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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Exploring large-scale networks is a time consuming and expensive task which is usually operated in a complex and uncertain environment. A crucial aspect of network exploration is the development of suitable strategies that decide which nodes and edges to probe at each stage of the process.
To model this process, we introduce the stochastic graph exploration problem. The input is an undirected graph G=(V,E) with a source vertex s, stochastic edge costs drawn from a distribution pi_e, e in E, and rewards on vertices of maximum value R. The goal is to find a set F of edges of total cost at most B such that the subgraph of G induced by F is connected, contains s, and maximizes the total reward. This problem generalizes the stochastic knapsack problem and other stochastic probing problems recently studied.
Our focus is on the development of efficient nonadaptive strategies that are competitive against the optimal adaptive strategy. A major challenge is the fact that the problem has an Omega(n) adaptivity gap even on a tree of n vertices. This is in sharp contrast with O(1) adaptivity gap of the stochastic knapsack problem, which is a special case of our problem. We circumvent this negative result by showing that O(log nR) resource augmentation suffices to obtain O(1) approximation on trees and O(log nR) approximation on general graphs. To achieve this result, we reduce stochastic graph exploration to a memoryless process - the minesweeper problem - which assigns to every edge a probability that the process terminates when the edge is probed. For this problem, interesting in its own, we present an optimal polynomial time algorithm on trees and an O(log nR) approximation for general graphs.
We study also the problem in which the maximum cost of an edge is a logarithmic fraction of the budget. We show that under this condition, there exist polynomial-time oblivious strategies that use 1+epsilon budget, whose adaptivity gaps on trees and general graphs are 1+epsilon and 8+epsilon, respectively. Finally, we provide additional results on the structure and the complexity of nonadaptive and adaptive strategies.

Aris Anagnostopoulos, Ilan R. Cohen, Stefano Leonardi, and Jakub Łącki. Stochastic Graph Exploration. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 136:1-136:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{anagnostopoulos_et_al:LIPIcs.ICALP.2019.136, author = {Anagnostopoulos, Aris and Cohen, Ilan R. and Leonardi, Stefano and {\L}\k{a}cki, Jakub}, title = {{Stochastic Graph Exploration}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {136:1--136:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.136}, URN = {urn:nbn:de:0030-drops-107122}, doi = {10.4230/LIPIcs.ICALP.2019.136}, annote = {Keywords: stochastic optimization, graph exploration, approximation algorithms} }

Document

**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [Kalyanasundaram and Pruhs, 2000], the allocation problem [Buchbinder et al., 2007], and the AdWords problem [Mehta et al., 2007] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [Naor and Wajc, 2018]. Such graphs model natural properties on the degrees of advertisers and queries in the allocation and AdWords problems. While previous work only considers the scenario where k ≥ d, we consider the interesting intermediate regime of k ≤ d and prove a tight competitive ratio as a function of k,d (under the small-bid assumption) of τ(k,d) = 1 - (1-k/d)⋅(1-1/d)^{d - k} for the b-matching and allocation problems. We exploit primal-dual schemes [Buchbinder et al., 2009; Azar et al., 2017] to design and analyze the corresponding tight upper and lower bounds. Finally, we show a separation between the allocation and AdWords problems. We demonstrate that τ(k,d) competitiveness is impossible for the AdWords problem even in (k,d)-bounded degree graphs.

Ilan Reuven Cohen and Binghui Peng. Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 35:1-35:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cohen_et_al:LIPIcs.ESA.2023.35, author = {Cohen, Ilan Reuven and Peng, Binghui}, title = {{Primal-Dual Schemes for Online Matching in Bounded Degree Graphs}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {35:1--35:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.35}, URN = {urn:nbn:de:0030-drops-186884}, doi = {10.4230/LIPIcs.ESA.2023.35}, annote = {Keywords: Online Matching, Primal-dual analysis, bounded-degree graph, the AdWords problem} }

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Track A: Algorithms, Complexity and Games

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

Online allocation is a broad class of problems where items arriving online have to be allocated to agents who have a fixed utility/cost for each assigned item so to maximize/minimize some objective. This framework captures a broad range of fundamental problems such as the Santa Claus problem (maximizing minimum utility), Nash welfare maximization (maximizing geometric mean of utilities), makespan minimization (minimizing maximum cost), minimization of 𝓁_p-norms, and so on. We focus on divisible items (i.e., fractional allocations) in this paper. Even for divisible items, these problems are characterized by strong super-constant lower bounds in the classical worst-case online model.
In this paper, we study online allocations in the learning-augmented setting, i.e., where the algorithm has access to some additional (machine-learned) information about the problem instance. We introduce a general algorithmic framework for learning-augmented online allocation that produces nearly optimal solutions for this broad range of maximization and minimization objectives using only a single learned parameter for every agent. As corollaries of our general framework, we improve prior results of Lattanzi et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan minimization, and obtain the first learning-augmented nearly-optimal algorithms for the other objectives such as Santa Claus, Nash welfare, 𝓁_p-minimization, etc. We also give tight bounds on the resilience of our algorithms to errors in the learned parameters, and study the learnability of these parameters.

