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Documents authored by Grandoni, Fabrizio


Document
Track A: Algorithms, Complexity and Games
A 4/3 Approximation for 2-Vertex-Connectivity

Authors: Miguel Bosch-Calvo, Fabrizio Grandoni, and Afrouz Jabal Ameli

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph G. Our goal is to find a subgraph S of G with the minimum number of edges which is 2-vertex-connected, namely S remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is 10/7 by Heeger and Vygen [SIDMA'17] (improving on earlier results by Khuller and Vishkin [STOC'92] and Garg, Vempala and Singla [SODA'93]). Here we present an improved 4/3 approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are "almost" 3-vertex-connected. The latter reduction might be helpful in future work.

Cite as

Miguel Bosch-Calvo, Fabrizio Grandoni, and Afrouz Jabal Ameli. A 4/3 Approximation for 2-Vertex-Connectivity. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 29:1-29:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{boschcalvo_et_al:LIPIcs.ICALP.2023.29,
  author =	{Bosch-Calvo, Miguel and Grandoni, Fabrizio and Jabal Ameli, Afrouz},
  title =	{{A 4/3 Approximation for 2-Vertex-Connectivity}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{29:1--29:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.29},
  URN =		{urn:nbn:de:0030-drops-180813},
  doi =		{10.4230/LIPIcs.ICALP.2023.29},
  annote =	{Keywords: Algorithm, Network Design, Vertex-Connectivity, Approximation}
}
Document
Unsplittable Euclidean Capacitated Vehicle Routing: A (2+ε)-Approximation Algorithm

Authors: Fabrizio Grandoni, Claire Mathieu, and Hang Zhou

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
In the unsplittable capacitated vehicle routing problem, we are given a metric space with a vertex called depot and a set of vertices called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the depot such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1. Our main result is a polynomial-time (2+ε)-approximation algorithm for this problem in the two-dimensional Euclidean plane, i.e., for the special case where the terminals and the depot are associated with points in the Euclidean plane and their distances are defined accordingly. This improves on recent work by Blauth, Traub, and Vygen [IPCO'21] and Friggstad, Mousavi, Rahgoshay, and Salavatipour [IPCO'22].

Cite as

Fabrizio Grandoni, Claire Mathieu, and Hang Zhou. Unsplittable Euclidean Capacitated Vehicle Routing: A (2+ε)-Approximation Algorithm. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 63:1-63:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{grandoni_et_al:LIPIcs.ITCS.2023.63,
  author =	{Grandoni, Fabrizio and Mathieu, Claire and Zhou, Hang},
  title =	{{Unsplittable Euclidean Capacitated Vehicle Routing: A (2+\epsilon)-Approximation Algorithm}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{63:1--63:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.63},
  URN =		{urn:nbn:de:0030-drops-175660},
  doi =		{10.4230/LIPIcs.ITCS.2023.63},
  annote =	{Keywords: capacitated vehicle routing, approximation algorithms, Euclidean plane}
}
Document
APPROX
Approximation Algorithms for Demand Strip Packing

Authors: Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, and Kamyar Khodamoradi

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)


Abstract
In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+ε)-approximation algorithm for any constant ε > 0. We also achieve best-possible approximation factors for some relevant special cases.

Cite as

Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, and Kamyar Khodamoradi. Approximation Algorithms for Demand Strip Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 20:1-20:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{galvez_et_al:LIPIcs.APPROX/RANDOM.2021.20,
  author =	{G\'{a}lvez, Waldo and Grandoni, Fabrizio and Ameli, Afrouz Jabal and Khodamoradi, Kamyar},
  title =	{{Approximation Algorithms for Demand Strip Packing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{20:1--20:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.20},
  URN =		{urn:nbn:de:0030-drops-147130},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.20},
  annote =	{Keywords: Strip Packing, Two-Dimensional Packing, Approximation Algorithms}
}
Document
Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back

Authors: Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)


