsplittable Flow Problem (UFP) and related variants, namely UFP with Bag Constraints and UFP with Rounds, on paths and trees. We provide improved constant factor approximation algorithms for all these problems under the no bottleneck assumption (NBA), which says that the maximum demand for any source-sink pair is at most the minimum capacity of any edge. We obtain these improved
results by expressing a feasible solution to a natural LP relaxation of the UFP as a near-convex combination of feasible integral solutions.

Khaled Elbassioni, Naveen Garg, Divya Gupta, Amit Kumar, Vishal Narula, and Arindam Pal. Approximation Algorithms for the Unsplittable Flow Problem on Paths and Trees. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 267-275, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{elbassioni_et_al:LIPIcs.FSTTCS.2012.267, author = {Elbassioni, Khaled and Garg, Naveen and Gupta, Divya and Kumar, Amit and Narula, Vishal and Pal, Arindam}, title = {{Approximation Algorithms for the Unsplittable Flow Problem on Paths and Trees}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {267--275}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.267}, URN = {urn:nbn:de:0030-drops-38650}, doi = {10.4230/LIPIcs.FSTTCS.2012.267}, annote = {Keywords: Approximation Algorithms, Integer Decomposition, Linear Programming, Scheduling, Unsplittable Flows} }

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