Ilan Reuven Cohen and Debmalya Panigrahi. A General Framework for Learning-Augmented Online Allocation. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 43:1-43:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cohen_et_al:LIPIcs.ICALP.2023.43, author = {Cohen, Ilan Reuven and Panigrahi, Debmalya}, title = {{A General Framework for Learning-Augmented Online Allocation}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {43:1--43:21}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.43}, URN = {urn:nbn:de:0030-drops-180952}, doi = {10.4230/LIPIcs.ICALP.2023.43}, annote = {Keywords: Algorithms with predictions, Scheduling algorithms, Online algorithms} }

Document

APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

In this work, we consider online d-dimensional vector bin packing. It is known that no algorithm can have a competitive ratio of o(d/log² d) in the absolute sense, although upper bounds for this problem have always been presented in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure, and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds of the asymptotic competitive ratio. The existing lower bounds prior to this work were known to be smaller than 3, even for very large d. Here, we significantly improved on the best known lower bounds of the asymptotic competitive ratio (and as a byproduct, on the absolute competitive ratio) for online vector packing of vectors with d ≥ 3 dimensions, for every dimension d. To obtain these results, we use several different constructions, one of which is an adaptive construction with a lower bound of Ω(√d). Our main result is that the lower bound of Ω(d/log² d) on the competitive ratio holds also in the asymptotic sense. This result holds also against randomized algorithms, and requires a careful adaptation of constructions for online coloring, rather than simple black-box reductions.

János Balogh, Ilan Reuven Cohen, Leah Epstein, and Asaf Levin. Truly Asymptotic Lower Bounds for Online Vector Bin Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 8:1-8:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{balogh_et_al:LIPIcs.APPROX/RANDOM.2021.8, author = {Balogh, J\'{a}nos and Cohen, Ilan Reuven and Epstein, Leah and Levin, Asaf}, title = {{Truly Asymptotic Lower Bounds for Online Vector Bin Packing}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.8}, URN = {urn:nbn:de:0030-drops-147013}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.8}, annote = {Keywords: Bin packing, online algorithms, approximation algorithms, vector packing} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

In this paper we consider the k-server problem where events are generated by selfish agents, known as the selfish k-server problem. In this setting, there is a set of k servers located in some metric space. Selfish agents arrive in an online fashion, each has a request located on some point in the metric space, and seeks to serve his request with the server of minimum distance to the request. If agents choose to serve their request with the nearest server, this mimics the greedy algorithm which has an unbounded competitive ratio. We propose an algorithm that associates a surcharge with each server independently of the agent to arrive (and therefore, yields a truthful online mechanism). An agent chooses to serve his request with the server that minimizes the distance to the request plus the associated surcharge to the server.
This paper extends [Ilan Reuven Cohen et al., 2015], which gave an optimal k-competitive dynamic pricing scheme for the selfish k-server problem on the line. We give a k-competitive dynamic pricing algorithm for the selfish k-server problem on tree metric spaces, which matches the optimal online (non truthful) algorithm. We show that an alpha-competitive dynamic pricing scheme exists on the tree if and only if there exists alpha-competitive online algorithm on the tree that is lazy and monotone. Given this characterization, the main technical difficulty is coming up with such an online algorithm.

Ilan Reuven Cohen, Alon Eden, Amos Fiat, and Łukasz Jeż. Dynamic Pricing of Servers on Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 10:1-10:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cohen_et_al:LIPIcs.APPROX-RANDOM.2019.10, author = {Cohen, Ilan Reuven and Eden, Alon and Fiat, Amos and Je\.{z}, {\L}ukasz}, title = {{Dynamic Pricing of Servers on Trees}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.10}, URN = {urn:nbn:de:0030-drops-112252}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.10}, annote = {Keywords: Online algorithms, Online mechanisms, k-server problem, Online pricing} }

Document

Track A: Algorithms, Complexity and Games

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

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O((log d)/(log log d)) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper:
- We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions.
- If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions.
- For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem.

Ilan Cohen, Sungjin Im, and Debmalya Panigrahi. Online Two-Dimensional Load Balancing. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 34:1-34:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cohen_et_al:LIPIcs.ICALP.2020.34, author = {Cohen, Ilan and Im, Sungjin and Panigrahi, Debmalya}, title = {{Online Two-Dimensional Load Balancing}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {34:1--34:21}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.34}, URN = {urn:nbn:de:0030-drops-124415}, doi = {10.4230/LIPIcs.ICALP.2020.34}, annote = {Keywords: Online algorithms, scheduling, multi-dimensional} }

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