65 Search Results for "Garg, Naveen"

Volume

LIPIcs, Volume 40

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

APPROX/RANDOM 2015, August 24-26, 2015, Princeton, USA

Editors: Naveen Garg, Klaus Jansen, Anup Rao, and José D. P. Rolim

Document
DOI: 10.4230/LIPIcs.ISAAC.2020.41
Multicommodity Flows in Planar Graphs with Demands on Faces

Authors: Nikhil Kumar

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
We consider the problem of multicommodity flows in planar graphs. Seymour [Seymour, 1981] showed that if the union of supply and demand graphs is planar, then the cut condition is also sufficient for routing demands. Okamura-Seymour [Okamura and Seymour, 1981] showed that if the supply graph is planar and all demands are incident on one face, then again the cut condition is sufficient for routing demands. We consider a common generalization of these settings where the end points of each demand are on the same face of the planar graph. We show that if the source sink pairs on each face of the graph are such that sources and sinks appear contiguously on the cycle bounding the face, then the flow cut gap is at most 3. We come up with a notion of approximating demands on a face by convex combination of laminar demands to prove this result.
Cite as

Nikhil Kumar. Multicommodity Flows in Planar Graphs with Demands on Faces. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 41:1-41:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kumar:LIPIcs.ISAAC.2020.41,
  author =	{Kumar, Nikhil},
  title =	{{Multicommodity Flows in Planar Graphs with Demands on Faces}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{41:1--41:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.41},
  URN =		{urn:nbn:de:0030-drops-133857},
  doi =		{10.4230/LIPIcs.ISAAC.2020.41},
  annote =	{Keywords: Combinatorial Optimization, Multicommodity Flow, Network Design}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.55
Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow

Authors: Naveen Garg and Nikhil Kumar

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

Abstract
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.
Cite as

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-dev.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}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.63
Non-Clairvoyant Precedence Constrained Scheduling

Authors: Naveen Garg, Anupam Gupta, Amit Kumar, and Sahil Singla

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2018.24
A 5-Approximation for Universal Facility Location

Authors: Manisha Bansal, Naveen Garg, and Neelima Gupta

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2018.25
On Fair Division for Indivisible Items

Authors: Bhaskar Ray Chaudhury, Yun Kuen Cheung, Jugal Garg, Naveen Garg, Martin Hoefer, and Kurt Mehlhorn

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

Abstract
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.
Cite as

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-dev.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}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015
LIPIcs, Volume 40, APPROX/RANDOM'15, Complete Volume

Authors: Naveen Garg, Klaus Jansen, Anup Rao, and José D. P. Rolim

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

Abstract
LIPIcs, Volume 40, APPROX/RANDOM'15, Complete Volume
Cite as

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-dev.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}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.i
Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors

Authors: Naveen Garg, Klaus Jansen, Anup Rao, and José D. P. Rolim

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

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

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.1
On Guillotine Cutting Sequences

Authors: Fidaa Abed, Parinya Chalermsook, José Correa, Andreas Karrenbauer, Pablo Pérez-Lantero, José A. Soto, and Andreas Wiese

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

Abstract
Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.
Cite as

Fidaa Abed, Parinya Chalermsook, José Correa, Andreas Karrenbauer, Pablo Pérez-Lantero, José A. Soto, and Andreas Wiese. On Guillotine Cutting Sequences. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{abed_et_al:LIPIcs.APPROX-RANDOM.2015.1,
  author =	{Abed, Fidaa and Chalermsook, Parinya and Correa, Jos\'{e} and Karrenbauer, Andreas and P\'{e}rez-Lantero, Pablo and Soto, Jos\'{e} A. and Wiese, Andreas},
  title =	{{On Guillotine Cutting Sequences}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{1--19},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.1},
  URN =		{urn:nbn:de:0030-drops-52917},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.1},
  annote =	{Keywords: Guillotine cuts, Rectangles, Squares, Independent Sets, Packing}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.20
Approximate Nearest Neighbor Search in Metrics of Planar Graphs

Authors: Ittai Abraham, Shiri Chechik, Robert Krauthgamer, and Udi Wieder

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

Abstract
We investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G=(V,E) on m vertices and a small "dataset" D \subset V of size n << m, so that given a query point q \in V, one can quickly approximate dist(q,D) (the distance from q to its closest vertex in D) and find a vertex a \in D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices. For planar graphs G with maximum degree Delta, we show how to efficiently construct a compact data structure -- of size ~O(n(Delta+1/epsilon)) -- that answers (1+epsilon)-NNS queries in time ~O(Delta+1/epsilon). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into l_2. We complement our algorithmic result by lower bounds showing that the access to an exact distance oracle (rather than an approximate one) and the dependency on Delta (in query time) are both essential.
Cite as

Ittai Abraham, Shiri Chechik, Robert Krauthgamer, and Udi Wieder. Approximate Nearest Neighbor Search in Metrics of Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 20-42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{abraham_et_al:LIPIcs.APPROX-RANDOM.2015.20,
  author =	{Abraham, Ittai and Chechik, Shiri and Krauthgamer, Robert and Wieder, Udi},
  title =	{{Approximate Nearest Neighbor Search in Metrics of Planar Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{20--42},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.20},
  URN =		{urn:nbn:de:0030-drops-52923},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.20},
  annote =	{Keywords: Data Structures, Nearest Neighbor Search, Planar Graphs, Planar Metrics, Planar Separator}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.43
How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking

Authors: Anna Adamaszek, Parinya Chalermsook, and Andreas Wiese

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

Abstract
In the Maximum Weight Independent Set of Rectangles (MWISR) problem, we are given a collection of weighted axis-parallel rectangles in the plane. Our goal is to compute a maximum weight subset of pairwise non-overlapping rectangles. Due to its various applications, as well as connections to many other problems in computer science, MWISR has received a lot of attention from the computational geometry and the approximation algorithms community. However, despite being extensively studied, MWISR remains not very well understood in terms of polynomial time approximation algorithms, as there is a large gap between the upper and lower bounds, i.e., O(log n\ loglog n) v.s. NP-hardness. Another important, poorly understood question is whether one can color rectangles with at most O(omega(R)) colors where omega(R) is the size of a maximum clique in the intersection graph of a set of input rectangles R. Asplund and Grünbaum obtained an upper bound of O(omega(R)^2) about 50 years ago, and the result has remained asymptotically best. This question is strongly related to the integrality gap of the canonical LP for MWISR. In this paper, we settle above three open problems in a relaxed model where we are allowed to shrink the rectangles by a tiny bit (rescaling them by a factor of 1-delta for an arbitrarily small constant delta > 0. Namely, in this model, we show (i) a PTAS for MWISR and (ii) a coloring with O(omega(R)) colors which implies a constant upper bound on the integrality gap of the canonical LP. For some applications of MWISR the possibility to shrink the rectangles has a natural, well-motivated meaning. Our results can be seen as an evidence that the shrinking model is a promising way to relax a geometric problem for the purpose of better algorithmic results.
Cite as

Anna Adamaszek, Parinya Chalermsook, and Andreas Wiese. How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 43-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{adamaszek_et_al:LIPIcs.APPROX-RANDOM.2015.43,
  author =	{Adamaszek, Anna and Chalermsook, Parinya and Wiese, Andreas},
  title =	{{How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{43--60},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.43},
  URN =		{urn:nbn:de:0030-drops-52936},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.43},
  annote =	{Keywords: Approximation algorithms, independent set, resource augmentation, rectangle intersection graphs, PTAS}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.61
Non-Uniform Robust Network Design in Planar Graphs

Authors: David Adjiashvili

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

Abstract
Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on Bulk-Robustness, a model recently introduced (Adjiashvili, Stiller, Zenklusen 2015) for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used by Adjiashvili et al. to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.
Cite as

David Adjiashvili. Non-Uniform Robust Network Design in Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 61-77, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{adjiashvili:LIPIcs.APPROX-RANDOM.2015.61,
  author =	{Adjiashvili, David},
  title =	{{Non-Uniform Robust Network Design in Planar Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{61--77},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.61},
  URN =		{urn:nbn:de:0030-drops-52948},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.61},
  annote =	{Keywords: Robust optimization, Network design, Planar graph, Approximation algorithm, LP rounding}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.78
Large Supports are Required for Well-Supported Nash Equilibria

Authors: Yogesh Anbalagan, Hao Huang, Shachar Lovett, Sergey Norin, Adrian Vetta, and Hehui Wu

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

Abstract
We prove that for any constant k and any epsilon < 1, there exist bimatrix win-lose games for which every epsilon-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess epsilon-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou and Myers.
Cite as

Yogesh Anbalagan, Hao Huang, Shachar Lovett, Sergey Norin, Adrian Vetta, and Hehui Wu. Large Supports are Required for Well-Supported Nash Equilibria. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 78-84, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{anbalagan_et_al:LIPIcs.APPROX-RANDOM.2015.78,
  author =	{Anbalagan, Yogesh and Huang, Hao and Lovett, Shachar and Norin, Sergey and Vetta, Adrian and Wu, Hehui},
  title =	{{Large Supports are Required for Well-Supported Nash Equilibria}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{78--84},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.78},
  URN =		{urn:nbn:de:0030-drops-52959},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.78},
  annote =	{Keywords: bimatrix games, well-supported Nash equilibria}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.85
Minimizing Maximum Flow-time on Related Machines

Authors: Nikhil Bansal and Bouke Cloostermans

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

Abstract
We consider the online problem of minimizing the maximum flow-time on related machines. This is a natural generalization of the extensively studied makespan minimization problem to the setting where jobs arrive over time. Interestingly, natural algorithms such as Greedy or Slow-fit that work for the simpler identical machines case or for makespan minimization on related machines, are not O(1)-competitive. Our main result is a new O(1)-competitive algorithm for the problem. Previously, O(1)-competitive algorithms were known only with resource augmentation, and in fact no O(1) approximation was known even in the offline case.
Cite as

Nikhil Bansal and Bouke Cloostermans. Minimizing Maximum Flow-time on Related Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 85-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bansal_et_al:LIPIcs.APPROX-RANDOM.2015.85,
  author =	{Bansal, Nikhil and Cloostermans, Bouke},
  title =	{{Minimizing Maximum Flow-time on Related Machines}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{85--95},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.85},
  URN =		{urn:nbn:de:0030-drops-52964},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.85},
  annote =	{Keywords: Related machines scheduling, Maximum flow-time minimization, On-line algorithm, Approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.96
A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs

Authors: Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein

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

Abstract
We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time is a location on the real line. At each integer time, a convex function arrives online. In response, the online algorithm picks a new location. The cost paid by the online algorithm for this response is the distance moved plus the value of the function at the final destination. The objective is then to minimize the aggregate cost over all time. The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.
Cite as

Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein. A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 96-109, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bansal_et_al:LIPIcs.APPROX-RANDOM.2015.96,
  author =	{Bansal, Nikhil and Gupta, Anupam and Krishnaswamy, Ravishankar and Pruhs, Kirk and Schewior, Kevin and Stein, Cliff},
  title =	{{A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{96--109},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.96},
  URN =		{urn:nbn:de:0030-drops-52970},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.96},
  annote =	{Keywords: Stochastic, Scheduling}
}
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