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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.92

Online Demand Scheduling with Failovers

Authors: Konstantina Mellou, Marco Molinaro, and Rudy Zhou

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

Motivated by cloud computing applications, we study the problem of how to optimally deploy new hardware subject to both power and robustness constraints. To model the situation observed in large-scale data centers, we introduce the Online Demand Scheduling with Failover problem. There are m identical devices with capacity constraints. Demands come one-by-one and, to be robust against a device failure, need to be assigned to a pair of devices. When a device fails (in a failover scenario), each demand assigned to it is rerouted to its paired device (which may now run at increased capacity). The goal is to assign demands to the devices to maximize the total utilization subject to both the normal capacity constraints as well as these novel failover constraints. These latter constraints introduce new decision tradeoffs not present in classic assignment problems such as the Multiple Knapsack problem and AdWords. In the worst-case model, we design a deterministic ≈ 1/2-competitive algorithm, and show this is essentially tight. To circumvent this constant-factor loss, which represents substantial capital losses for big cloud providers, we consider the stochastic arrival model, where all demands come i.i.d. from an unknown distribution. In this model we design an algorithm that achieves sub-linear additive regret (i.e. as OPT or m increases, the multiplicative competitive ratio goes to 1). This requires a combination of different techniques, including a configuration LP with a non-trivial post-processing step and an online monotone matching procedure introduced by Rhee and Talagrand.

Cite as

Konstantina Mellou, Marco Molinaro, and Rudy Zhou. Online Demand Scheduling with Failovers. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 92:1-92:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mellou_et_al:LIPIcs.ICALP.2023.92,
  author =	{Mellou, Konstantina and Molinaro, Marco and Zhou, Rudy},
  title =	{{Online Demand Scheduling with Failovers}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{92:1--92:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.92},
  URN =		{urn:nbn:de:0030-drops-181443},
  doi =		{10.4230/LIPIcs.ICALP.2023.92},
  annote =	{Keywords: online algorithms, approximation algorithms, resource allocation}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2020.72

Knapsack Secretary with Bursty Adversary

Authors: Thomas Kesselheim and Marco Molinaro

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract

The random-order or secretary model is one of the most popular beyond-worst case model for online algorithms. While this model avoids the pessimism of the traditional adversarial model, in practice we cannot expect the input to be presented in perfectly random order. This has motivated research on best of both worlds (algorithms with good performance on both purely stochastic and purely adversarial inputs), or even better, on inputs that are a mix of both stochastic and adversarial parts. Unfortunately the latter seems much harder to achieve and very few results of this type are known. Towards advancing our understanding of designing such robust algorithms, we propose a random-order model with bursts of adversarial time steps. The assumption of burstiness of unexpected patterns is reasonable in many contexts, since changes (e.g. spike in a demand for a good) are often triggered by a common external event. We then consider the Knapsack Secretary problem in this model: there is a knapsack of size k (e.g., available quantity of a good), and in each of the n time steps an item comes with its value and size in [0,1] and the algorithm needs to make an irrevocable decision whether to accept or reject the item. We design an algorithm that gives an approximation of 1 - Õ(Γ/k) when the adversarial time steps can be covered by Γ ≥ √k intervals of size Õ(n/k). In particular, setting Γ = √k gives a (1 - O((ln² k)/√k))-approximation that is resistant to up to a (ln k)/√k-fraction of the items being adversarial, which is almost optimal even in the absence of adversarial items. Also, setting Γ = Ω̃(k) gives a constant approximation that is resistant to up to a constant fraction of items being adversarial. While the algorithm is a simple "primal" one it does not possess the crucial symmetry properties exploited in the traditional analyses. The strategy of our analysis is more robust and significantly different from previous ones, and we hope it can be useful for other beyond-worst-case models.

