150 Search Results for "Kumar, Amit"

Volume

LIPIcs, Volume 317

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

APPROX/RANDOM 2024, August 28-30, 2024, London School of Economics, London, UK

Editors: Amit Kumar and Noga Ron-Zewi

Volume

LIPIcs, Volume 13

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

FSTTCS 2011, December 12-14, 2011, Mumbai, India

Editors: Supratik Chakraborty and Amit Kumar

Document
Complete Volume
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024
LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume

Authors: Amit Kumar and Noga Ron-Zewi

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume
Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 1-1496, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Proceedings{kumar_et_al:LIPIcs.APPROX/RANDOM.2024,
  title =	{{LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{1--1496},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024},
  URN =		{urn:nbn:de:0030-drops-209921},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024},
  annote =	{Keywords: LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Amit Kumar and Noga Ron-Zewi

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

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

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 0:i-0:xxvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kumar_et_al:LIPIcs.APPROX/RANDOM.2024.0,
  author =	{Kumar, Amit and Ron-Zewi, Noga},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{0:i--0:xxvi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.0},
  URN =		{urn:nbn:de:0030-drops-209933},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.1
A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP

Authors: Susanne Armbruster, Matthias Mnich, and Martin Nägele

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We present a new (3/2 + 1/e)-approximation algorithm for the Ordered Traveling Salesperson Problem (Ordered TSP). Ordered TSP is a variant of the classic metric Traveling Salesperson Problem (TSP) where a specified subset of vertices needs to appear on the output Hamiltonian cycle in a given order, and the task is to compute a cheapest such cycle. Our approximation guarantee of approximately 1.868 holds with respect to the value of a natural new linear programming (LP) relaxation for Ordered TSP. Our result significantly improves upon the previously best known guarantee of 5/2 for this problem and thereby considerably reduces the gap between approximability of Ordered TSP and metric TSP. Our algorithm is based on a decomposition of the LP solution into weighted trees that serve as building blocks in our tour construction.
Cite as

Susanne Armbruster, Matthias Mnich, and Martin Nägele. A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{armbruster_et_al:LIPIcs.APPROX/RANDOM.2024.1,
  author =	{Armbruster, Susanne and Mnich, Matthias and N\"{a}gele, Martin},
  title =	{{A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{1:1--1:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.1},
  URN =		{urn:nbn:de:0030-drops-209943},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.1},
  annote =	{Keywords: Travelling Salesperson Problem, precedence constraints, linear programming, approximation algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.2
Online Time-Windows TSP with Predictions

Authors: Shuchi Chawla and Dimitris Christou

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
In the Time-Windows TSP (TW-TSP) we are given requests at different locations on a network; each request is endowed with a reward and an interval of time; the goal is to find a tour that visits as much reward as possible during the corresponding time window. For the online version of this problem, where each request is revealed at the start of its time window, no finite competitive ratio can be obtained. We consider a version of the problem where the algorithm is presented with predictions of where and when the online requests will appear, without any knowledge of the quality of this side information. Vehicle routing problems such as the TW-TSP can be very sensitive to errors or changes in the input due to the hard time-window constraints, and it is unclear whether imperfect predictions can be used to obtain a finite competitive ratio. We show that good performance can be achieved by explicitly building slack into the solution. Our main result is an online algorithm that achieves a competitive ratio logarithmic in the diameter of the underlying network, matching the performance of the best offline algorithm to within factors that depend on the quality of the provided predictions. The competitive ratio degrades smoothly as a function of the quality and we show that this dependence is tight within constant factors.
Cite as

Shuchi Chawla and Dimitris Christou. Online Time-Windows TSP with Predictions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chawla_et_al:LIPIcs.APPROX/RANDOM.2024.2,
  author =	{Chawla, Shuchi and Christou, Dimitris},
  title =	{{Online Time-Windows TSP with Predictions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.2},
  URN =		{urn:nbn:de:0030-drops-209954},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.2},
  annote =	{Keywords: Travelling Salesman Problem, Predictions, Learning-Augmented Algorithms, Approximation}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.3
Degrees and Network Design: New Problems and Approximations

