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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server.
Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an O(log n)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm.
We also consider the case of non-convex cost functions. Using a simple 𝓁₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.

Nikhil Bansal and Christian Coester. Online Metric Allocation and Time-Varying Regularization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ESA.2022.13, author = {Bansal, Nikhil and Coester, Christian}, title = {{Online Metric Allocation and Time-Varying Regularization}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.13}, URN = {urn:nbn:de:0030-drops-169515}, doi = {10.4230/LIPIcs.ESA.2022.13}, annote = {Keywords: Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

The k-server conjecture, first posed by Manasse, McGeoch and Sleator in 1988, states that a k-competitive deterministic algorithm for the k-server problem exists. It is conjectured that the work function algorithm (WFA) achieves this guarantee, a multi-purpose algorithm with applications to various online problems. This has been shown for several special cases: k = 2, (k+1)-point metrics, (k+2)-point metrics, the line metric, weighted star metrics, and k = 3 in the Manhattan plane.
The known proofs of these results are based on potential functions tied to each particular special case, thus requiring six different potential functions for the six cases. We present a single potential function proving k-competitiveness of WFA for all these cases. We also use this potential to show k-competitiveness of WFA on multiray spaces and for k = 3 on trees. While the DoubleCoverage algorithm was known to be k-competitive for these latter cases, it has been open for WFA. Our potential captures a type of lazy adversary and thus shows that in all settled cases, the worst-case adversary is lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the lazy adversary would resolve the k-server conjecture.
To our major surprise, this is not the case, as we show (using connections to the k-taxi problem) that our potential fails for three servers on the circle. Thus, our potential highlights laziness of the adversary as a fundamental property that is shared by all settled cases but violated in general. On the one hand, this weakens our confidence in the validity of the k-server conjecture. On the other hand, if the k-server conjecture holds, then we believe it can be proved by a variant of our potential.

Christian Coester and Elias Koutsoupias. Towards the k-Server Conjecture: A Unifying Potential, Pushing the Frontier to the Circle. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 57:1-57:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{coester_et_al:LIPIcs.ICALP.2021.57, author = {Coester, Christian and Koutsoupias, Elias}, title = {{Towards the k-Server Conjecture: A Unifying Potential, Pushing the Frontier to the Circle}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {57:1--57:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.57}, URN = {urn:nbn:de:0030-drops-141263}, doi = {10.4230/LIPIcs.ICALP.2021.57}, annote = {Keywords: Online algorithms, competitive analysis, k-server, work function algorithm} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) f_t: A_t → B_t between subsets A_t and B_t of the metric space. To serve it, the algorithm has to go to a point a_t ∈ A_t, paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm’s state to f_t(a_t). Such transformations can model, e.g., changes to the environment that are outside of an algorithm’s control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the k-taxi problem.
We show that for α-Lipschitz transformations, the competitive ratio is Θ(α)^{n-2} on n-point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the k-taxi problem, we prove a competitive ratio of Õ((nlog k)²). For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists.
The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M, what is the required cardinality of an extension M̂ ⊇ M where each partial isometry on M extends to an automorphism? We give partial answers for special cases.

Sébastien Bubeck, Niv Buchbinder, Christian Coester, and Mark Sellke. Metrical Service Systems with Transformations. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bubeck_et_al:LIPIcs.ITCS.2021.21, author = {Bubeck, S\'{e}bastien and Buchbinder, Niv and Coester, Christian and Sellke, Mark}, title = {{Metrical Service Systems with Transformations}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {21:1--21:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.21}, URN = {urn:nbn:de:0030-drops-135608}, doi = {10.4230/LIPIcs.ITCS.2021.21}, annote = {Keywords: Online algorithms, competitive analysis, metrical task systems, k-taxi} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We study online competitive algorithms for the line chasing problem in Euclidean spaces R^d, where the input consists of an initial point P_0 and a sequence of lines X_1, X_2, ..., X_m, revealed one at a time. At each step t, when the line X_t is revealed, the algorithm must determine a point P_t in X_t. An online algorithm is called c-competitive if for any input sequence the path P_0, P_1 , ..., P_m it computes has length at most c times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets.
To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We improve this bound by providing a simple 3-competitive algorithm for any dimension d. We complement this bound by a matching lower bound for algorithms that are memoryless in the sense of our algorithm, and a lower bound of 1.5358 for arbitrary algorithms. The latter bound also improves upon the previous lower bound of sqrt{2}~=1.412 for convex body chasing in 2 dimensions.

Marcin Bienkowski, Jarosław Byrka, Marek Chrobak, Christian Coester, Łukasz Jeż, and Elias Koutsoupias. Better Bounds for Online Line Chasing. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bienkowski_et_al:LIPIcs.MFCS.2019.8, author = {Bienkowski, Marcin and Byrka, Jaros{\l}aw and Chrobak, Marek and Coester, Christian and Je\.{z}, {\L}ukasz and Koutsoupias, Elias}, title = {{Better Bounds for Online Line Chasing}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.8}, URN = {urn:nbn:de:0030-drops-109521}, doi = {10.4230/LIPIcs.MFCS.2019.8}, annote = {Keywords: convex body chasing, line chasing, competitive analysis} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which implies the same lower bound for the (h,k)-server problem even when k>>h and holds also for the line metric; the previous known bounds were 2.4 for general metric spaces and 2 for the line. For weighted trees and layered graphs we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval away from the original position of the servers. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

Christian Coester, Elias Koutsoupias, and Philip Lazos. The Infinite Server Problem. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{coester_et_al:LIPIcs.ICALP.2017.14, author = {Coester, Christian and Koutsoupias, Elias and Lazos, Philip}, title = {{The Infinite Server Problem}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.14}, URN = {urn:nbn:de:0030-drops-74563}, doi = {10.4230/LIPIcs.ICALP.2017.14}, annote = {Keywords: Online Algorithms, k-Server, Resource Augmentation} }

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