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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.21

On Minimizing Generalized Makespan on Unrelated Machines

Authors: Nikhil Ayyadevara, Nikhil Bansal, and Milind Prabhu

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract

We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given n jobs and m machines and each job j has arbitrary processing time p_{ij} on machine i. Additionally, there is a general symmetric monotone norm ψ_i for each machine i, that determines the load on machine i as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load. Recently, Deng, Li, and Rabani [Deng et al., 2023] gave a 3 approximation for GMP when the ψ_i are top-k norms, and they ask the question whether an O(1) approximation exists for general norms ψ? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant δ > 0, such that GMP is Ω(log^δ n) hard to approximate. We also give an Ω(log^{1/2} n) integrality gap for the natural configuration LP.

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Nikhil Ayyadevara, Nikhil Bansal, and Milind Prabhu. On Minimizing Generalized Makespan on Unrelated Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ayyadevara_et_al:LIPIcs.APPROX/RANDOM.2023.21,
  author =	{Ayyadevara, Nikhil and Bansal, Nikhil and Prabhu, Milind},
  title =	{{On Minimizing Generalized Makespan on Unrelated Machines}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{21:1--21:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.21},
  URN =		{urn:nbn:de:0030-drops-188462},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.21},
  annote =	{Keywords: Hardness of Approximation, Generalized Makespan}
}

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DOI: 10.4230/LIPIcs.ESA.2021.9

The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential

Authors: Nikhil Ayyadevara and Ashish Chiplunkar

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

Abstract

The weighted k-server problem is a natural generalization of the k-server problem in which the cost incurred in moving a server is the distance traveled times the weight of the server. Even after almost three decades since the seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem remains poorly understood even on the simplest class of metric spaces - the uniform metric spaces. In particular, in the case of randomized algorithms against the oblivious adversary, neither a better upper bound that the doubly exponential deterministic upper bound, nor a better lower bound than the logarithmic lower bound of unweighted k-server, is known. In this paper, we make significant progress towards understanding the randomized competitive ratio of weighted k-server on uniform metrics. We cut down the triply exponential gap between the upper and lower bound to a singly exponential gap by proving that the competitive ratio is at least exponential in k, substantially improving on the previously known lower bound of about ln k.

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Nikhil Ayyadevara and Ashish Chiplunkar. The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ayyadevara_et_al:LIPIcs.ESA.2021.9,
  author =	{Ayyadevara, Nikhil and Chiplunkar, Ashish},
  title =	{{The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{9:1--9:11},
  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.2021.9},
  URN =		{urn:nbn:de:0030-drops-145904},
  doi =		{10.4230/LIPIcs.ESA.2021.9},
  annote =	{Keywords: weighted k-server, competitive analysis}
}

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Documents authored by Ayyadevara, Nikhil

On Minimizing Generalized Makespan on Unrelated Machines

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The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential

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