Authors: Michael Dinitz, Guy Kortsarz, and Shi Li

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints that might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the 𝓁_p-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the 𝓁_∞-norm) and the number of edges (the 𝓁₁-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogarithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Guy Kortsarz and Zeev Nutov, 2022] and [X. Guo et al., 2022].
Cite as

Michael Dinitz, Guy Kortsarz, and Shi Li. Degrees and Network Design: New Problems and Approximations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dinitz_et_al:LIPIcs.APPROX/RANDOM.2024.3,
  author =	{Dinitz, Michael and Kortsarz, Guy and Li, Shi},
  title =	{{Degrees and Network Design: New Problems and Approximations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{3:1--3:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.3},
  URN =		{urn:nbn:de:0030-drops-209969},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.3},
  annote =	{Keywords: Network Design, Degrees}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.4
Hybrid k-Clustering: Blending k-Median and k-Center

Authors: Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clustering problem, given a set P of points in ℝ^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L₁-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r = 0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given ε > 0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+ε of the optimum, and it operates in time 2^{(kd/ε)^𝒪(1)} ⋅ n^𝒪(1). Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clustering.
Cite as

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Hybrid k-Clustering: Blending k-Median and k-Center. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fomin_et_al:LIPIcs.APPROX/RANDOM.2024.4,
  author =	{Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Hybrid k-Clustering: Blending k-Median and k-Center}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.4},
  URN =		{urn:nbn:de:0030-drops-209975},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.4},
  annote =	{Keywords: clustering, k-center, k-median, Euclidean space, fpt approximation}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.5
Asynchronous Majority Dynamics on Binomial Random Graphs

Authors: Divyarthi Mohan and Paweł Prałat

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We study information aggregation in networks when agents interact to learn a binary state of the world. Initially each agent privately observes an independent signal which is correct with probability 1/2+δ for some δ > 0. At each round, a node is selected uniformly at random to update their public opinion to match the majority of their neighbours (breaking ties in favour of their initial private signal). Our main result shows that for sparse and connected binomial random graphs G(n,p) the process stabilizes in a correct consensus in 𝒪(nlog² n/log log n) steps with high probability. In fact, when log n/n ≪ p = o(1) the process terminates at time T^ = (1+o(1))nlog n, where T^ is the first time when all nodes have been selected at least once. However, in dense binomial random graphs with p = Ω(1), there is an information cascade where the process terminates in the incorrect consensus with probability bounded away from zero.
Cite as

Divyarthi Mohan and Paweł Prałat. Asynchronous Majority Dynamics on Binomial Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mohan_et_al:LIPIcs.APPROX/RANDOM.2024.5,
  author =	{Mohan, Divyarthi and Pra{\l}at, Pawe{\l}},
  title =	{{Asynchronous Majority Dynamics on Binomial Random Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.5},
  URN =		{urn:nbn:de:0030-drops-209985},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.5},
  annote =	{Keywords: Opinion dynamics, Social learning, Stochastic processes, Random Graphs, Consensus}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.6
Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3

Authors: Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
In a disk graph, every vertex corresponds to a disk in ℝ² and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) 3-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a (3-α)-approximation algorithm for Bipartization on disk graphs, for some constant α > 0. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.
Cite as

Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lokshtanov_et_al:LIPIcs.APPROX/RANDOM.2024.6,
  author =	{Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Xue, Jie and Zehavi, Meirav},
  title =	{{Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.6},
  URN =		{urn:nbn:de:0030-drops-209990},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.6},
  annote =	{Keywords: bipartization, geometric intersection graphs, approximation algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.7
A Logarithmic Approximation of Linearly-Ordered Colourings

Authors: Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
A linearly ordered (LO) k-colouring of a hypergraph assigns to each vertex a colour from the set {0,1,…,k-1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO k-colouring of an LO 2-colourable 3-uniform hypergraph for any constant k ≥ 2 [STACS'21] but even the case k = 3 is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with O^*(√n) colours [ICALP'22] and an LO colouring with O^*(n^(1/3)) colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with O^*(n^(1/5)) colours. We present two simple polynomial-time algorithms that find an LO colouring with O(log₂(n)) colours, which is an exponential improvement.
Cite as

Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný. A Logarithmic Approximation of Linearly-Ordered Colourings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 7:1-7:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hastad_et_al:LIPIcs.APPROX/RANDOM.2024.7,
  author =	{H\r{a}stad, Johan and Martinsson, Bj\"{o}rn and Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav},
  title =	{{A Logarithmic Approximation of Linearly-Ordered Colourings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{7:1--7:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.7},
  URN =		{urn:nbn:de:0030-drops-210006},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.7},
  annote =	{Keywords: Linear ordered colouring, Hypergraph, Approximation, Promise Constraint Satisfaction Problems}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.8
Speed-Robust Scheduling Revisited

Authors: Josef Minařík and Jiří Sgall

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Speed-robust scheduling is the following two-stage problem of scheduling n jobs on m uniformly related machines. In the first stage, the algorithm receives the value of m and the processing times of n jobs; it has to partition the jobs into b groups called bags. In the second stage, the machine speeds are revealed and the bags are assigned to the machines, i.e., the algorithm produces a schedule where all the jobs in the same bag are assigned to the same machine. The objective is to minimize the makespan (the length of the schedule). The algorithm is compared to the optimal schedule and it is called ρ-robust, if its makespan is always at most ρ times the optimal one. Our main result is an improved bound for equal-size jobs for b = m. We give an upper bound of 1.6. This improves previous bound of 1.8 and it is almost tight in the light of previous lower bound of 1.58. Second, for infinitesimally small jobs, we give tight upper and lower bounds for the case when b ≥ m. This generalizes and simplifies the previous bounds for b = m. Finally, we introduce a new special case with relatively small jobs for which we give an algorithm whose robustness is close to that of infinitesimal jobs and thus gives better than 2-robust for a large class of inputs.
Cite as

Josef Minařík and Jiří Sgall. Speed-Robust Scheduling Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{minarik_et_al:LIPIcs.APPROX/RANDOM.2024.8,
  author =	{Mina\v{r}{\'\i}k, Josef and Sgall, Ji\v{r}{\'\i}},
  title =	{{Speed-Robust Scheduling Revisited}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.8},
  URN =		{urn:nbn:de:0030-drops-210010},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.8},
  annote =	{Keywords: scheduling, approximation algorithms, makespan, uniform speeds}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.9
On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms

Authors: Karthekeyan Chandrasekaran, Chandra Chekuri, Manuel R. Torres, and Weihao Zhu

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Dense subgraph discovery is an important problem in graph mining and network analysis with several applications. Two canonical polynomial-time solvable problems here are to find a maxcore (subgraph of maximum min degree) and to find a densest subgraph (subgraph of maximum average degree). Both of these problems can be solved in polynomial time. Veldt, Benson, and Kleinberg [Veldt et al., 2021] introduced the generalized p-mean densest subgraph problem which captures the maxcore problem when p = -∞ and the densest subgraph problem when p = 1. They observed that for p ≥ 1, the objective function is supermodular and hence the problem can be solved in polynomial time. In this work, we focus on the p-mean densest subgraph problem for p ∈ (-∞, 1). We prove that for every p ∈ (-∞,1), the problem is NP-hard, thus resolving an open question from [Veldt et al., 2021]. We also show that for every p ∈ (0,1), the weighted version of the problem is APX-hard. On the algorithmic front, we describe two simple 1/2-approximation algorithms for every p ∈ (-∞, 1). We complement the approximation algorithms by exhibiting non-trivial instances on which the algorithms simultaneously achieve an approximation factor of at most 1/2.
Cite as

