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DOI: 10.4230/LIPIcs.ESA.2023.17

Approximating Min-Diameter: Standard and Bichromatic

Authors: Aaron Berger, Jenny Kaufmann, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract

The min-diameter of a directed graph G is a measure of the largest distance between nodes. It is equal to the maximum min-distance d_{min}(u,v) across all pairs u,v ∈ V(G), where d_{min}(u,v) = min(d(u,v), d(v,u)). Min-diameter approximation in directed graphs has attracted attention recently as an offshoot of the classical and well-studied diameter approximation problem. Our work provides a 3/2-approximation algorithm for min-diameter in DAGs running in time O(m^{1.426} n^{0.288}), and a faster almost-3/2-approximation variant which runs in time O(m^{0.713} n). (An almost-α-approximation algorithm determines the min-diameter to within a multiplicative factor of α plus constant additive error.) This is the first known algorithm to solve 3/2-approximation for min-diameter in sparse DAGs in truly subquadratic time O(m^{2-ε}) for ε > 0; previously only a 2-approximation was known. By a conditional lower bound result of [Abboud et al, SODA 2016], a better than 3/2-approximation can't be achieved in truly subquadratic time under the Strong Exponential Time Hypothesis (SETH), so our result is conditionally tight. We additionally obtain a new conditional lower bound for min-diameter approximation in general directed graphs, showing that under SETH, one cannot achieve an approximation factor below 2 in truly subquadratic time. Our work also presents the first study of approximating bichromatic min-diameter, which is the maximum min-distance between oppositely colored vertices in a 2-colored graph. We show that SETH implies that in DAGs, a better than 2 approximation cannot be achieved in truly subquadratic time, and that in general graphs, an approximation within a factor below 5/2 is similarly out of reach. We then obtain an O(m)-time algorithm which determines if bichromatic min-diameter is finite, and an almost-2-approximation algorithm for bichromatic min-diameter with runtime Õ(min(m^{4/3} n^{1/3}, m^{1/2} n^{3/2})).

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Aaron Berger, Jenny Kaufmann, and Virginia Vassilevska Williams. Approximating Min-Diameter: Standard and Bichromatic. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berger_et_al:LIPIcs.ESA.2023.17,
  author =	{Berger, Aaron and Kaufmann, Jenny and Vassilevska Williams, Virginia},
  title =	{{Approximating Min-Diameter: Standard and Bichromatic}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.17},
  URN =		{urn:nbn:de:0030-drops-186705},
  doi =		{10.4230/LIPIcs.ESA.2023.17},
  annote =	{Keywords: diameter, min distances, fine-grained, approximation algorithm}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2022.19

Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost

Authors: Aaron Berger, William Kuszmaul, Adam Polak, Jonathan Tidor, and Nicole Wein

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract

We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of w workers and a multiset T ⊆ [t] of |T| = w tasks, a memoryless worker-task assignment function is any function ϕ that assigns the workers [w] to the tasks T based only on the current value of T. The assignment function ϕ is said to have switching cost at most k if, for every task multiset T, changing the contents of T by one task changes ϕ(T) by at most k worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost. In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020). We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost O(log w log (wt)) and an explicit construction that achieves switching cost polylog (wt). We also prove a super-constant lower bound on switching cost: we show that for any value of w, there exists a value of t for which the optimal switching cost is w. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of w. Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.

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Aaron Berger, William Kuszmaul, Adam Polak, Jonathan Tidor, and Nicole Wein. Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berger_et_al:LIPIcs.ICALP.2022.19,
  author =	{Berger, Aaron and Kuszmaul, William and Polak, Adam and Tidor, Jonathan and Wein, Nicole},
  title =	{{Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.19},
  URN =		{urn:nbn:de:0030-drops-163608},
  doi =		{10.4230/LIPIcs.ICALP.2022.19},
  annote =	{Keywords: Distributed Task Allocation, Metric Embeddings, Probabilistic Method}
}

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Documents authored by Berger, Aaron

Approximating Min-Diameter: Standard and Bichromatic

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Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost

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