7 Search Results for "Sharma, Eklavya"

Document
DOI: 10.4230/LIPIcs.ESA.2024.58
Random-Order Online Independent Set of Intervals and Hyperrectangles

Authors: Mohit Garg, Debajyoti Kar, and Arindam Khan

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Maximum Independent Set of Hyperrectangles problem, we are given a set of n (possibly overlapping) d-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For d = 1, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for d-dimensional hyperrectangles, polynomial time (log n)^{O(d)}-approximation algorithms are known [Chalermsook and Chuzhoy, 2009]. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an Ω(n) lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis (see the survey by Gupta and Singla [Anupam Gupta and Sahil Singla, 2020]). Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple (log n)^{O(d)}-competitive algorithm for d-dimensional hyperrectangles in this model, which runs in O_d̃(n) time. Our approach also yields (log n)^{O(d)}-competitive algorithms in the random-order model for more general objects such as d-dimensional fat objects and ellipsoids. Furthermore, all our competitiveness guarantees hold with high probability, and not just in expectation.
Cite as

Mohit Garg, Debajyoti Kar, and Arindam Khan. Random-Order Online Independent Set of Intervals and Hyperrectangles. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{garg_et_al:LIPIcs.ESA.2024.58,
  author =	{Garg, Mohit and Kar, Debajyoti and Khan, Arindam},
  title =	{{Random-Order Online Independent Set of Intervals and Hyperrectangles}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{58:1--58:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.58},
  URN =		{urn:nbn:de:0030-drops-211298},
  doi =		{10.4230/LIPIcs.ESA.2024.58},
  annote =	{Keywords: Online Algorithms, Random-Order Model, Maximum Independent Set of Rectangles, Hyperrectangles, Fat Objects, Interval Scheduling}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.23
Splitting-Off in Hypergraphs

Authors: Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász [Lov{á}sz, 1974; Lov{á}sz, 1993] and Mader [Mader, 1978] showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature [Lovász, 1976; Mader, 1978; Frank, 1993; Frank and Király, 2002; Király and Lau, 2008; Frank, 1992; Goemans and Bertsimas, 1993; Frank, 1994; Bang-Jensen et al., 1995; Frank, 2011; Nagamochi and Ibaraki, 2008; Nagamochi et al., 1997; Henzinger and Williamson, 1996; Goemans, 2001; Jordán, 2003; Kriesell, 2003; Jain et al., 2003; Chan et al., 2011; Bhalgat et al., 2008; Lau, 2007; Chekuri and Shepherd, 2008; Nägele and Zenklusen, 2020; Blauth and Nägele, 2023]. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of k-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau [Király and Lau, 2008]). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.
Cite as

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni. Splitting-Off in Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berczi_et_al:LIPIcs.ICALP.2024.23,
  author =	{B\'{e}rczi, Krist\'{o}f and Chandrasekaran, Karthekeyan and Kir\'{a}ly, Tam\'{a}s and Kulkarni, Shubhang},
  title =	{{Splitting-Off in Hypergraphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.23},
  URN =		{urn:nbn:de:0030-drops-201660},
  doi =		{10.4230/LIPIcs.ICALP.2024.23},
  annote =	{Keywords: Hypergraphs, Hypergraph Connectivity, Splitting-off, Constructive Characterizations, Hypergraph Orientations, Submodular Functions, Combinatorial Optimization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.8
Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects

Authors: Pritam Acharya, Sujoy Bhore, Aaryan Gupta, Arindam Khan, Bratin Mondal, and Andreas Wiese

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We study the geometric knapsack problem in which we are given a set of d-dimensional objects (each with associated profits) and the goal is to find the maximum profit subset that can be packed non-overlappingly into a given d-dimensional (unit hypercube) knapsack. Even if d = 2 and all input objects are disks, this problem is known to be NP-hard [Demaine, Fekete, Lang, 2010]. In this paper, we give polynomial time (1+ε)-approximation algorithms for the following types of input objects in any constant dimension d: - disks and hyperspheres, - a class of fat convex polygons that generalizes regular k-gons for k ≥ 5 (formally, polygons with a constant number of edges, whose lengths are in a bounded range, and in which each angle is strictly larger than π/2), - arbitrary fat convex objects that are sufficiently small compared to the knapsack. We remark that in our PTAS for disks and hyperspheres, we output the computed set of objects, but for a O_ε(1) of them we determine their coordinates only up to an exponentially small error. However, it is not clear whether there always exists a (1+ε)-approximate solution that uses only rational coordinates for the disks' centers. We leave this as an open problem which is related to well-studied geometric questions in the realm of circle packing.
Cite as

Pritam Acharya, Sujoy Bhore, Aaryan Gupta, Arindam Khan, Bratin Mondal, and Andreas Wiese. Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{acharya_et_al:LIPIcs.ICALP.2024.8,
  author =	{Acharya, Pritam and Bhore, Sujoy and Gupta, Aaryan and Khan, Arindam and Mondal, Bratin and Wiese, Andreas},
  title =	{{Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.8},
  URN =		{urn:nbn:de:0030-drops-201511},
  doi =		{10.4230/LIPIcs.ICALP.2024.8},
  annote =	{Keywords: Approximation Algorithms, Polygon Packing, Circle Packing, Sphere Packing, Geometric Knapsack, Resource Augmentation}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.18
Nash Equilibria of Two-Player Matrix Games Repeated Until Collision