Abstract
Unsplittable flow on a path (UFP) is an important and well-studied problem. We are given a path with capacities on its edges, and a set of tasks where for each task we are given a demand, a subpath, and a weight. The goal is to select the set of tasks of maximum total weight whose total demands do not exceed the capacity on any edge. UFP admits an (1+ε)-approximation with a running time of n^{O_{ε}(poly(log n))}, i.e., a QPTAS {[}Bansal et al., STOC 2006; Batra et al., SODA 2015{]} and it is considered an important open problem to construct a PTAS. To this end, in a series of papers polynomial time approximation algorithms have been developed, which culminated in a (5/3+ε)-approximation {[}Grandoni et al., STOC 2018{]} and very recently an approximation ratio of (1+1/(e+1)+ε) < 1.269 {[}Grandoni et al., 2020{]}. In this paper, we address the search for a PTAS from a different angle: we present a faster (1+ε)-approximation with a running time of only n^{O_{ε}(log log n)}. We first give such a result in the relaxed setting of resource augmentation and then transform it to an algorithm without resource augmentation. For this, we present a framework which transforms algorithms for (a slight generalization of) UFP under resource augmentation in a black-box manner into algorithms for UFP without resource augmentation, with only negligible loss.

Cite as

Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 49:1-49:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{grandoni_et_al:LIPIcs.ESA.2021.49,
  author =	{Grandoni, Fabrizio and M\"{o}mke, Tobias and Wiese, Andreas},
  title =	{{Faster (1+\epsilon)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.49},
  URN =		{urn:nbn:de:0030-drops-146301},
  doi =		{10.4230/LIPIcs.ESA.2021.49},
  annote =	{Keywords: Approximation Algorithms, Unsplittable Flow, Dynamic Programming}
}
Document
Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

Authors: Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas Wiese

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)


Abstract
In the 2-Dimensional Knapsack problem (2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is 17/9+ε < 1.89 and there is a (3/2+ε)-approximation algorithm if we are allowed to rotate items by 90 degrees [Gálvez et al., FOCS 2017]. In this paper, we give (4/3+ε)-approximation algorithms in polynomial time for both cases, assuming that all input data are integers polynomially bounded in n. Gálvez et al.’s algorithm for 2DK partitions the knapsack into a constant number of rectangular regions plus one L-shaped region and packs items into those in a structured way. We generalize this approach by allowing up to a constant number of more general regions that can have the shape of an L, a U, a Z, a spiral, and more, and therefore obtain an improved approximation ratio. In particular, we present an algorithm that computes the essentially optimal structured packing into these regions.

Cite as

Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas Wiese. Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 39:1-39:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{galvez_et_al:LIPIcs.SoCG.2021.39,
  author =	{G\'{a}lvez, Waldo and Grandoni, Fabrizio and Khan, Arindam and Ram{\'\i}rez-Romero, Diego and Wiese, Andreas},
  title =	{{Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{39:1--39:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.39},
  URN =		{urn:nbn:de:0030-drops-138386},
  doi =		{10.4230/LIPIcs.SoCG.2021.39},
  annote =	{Keywords: Approximation algorithms, two-dimensional knapsack, geometric packing}
}
Document
Complete Volume
LIPIcs, Volume 173, ESA 2020, Complete Volume

Authors: Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders

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


Abstract
LIPIcs, Volume 173, ESA 2020, Complete Volume

Cite as

Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders. LIPIcs, Volume 173, ESA 2020, Complete Volume. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 1-1598, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@Proceedings{grandoni_et_al:LIPIcs.ESA.2020,
  title =	{{LIPIcs, Volume 173, ESA 2020, Complete Volume}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{1--1598},
  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},
  URN =		{urn:nbn:de:0030-drops-128651},
  doi =		{10.4230/LIPIcs.ESA.2020},
  annote =	{Keywords: LIPIcs, Volume 173, ESA 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders

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


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders. Front Matter, Table of Contents, Preface, Conference Organization. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 0:i-0:xx, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{grandoni_et_al:LIPIcs.ESA.2020.0,
  author =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{0:i--0:xx},
  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.2020.0},
  URN =		{urn:nbn:de:0030-drops-128669},
  doi =		{10.4230/LIPIcs.ESA.2020.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
APPROX
A Tight (3/2+ε) Approximation for Skewed Strip Packing

Authors: Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, Klaus Jansen, Arindam Khan, and Malin Rau

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)


Abstract
In the Strip Packing problem, we are given a vertical half-strip [0,W]× [0,+∞) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. Strip Packing generalizes classical well-studied problems such as Makespan Minimization on identical machines (when rectangle widths are identical) and Bin Packing (when rectangle heights are identical). It has applications in manufacturing, scheduling and energy consumption in smart grids among others. It is NP-hard to approximate this problem within a factor (3/2-ε) for any constant ε > 0 by a simple reduction from the Partition problem. The current best approximation factor for Strip Packing is (5/3+ε) by Harren et al. [Computational Geometry '14], and it is achieved with a fairly complex algorithm and analysis. It seems plausible that Strip Packing admits a (3/2+ε)-approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ > 0, we call large the rectangles with width at least δ W and height at least δ OPT, and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, it is NP-hard to approximate within a factor (3/2-ε) and we provide an (almost) tight (3/2+ε)-approximation algorithm.