Karthekeyan Chandrasekaran, Chandra Chekuri, Manuel R. Torres, and Weihao Zhu. On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chandrasekaran_et_al:LIPIcs.APPROX/RANDOM.2024.9,
  author =	{Chandrasekaran, Karthekeyan and Chekuri, Chandra and Torres, Manuel R. and Zhu, Weihao},
  title =	{{On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.9},
  URN =		{urn:nbn:de:0030-drops-210025},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.9},
  annote =	{Keywords: Densest subgraph problem, Hardness of approximation, Approximation algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.10
Improved Online Load Balancing with Known Makespan

Authors: Martin Böhm, Matej Lieskovský, Sören Schmitt, Jiří Sgall, and Rob van Stee

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We break the barrier of 3/2 for the problem of online load balancing with known makespan, also known as bin stretching. In this problem, m identical machines and the optimal makespan are given. The load of a machine is the total size of all the jobs assigned to it and the makespan is the maximum load of all the machines. Jobs arrive online and the goal is to assign each job to a machine while staying within a small factor (the competitive ratio) of the optimal makespan. We present an algorithm that maintains a competitive ratio of 139/93 < 1.495 for sufficiently large values of m, improving the previous bound of 3/2. The value 3/2 represents a natural bound for this problem: as long as the online bins are of size at least 3/2 of the offline bin, all items that fit at least two times in an offline bin have two nice properties. They fit three times in an online bin and a single such item can be packed together with an item of any size in an online bin. These properties are now both lost, which means that putting even one job on a wrong machine can leave some job unassigned at the end. It also makes it harder to determine good thresholds for the item types. This was one of the main technical issues in getting below 3/2. The analysis consists of an intricate mixture of size and weight arguments.
Cite as

Martin Böhm, Matej Lieskovský, Sören Schmitt, Jiří Sgall, and Rob van Stee. Improved Online Load Balancing with Known Makespan. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bohm_et_al:LIPIcs.APPROX/RANDOM.2024.10,
  author =	{B\"{o}hm, Martin and Lieskovsk\'{y}, Matej and Schmitt, S\"{o}ren and Sgall, Ji\v{r}{\'\i} and van Stee, Rob},
  title =	{{Improved Online Load Balancing with Known Makespan}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{10:1--10:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.10},
  URN =		{urn:nbn:de:0030-drops-210032},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.10},
  annote =	{Keywords: Online algorithms, bin stretching, bin packing}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.11
On the NP-Hardness Approximation Curve for Max-2Lin(2)

Authors: Björn Martinsson

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
In the Max-2Lin(2) problem you are given a system of equations on the form x_i + x_j ≡ b mod 2, and your objective is to find an assignment that satisfies as many equations as possible. Let c ∈ [0.5, 1] denote the maximum fraction of satisfiable equations. In this paper we construct a curve s (c) such that it is NP-hard to find a solution satisfying at least a fraction s of equations. This curve either matches or improves all of the previously known inapproximability NP-hardness results for Max-2Lin(2). In particular, we show that if c ⩾ 0.9232 then (1 - s(c))/(1 - c) > 1.48969, which improves the NP-hardness inapproximability constant for the min deletion version of Max-2Lin(2). Our work complements the work of O'Donnell and Wu that studied the same question assuming the Unique Games Conjecture. Similar to earlier inapproximability results for Max-2Lin(2), we use a gadget reduction from the (2^k - 1)-ary Hadamard predicate. Previous works used k ranging from 2 to 4. Our main result is a procedure for taking a gadget for some fixed k, and use it as a building block to construct better and better gadgets as k tends to infinity. Our method can be used to boost the result of both smaller gadgets created by hand (k = 3) or larger gadgets constructed using a computer (k = 4).
Cite as

Björn Martinsson. On the NP-Hardness Approximation Curve for Max-2Lin(2). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 11:1-11:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{martinsson:LIPIcs.APPROX/RANDOM.2024.11,
  author =	{Martinsson, Bj\"{o}rn},
  title =	{{On the NP-Hardness Approximation Curve for Max-2Lin(2)}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{11:1--11:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.11},
  URN =		{urn:nbn:de:0030-drops-210049},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.11},
  annote =	{Keywords: Inapproximability, NP-hardness, 2Lin(2), Max-Cut, Gadget}
}
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