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DOI: 10.4230/LIPIcs.STACS.2021.14

A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location

Authors: Marcin Bienkowski, Björn Feldkord, and Paweł Schmidt

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract

In the online non-metric variant of the facility location problem, there is a given graph consisting of a set F of facilities (each with a certain opening cost), a set C of potential clients, and weighted connections between them. The online part of the input is a sequence of clients from C, and in response to any requested client, an online algorithm may open an additional subset of facilities and must connect the given client to an open facility. We give an online, polynomial-time deterministic algorithm for this problem, with a competitive ratio of O(log |F| ⋅ (log |C| + log log |F|)). The result is optimal up to loglog factors. Our algorithm improves over the O((log |C| + log |F|) ⋅ (log |C| + log log |F|))-competitive construction that first reduces the facility location instance to a set cover one and then later solves such instance using the deterministic algorithm by Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical scenario where |F| ≪ |C|. We achieve this by a more direct approach: we design an algorithm for a fractional relaxation of the non-metric facility location problem with clustered facilities. To handle the constraints of such non-covering LP, we combine the dual fitting and multiplicative weight updates approach. By maintaining certain additional monotonicity properties of the created fractional solution, we can handle the dependencies between facilities and connections in a rounding routine. Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the first deterministic algorithm for the online node-weighted Steiner tree problem. The resulting competitive ratio is O(log k ⋅ log² 𝓁) on graphs of 𝓁 nodes and k terminals.

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Marcin Bienkowski, Björn Feldkord, and Paweł Schmidt. A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bienkowski_et_al:LIPIcs.STACS.2021.14,
  author =	{Bienkowski, Marcin and Feldkord, Bj\"{o}rn and Schmidt, Pawe{\l}},
  title =	{{A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.14},
  URN =		{urn:nbn:de:0030-drops-136598},
  doi =		{10.4230/LIPIcs.STACS.2021.14},
  annote =	{Keywords: Online algorithms, deterministic rounding, linear programming, facility location, set cover}
}

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DOI: 10.4230/LIPIcs.ISAAC.2019.14

Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics

Authors: Marcin Bienkowski, Łukasz Jeż, and Paweł Schmidt

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract

The generalized k-server problem is an extension of the weighted k-server problem, which in turn extends the classic k-server problem. In the generalized k-server problem, each of k servers s_1, ..., s_k remains in its own metric space M_i. A request is a tuple (r_1,...,r_k), where r_i in M_i, and to service it, an algorithm needs to move at least one server s_i to the point r_i. The objective is to minimize the total distance traveled by all servers. In this paper, we focus on the generalized k-server problem for the case where all M_i are uniform metrics. We show an O(k^2 * log k)-competitive randomized algorithm improving over a recent result by Bansal et al. [SODA 2018], who gave an O(k^3 * log k)-competitive algorithm. To this end, we define an abstract online problem, called Hydra game, and we show that a randomized solution of low cost to this game implies a randomized algorithm to the generalized k-server problem with low competitive ratio. We also show that no randomized algorithm can achieve competitive ratio lower than Omega(k), thus improving the lower bound of Omega(k / log^2 k) by Bansal et al.

Cite as

Marcin Bienkowski, Łukasz Jeż, and Paweł Schmidt. Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bienkowski_et_al:LIPIcs.ISAAC.2019.14,
  author =	{Bienkowski, Marcin and Je\.{z}, {\L}ukasz and Schmidt, Pawe{\l}},
  title =	{{Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.14},
  URN =		{urn:nbn:de:0030-drops-115104},
  doi =		{10.4230/LIPIcs.ISAAC.2019.14},
  annote =	{Keywords: k-server, generalized k-server, competitive analysis}
}

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Documents authored by Schmidt, Paweł

A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location

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Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics

Abstract

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