Cite as

Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, Klaus Jansen, Arindam Khan, and Malin Rau. A Tight (3/2+ε) Approximation for Skewed Strip Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 44:1-44:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{galvez_et_al:LIPIcs.APPROX/RANDOM.2020.44,
  author =	{G\'{a}lvez, Waldo and Grandoni, Fabrizio and Ameli, Afrouz Jabal and Jansen, Klaus and Khan, Arindam and Rau, Malin},
  title =	{{A Tight (3/2+\epsilon) Approximation for Skewed Strip Packing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{44:1--44:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.44},
  URN =		{urn:nbn:de:0030-drops-126478},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.44},
  annote =	{Keywords: strip packing, approximation algorithm}
}
Document
Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

Authors: Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)


Abstract
The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability: - In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k. - In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA'04]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR. For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f'(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time. For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC'17].

Cite as

Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 53:1-53:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{grandoni_et_al:LIPIcs.ESA.2019.53,
  author =	{Grandoni, Fabrizio and Kratsch, Stefan and Wiese, Andreas},
  title =	{{Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.53},
  URN =		{urn:nbn:de:0030-drops-111741},
  doi =		{10.4230/LIPIcs.ESA.2019.53},
  annote =	{Keywords: parameterized approximation, parameterized intractability, independent set of rectangles, geometric knapsack}
}
Document
Packing Cars into Narrow Roads: PTASs for Limited Supply Highway

Authors: Fabrizio Grandoni and Andreas Wiese

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)


Abstract
In the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers’s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA'11, SICOMP'16]. In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway. As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem.

Cite as

Fabrizio Grandoni and Andreas Wiese. Packing Cars into Narrow Roads: PTASs for Limited Supply Highway. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{grandoni_et_al:LIPIcs.ESA.2019.54,
  author =	{Grandoni, Fabrizio and Wiese, Andreas},
  title =	{{Packing Cars into Narrow Roads: PTASs for Limited Supply Highway}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.54},
  URN =		{urn:nbn:de:0030-drops-111751},
  doi =		{10.4230/LIPIcs.ESA.2019.54},
  annote =	{Keywords: approximation algorithms, pricing problems, highway problem, unsplittable flow on a path}
}
Document
When the Optimum is also Blind: a New Perspective on Universal Optimization

Authors: Marek Adamczyk, Fabrizio Grandoni, Stefano Leonardi, and Michal Wlodarczyk

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
Consider the following variant of the set cover problem. We are given a universe U={1,...,n} and a collection of subsets C = {S_1,...,S_m} where each S_i is a subset of U. For every element u from U we need to find a set phi(u) from collection C such that u belongs to phi(u). Once we construct and fix the mapping phi from U to C a subset X from the universe U is revealed, and we need to cover all elements from X with exactly phi(X), that is {phi(u)}_{all u from X}. The goal is to find a mapping such that the cover phi(X) is as cheap as possible. This is an example of a universal problem where the solution has to be created before the actual instance to deal with is revealed. Such problems appear naturally in some settings when we need to optimize under uncertainty and it may be actually too expensive to begin finding a good solution once the input starts being revealed. A rich body of work was devoted to investigate such problems under the regime of worst case analysis, i.e., when we measure how good the solution is by looking at the worst-case ratio: universal solution for a given instance vs optimum solution for the same instance. As the universal solution is significantly more constrained, it is typical that such a worst-case ratio is actually quite big. One way to give a viewpoint on the problem that would be less vulnerable to such extreme worst-cases is to assume that the instance, for which we will have to create a solution, will be drawn randomly from some probability distribution. In this case one wants to minimize the expected value of the ratio: universal solution vs optimum solution. Here the bounds obtained are indeed smaller than when we compare to the worst-case ratio. But even in this case we still compare apples to oranges as no universal solution is able to construct the optimum solution for every possible instance. What if we would compare our approximate universal solution against an optimal universal solution that obeys the same rules as we do? We show that under this viewpoint, but still in the stochastic variant, we can indeed obtain better bounds than in the expected ratio model. For example, for the set cover problem we obtain $H_n$ approximation which matches the approximation ratio from the classic deterministic setup. Moreover, we show this for all possible probability distributions over $U$ that have a polynomially large carrier, while all previous results pertained to a model in which elements were sampled independently. Our result is based on rounding a proper configuration IP that captures the optimal universal solution, and using tools from submodular optimization. The same basic approach leads to improved approximation algorithms for other related problems, including Vertex Cover, Edge Cover, Directed Steiner Tree, Multicut, and Facility Location.