Cite as

Thomas Kesselheim and Marco Molinaro. Knapsack Secretary with Bursty Adversary. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 72:1-72:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kesselheim_et_al:LIPIcs.ICALP.2020.72,
  author =	{Kesselheim, Thomas and Molinaro, Marco},
  title =	{{Knapsack Secretary with Bursty Adversary}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{72:1--72:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.72},
  URN =		{urn:nbn:de:0030-drops-124798},
  doi =		{10.4230/LIPIcs.ICALP.2020.72},
  annote =	{Keywords: Beyond worst-case, secretary problem, random order, online algorithms, knapsack}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2020.32

Robust Algorithms for the Secretary Problem

Authors: Domagoj Bradac, Anupam Gupta, Sahil Singla, and Goran Zuzic

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract

In classical secretary problems, a sequence of n elements arrive in a uniformly random order, and we want to choose a single item, or a set of size K. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin’s popular 1/e-secretary algorithm is sensitive to even a single adversarial arrival: if the adversary gives one large bid at the beginning of the stream, the algorithm does not select any element at all. We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from [0,1]. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems? We show that selecting the highest value red set, or the single largest green element is not possible with even a small fraction of red items. However, on the positive side, we show that these are the only bad cases, by giving algorithms which get value comparable to the value of the optimal green set minus the largest green item. (This benchmark reminds us of regret minimization and digital auctions, where we subtract an additive term depending on the "scale" of the problem.) Specifically, we give an algorithm to pick K elements, which gets within (1-ε) factor of the above benchmark, as long as K ≥ poly(ε^{-1} log n). We extend this to the knapsack secretary problem, for large knapsack size K. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a poly log^* n-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle. We hope that this work will spur further research on robust algorithms for the secretary problem, and for other problems in sequential decision-making, where the existing algorithms are not robust and often tend to overfit to the model.

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Domagoj Bradac, Anupam Gupta, Sahil Singla, and Goran Zuzic. Robust Algorithms for the Secretary Problem. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 32:1-32:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bradac_et_al:LIPIcs.ITCS.2020.32,
  author =	{Bradac, Domagoj and Gupta, Anupam and Singla, Sahil and Zuzic, Goran},
  title =	{{Robust Algorithms for the Secretary Problem}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{32:1--32:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.32},
  URN =		{urn:nbn:de:0030-drops-117171},
  doi =		{10.4230/LIPIcs.ITCS.2020.32},
  annote =	{Keywords: stochastic optimization, robust optimization, secretary problem, matroid secretary, robust secretary}
}

Document

DOI: 10.4230/LIPIcs.ICALP.2018.71

Maximizing Profit with Convex Costs in the Random-order Model

Authors: Anupam Gupta, Ruta Mehta, and Marco Molinaro

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract

Suppose a set of requests arrives online: each request gives some value v_i if accepted, but requires using some amount of each of d resources. Our cost is a convex function of the vector of total utilization of these d resources. Which requests should be accept to maximize our profit, i.e., the sum of values of the accepted demands, minus the convex cost? We consider this problem in the random-order a.k.a. secretary model, and show an O(d)-competitive algorithm for the case where the convex cost function is also supermodular. If the set of accepted demands must also be independent in a given matroid, we give an O(d^3 alpha)-competitive algorithm for the supermodular case, and an improved O(d^2 alpha) if the convex cost function is also separable. Here alpha is the competitive ratio of the best algorithm for the submodular secretary problem. These extend and improve previous results known for this problem. Our techniques are simple but use powerful ideas from convex duality, which give clean interpretations of existing work, and allow us to give the extensions and improvements.

Cite as

Anupam Gupta, Ruta Mehta, and Marco Molinaro. Maximizing Profit with Convex Costs in the Random-order Model. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gupta_et_al:LIPIcs.ICALP.2018.71,
  author =	{Gupta, Anupam and Mehta, Ruta and Molinaro, Marco},
  title =	{{Maximizing Profit with Convex Costs in the Random-order Model}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{71:1--71:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.71},
  URN =		{urn:nbn:de:0030-drops-90751},
  doi =		{10.4230/LIPIcs.ICALP.2018.71},
  annote =	{Keywords: Online algorithms, secretary problem, random order, convex duality}
}

4 Search Results for "Molinaro, Marco"

Online Demand Scheduling with Failovers

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Knapsack Secretary with Bursty Adversary

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Robust Algorithms for the Secretary Problem

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Maximizing Profit with Convex Costs in the Random-order Model

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