Authors: Aniket Murhekar and Eklavya Sharma

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract
We introduce and initiate the study of a natural class of repeated two-player matrix games, called Repeated-Until-Collision (RUC) games. In each round, both players simultaneously pick an action from a common action set {1, 2, … , n}. Depending on their chosen actions, they derive payoffs given by n × n matrices A and B, respectively. If their actions collide (i.e., they pick the same action), the game ends, otherwise, it proceeds to the next round. Both players want to maximize their total payoff until the game ends. RUC games can be interpreted as pursuit-evasion games or repeated hide-and-seek games. They also generalize hand cricket, a popular game among children in India. We show that under mild assumptions on the payoff matrices, every RUC game admits a Nash equilibrium (NE). Moreover, we show the existence of a stationary NE, where each player chooses their action according to a probability distribution over the action set that does not change across rounds. Remarkably, we show that all NE are effectively the same as the stationary NE, thus showing that RUC games admit an almost unique NE. Lastly, we also show how to compute (approximate) NE for RUC games.
Cite as

Aniket Murhekar and Eklavya Sharma. Nash Equilibria of Two-Player Matrix Games Repeated Until Collision. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{murhekar_et_al:LIPIcs.FSTTCS.2023.18,
  author =	{Murhekar, Aniket and Sharma, Eklavya},
  title =	{{Nash Equilibria of Two-Player Matrix Games Repeated Until Collision}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.18},
  URN =		{urn:nbn:de:0030-drops-193913},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.18},
  annote =	{Keywords: Two player games, Nash equilibrium, Repeated games}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.23
Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

Authors: Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i-th item has width w(i), height h(i), and d nonnegative weights v₁(i), v₂(i), …, v_d(i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j-th weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well-studied in practice, approximation algorithms for this problem have rarely been explored. We first obtain two simple algorithms for GVBP having asymptotic approximation ratios 6(d+1) and 3(1 + ln(d+1) + ε). We then extend the Round-and-Approx (R&A) framework [Bansal et al., 2009; Bansal and Khan, 2014] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of 2(1 + ln((d+4)/2)) + ε for GVBP, which improves to 2.919+ε for the special case of d = 1. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids.
Cite as

Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas. Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{khan_et_al:LIPIcs.FSTTCS.2022.23,
  author =	{Khan, Arindam and Sharma, Eklavya and Sreenivas, K. V. N.},
  title =	{{Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{23:1--23:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.23},
  URN =		{urn:nbn:de:0030-drops-174151},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.23},
  annote =	{Keywords: Bin packing, rectangle packing, multidimensional packing, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.32
Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins

Authors: Eklavya Sharma

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
We explore approximation algorithms for the d-dimensional geometric bin packing problem (dBP). Caprara [Caprara, 2008] gave a harmonic-based algorithm for dBP having an asymptotic approximation ratio (AAR) of (T_∞)^{d-1} (where T_∞ ≈ 1.691). However, their algorithm doesn't allow items to be rotated. This is in contrast to some common applications of dBP, like packing boxes into shipping containers. We give approximation algorithms for dBP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm having AAR T_∞^d. We next give a more sophisticated harmonic-based algorithm, which we call HGaP_k, having AAR (T_∞)^{d-1}(1+ε). This gives an AAR of roughly 2.860 + ε for 3BP with rotations, which improves upon the best-known AAR of 4.5. In addition, we study the multiple-choice bin packing problem that generalizes the rotational case. Here we are given n sets of d-dimensional cuboidal items and we have to choose exactly one item from each set and then pack the chosen items. Our algorithms also work for the multiple-choice bin packing problem. We also give fast and simple approximation algorithms for the multiple-choice versions of dD strip packing and dD geometric knapsack.
Cite as

Eklavya Sharma. Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{sharma:LIPIcs.FSTTCS.2021.32,
  author =	{Sharma, Eklavya},
  title =	{{Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{32:1--32:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.32},
  URN =		{urn:nbn:de:0030-drops-155432},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.32},
  annote =	{Keywords: Geometric bin packing}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.22
Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items

Authors: Arindam Khan and Eklavya Sharma

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

Abstract
In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let λ be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by λ opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ≤ λ ≤ 1.692. Bansal and Khan [SODA'14] conjectured that λ = 4/3. The conjecture, if true, will imply a (4/3+ε)-approximation algorithm for 2BP. According to convention, for a given constant δ > 0, a rectangle is large if both its height and width are at least δ, and otherwise it is called skewed. We make progress towards the conjecture by showing λ = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on λ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS.
Cite as

Arindam Khan and Eklavya Sharma. Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{khan_et_al:LIPIcs.APPROX/RANDOM.2021.22,
  author =	{Khan, Arindam and Sharma, Eklavya},
  title =	{{Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{22:1--22:23},
  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.2021.22},
  URN =		{urn:nbn:de:0030-drops-147151},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.22},
  annote =	{Keywords: Geometric bin packing, guillotine separability, approximation algorithms}
}
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