Cite as

Marek Adamczyk, Fabrizio Grandoni, Stefano Leonardi, and Michal Wlodarczyk. When the Optimum is also Blind: a New Perspective on Universal Optimization. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 35:1-35:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{adamczyk_et_al:LIPIcs.ICALP.2017.35,
  author =	{Adamczyk, Marek and Grandoni, Fabrizio and Leonardi, Stefano and Wlodarczyk, Michal},
  title =	{{When the Optimum is also Blind: a New Perspective on Universal Optimization}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{35:1--35:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.35},
  URN =		{urn:nbn:de:0030-drops-74436},
  doi =		{10.4230/LIPIcs.ICALP.2017.35},
  annote =	{Keywords: approximation algorithms, stochastic optimization, submodularity}
}
Document
Preserving Distances in Very Faulty Graphs

Authors: Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain replacement paths around the faulted set. Unfortunately, the complexity of the interaction between replacement paths blows up significantly, even from 1 to 2 faults, and the structure of optimal preservers and spanners is poorly understood. In particular, no nontrivial bounds for preservers and additive spanners are known when the number of faults is bigger than 2. Even the answer to the following innocent question is completely unknown: what is the worst-case size of a preserver for a single pair of nodes in the presence of f edge faults? There are no super-linear lower bounds, nor subquadratic upper bounds for f>2. In this paper we make substantial progress on this and other fundamental questions: - We present the first truly sub-quadratic size fault-tolerant single-pair preserver in unweighted (possibly directed) graphs: for any n node graph and any fixed number f of faults, O~(fn^{2-1/2^f}) size suffices. Our result also generalizes to the single-source (all targets) case, and can be used to build new fault-tolerant additive spanners (for all pairs). - The size of the above single-pair preserver grows to O(n^2) for increasing f. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size O(n^{2-eps}) for \eps>0, then the additive error has to be \Omega(eps f). This surprisingly matches known upper bounds in the literature. - For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is Theta(n^2) for f > 1 in both directed and undirected graphs, while for f=1 the size is Theta(n) in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound. Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure.

Cite as

Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. Preserving Distances in Very Faulty Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 73:1-73:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{bodwin_et_al:LIPIcs.ICALP.2017.73,
  author =	{Bodwin, Greg and Grandoni, Fabrizio and Parter, Merav and Vassilevska Williams, Virginia},
  title =	{{Preserving Distances in Very Faulty Graphs}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{73:1--73:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.73},
  URN =		{urn:nbn:de:0030-drops-74906},
  doi =		{10.4230/LIPIcs.ICALP.2017.73},
  annote =	{Keywords: Fault Tolerance, shortest paths, replacement paths}
}
Document
Improved Pseudo-Polynomial-Time Approximation for Strip Packing

Authors: Waldo Gálvez, Fabrizio Grandoni, Salvatore Ingala, and Arindam Khan

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


Abstract
We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P!=NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a (7/5+epsilon)-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm (though a PPT approximation scheme might exist). In this paper we make further progress on the PPT approximability of strip packing, by presenting a (4/3+epsilon)-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the "empty space" left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by 90 degrees, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well.

