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Documents authored by Böhm, Martin

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.2022.45
Online Facility Location with Linear Delay

Authors: Marcin Bienkowski, Martin Böhm, Jarosław Byrka, and Jan Marcinkowski

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
In the problem of online facility location with delay, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a certain penalty: each client incurs a waiting cost (equal to the difference between its arrival and its connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. That is, an algorithm needs to balance three types of costs: cost of opening facilities, costs of connecting clients, and the waiting costs of clients. We study a natural variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay, where clients may connect to a facility only at its opening time. We present an O(1)-competitive deterministic algorithm for the two-sided delay variant. Our approach is an extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location. To this end, we substantially simplify the part of the original argument in which a bound on the sequence of factor-revealing LPs is derived. We then show how to transform our O(1)-competitive algorithm for the two-sided delay variant to O(log n / log log n)-competitive deterministic algorithm for one-sided delays. This improves the known O(log n) bound by Azar and Touitou [FOCS 2020]. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios.
Cite as

Marcin Bienkowski, Martin Böhm, Jarosław Byrka, and Jan Marcinkowski. Online Facility Location with Linear Delay. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bienkowski_et_al:LIPIcs.APPROX/RANDOM.2022.45,
  author =	{Bienkowski, Marcin and B\"{o}hm, Martin and Byrka, Jaros{\l}aw and Marcinkowski, Jan},
  title =	{{Online Facility Location with Linear Delay}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{45:1--45:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.45},
  URN =		{urn:nbn:de:0030-drops-171678},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.45},
  annote =	{Keywords: online facility location, network design problems, facility location with delay, JMS algorithm, competitive analysis, factor revealing LP}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.18
Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics

Authors: Martin Böhm, Ruben Hoeksma, Nicole Megow, Lukas Nölke, and Bertrand Simon

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
We consider the problem of computing a Steiner tree of minimum cost under a k-hop constraint which requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time n^O(k). For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension.
Cite as

Martin Böhm, Ruben Hoeksma, Nicole Megow, Lukas Nölke, and Bertrand Simon. Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bohm_et_al:LIPIcs.MFCS.2020.18,
  author =	{B\"{o}hm, Martin and Hoeksma, Ruben and Megow, Nicole and N\"{o}lke, Lukas and Simon, Bertrand},
  title =	{{Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.18},
  URN =		{urn:nbn:de:0030-drops-126870},
  doi =		{10.4230/LIPIcs.MFCS.2020.18},
  annote =	{Keywords: k-hop Steiner tree, dynamic programming, bounded treewidth}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2016.21
Online Packet Scheduling with Bounded Delay and Lookahead

Authors: Martin Böhm, Marek Chrobak, Lukasz Jez, Fei Li, Jirí Sgall, and Pavel Veselý

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
We study the online bounded-delay packet scheduling problem (PacketScheduling), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission. The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature, yet its optimal competitive ratio remains unknown: the best upper bound is 1.828 [Englert and Westermann, SODA 2007], still quite far from the best lower bound of phi approx 1.618 [Hajek, CISS 2001; Andelman et al, SODA 2003; Chin and Fung, Algorithmica, 2003]. In the variant of PacketScheduling with s-bounded instances, each packet can be scheduled in at most s consecutive slots, starting at its release time. The lower bound of phi applies even to the special case of 2-bounded instances, and a phi-competitive algorithm for 3-bounded instances was given in [Chin et al, JDA, 2006]. Improving that result, and addressing a question posed by Goldwasser [SIGACT News, 2010], we present a phi-competitive algorithm for 4-bounded instances. We also study a variant of PacketScheduling where an online algorithm has the additional power of 1-lookahead, knowing at time t which packets will arrive at time t+1. For PacketScheduling with 1-lookahead restricted to 2-bounded instances, we present an online algorithm with competitive ratio frac{1}{2}(sqrt{13} - 1) approx 1.303 and we prove a nearly tight lower bound of frac{1}{4}(1 + sqrt{17}) approx 1.281.
Cite as

Martin Böhm, Marek Chrobak, Lukasz Jez, Fei Li, Jirí Sgall, and Pavel Veselý. Online Packet Scheduling with Bounded Delay and Lookahead. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bohm_et_al:LIPIcs.ISAAC.2016.21,
  author =	{B\"{o}hm, Martin and Chrobak, Marek and Jez, Lukasz and Li, Fei and Sgall, Jir{\'\i} and Vesel\'{y}, Pavel},
  title =	{{Online Packet Scheduling with Bounded Delay and Lookahead}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{21:1--21:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.21},
  URN =		{urn:nbn:de:0030-drops-67901},
  doi =		{10.4230/LIPIcs.ISAAC.2016.21},
  annote =	{Keywords: buffer management, online scheduling, online algorithm, lookahead}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.12
Online Algorithms for Multi-Level Aggregation

Authors: Marcin Bienkowski, Martin Böhm, Jaroslaw Byrka, Marek Chrobak, Christoph Dürr, Lukas Folwarczny, Lukasz Jez, Jiri Sgall, Nguyen Kim Thang, and Pavel Vesely

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4*2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting.
Cite as

Marcin Bienkowski, Martin Böhm, Jaroslaw Byrka, Marek Chrobak, Christoph Dürr, Lukas Folwarczny, Lukasz Jez, Jiri Sgall, Nguyen Kim Thang, and Pavel Vesely. Online Algorithms for Multi-Level Aggregation. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bienkowski_et_al:LIPIcs.ESA.2016.12,
  author =	{Bienkowski, Marcin and B\"{o}hm, Martin and Byrka, Jaroslaw and Chrobak, Marek and D\"{u}rr, Christoph and Folwarczny, Lukas and Jez, Lukasz and Sgall, Jiri and Kim Thang, Nguyen and Vesely, Pavel},
  title =	{{Online Algorithms for Multi-Level Aggregation}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{12:1--12:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.12},
  URN =		{urn:nbn:de:0030-drops-63637},
  doi =		{10.4230/LIPIcs.ESA.2016.12},
  annote =	{Keywords: algorithmic aspects of networks, online algorithms, scheduling and resource allocation}
}
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