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

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.

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

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

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.

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} }

Document

**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

In 2005 Li~et~al. gave a \(\phi\)-competitive deterministic online algorithm for scheduling of packets with agreeable deadlines~\cite{DBLP:conf/soda/LiSS05} with a very interesting analysis. This is known to be optimal due to a lower bound by Hajek~\cite{Hajek-det-lb}. We claim that the algorithm by Li~et~al. can be slightly simplified, while retaining its competitive ratio. Then we introduce randomness to the modified algorithm and argue that the competitive ratio against oblivious adversary is at most (\frac{4}{3}\). Note that this still leaves a gap between the best known lower bound of \(\frac{5}{4}\) by Chin~et~al.~\cite{DBLP:journals/algorithmica/ChinF03} for randomized algorithms against oblivious adversary.

Lukasz Jez. Randomized Algorithm for Agreeable Deadlines Packet Scheduling. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 489-500, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{jez:LIPIcs.STACS.2010.2479, author = {Jez, Lukasz}, title = {{Randomized Algorithm for Agreeable Deadlines Packet Scheduling}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {489--500}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2479}, URN = {urn:nbn:de:0030-drops-24795}, doi = {10.4230/LIPIcs.STACS.2010.2479}, annote = {Keywords: Online algorithms, scheduling, buffer management} }

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