Cite as

Waldo Gálvez, Fabrizio Grandoni, Salvatore Ingala, and Arindam Khan. Improved Pseudo-Polynomial-Time Approximation for Strip Packing. 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. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{galvez_et_al:LIPIcs.FSTTCS.2016.9,
  author =	{G\'{a}lvez, Waldo and Grandoni, Fabrizio and Ingala, Salvatore and Khan, Arindam},
  title =	{{Improved Pseudo-Polynomial-Time Approximation for Strip Packing}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{9:1--9:14},
  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.2016.9},
  URN =		{urn:nbn:de:0030-drops-68445},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.9},
  annote =	{Keywords: approximation algorithms, strip packing, rectangle packing, cutting-stock problem}
}
Document
Complete Volume
LIPIcs, Volume 49, FUN'16, Complete Volume

Authors: Erik D. Demaine and Fabrizio Grandoni

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)


Abstract
LIPIcs, Volume 49, FUN'16, Complete Volume

Cite as

Erik D. Demaine and Fabrizio Grandoni. LIPIcs, Volume 49, FUN'16, Complete Volume. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@Proceedings{demaine_et_al:LIPIcs.FUN.2016,
  title =	{{LIPIcs, Volume 49, FUN'16, Complete Volume}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016},
  URN =		{urn:nbn:de:0030-drops-60579},
  doi =		{10.4230/LIPIcs.FUN.2016},
  annote =	{Keywords: Nonnumerical Algorithms and Problems, Discrete Mathematics, Complexity Measures and Classes}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Erik D. Demaine and Fabrizio Grandoni

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

Erik D. Demaine and Fabrizio Grandoni. Front Matter, Table of Contents, Preface, Conference Organization. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 0:i-0:xvi, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{demaine_et_al:LIPIcs.FUN.2016.0,
  author =	{Demaine, Erik D. and Grandoni, Fabrizio},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{0:i--0:xvi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.0},
  URN =		{urn:nbn:de:0030-drops-58624},
  doi =		{10.4230/LIPIcs.FUN.2016.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
On Pairwise Spanners

Authors: Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)


Abstract
Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(.) is a subgraph H \subseteq G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the u-v distance only for pairs (u,v) in a given set P \subseteq V x V. Such P-spanners were studied before [Coppersmith,Elkin'05] only in the special case that f(.) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(.)). In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P|log n)^{1/4}) with f(d) = d + 4 log n. Alternatively, for any epsislon > 0, there exists a P-spanner of size O(n|P|^{1/4} sqrt{(log n) / epsilon}) with f(d) = (1 + epsilon)d + 4. We also consider the relevant special case that there is a critical set of nodes S \subseteq V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S x S)-spanner of size O(n sqrt{|S|}) with f(d) = d + 2, and an (S x V)-spanner of size O(n sqrt{|S| log n}) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.

Cite as

Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha. On Pairwise Spanners. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 209-220, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)


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@InProceedings{cygan_et_al:LIPIcs.STACS.2013.209,
  author =	{Cygan, Marek and Grandoni, Fabrizio and Kavitha, Telikepalli},
  title =	{{On Pairwise Spanners}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{209--220},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.209},
  URN =		{urn:nbn:de:0030-drops-39353},
  doi =		{10.4230/LIPIcs.STACS.2013.209},
  annote =	{Keywords: Undirected graphs, shortest paths, additive spanners, distance preservers}
}
Document
Approximation Algorithms for Union and Intersection Covering Problems

Authors: Marek Cygan, Fabrizio Grandoni, Stefano Leonardi, Marcin Mucha, Marcin Pilipczuk, and Piotr Sankowski

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)


Abstract
In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here, nodes represent requests whereas edges correspond to items. In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems: - in an union multi-layer problem, a request is satisfied if it is satisfied in at least one layer; - in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers. To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, Intersection k-MST can formalize the problem of providing both electricity and water to at least k users.

Cite as

Marek Cygan, Fabrizio Grandoni, Stefano Leonardi, Marcin Mucha, Marcin Pilipczuk, and Piotr Sankowski. Approximation Algorithms for Union and Intersection Covering Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 28-40, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)


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@InProceedings{cygan_et_al:LIPIcs.FSTTCS.2011.28,
  author =	{Cygan, Marek and Grandoni, Fabrizio and Leonardi, Stefano and Mucha, Marcin and Pilipczuk, Marcin and Sankowski, Piotr},
  title =	{{Approximation Algorithms for Union and Intersection Covering Problems}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{28--40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.28},
  URN =		{urn:nbn:de:0030-drops-33213},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.28},
  annote =	{Keywords: Approximation algorithms, Partial covering problems}
}
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