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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We study the classic problem of searching for a hidden target in the line and the m-ray star, in a setting in which the searcher has some prediction on the hider’s position. We first focus on the main metric for comparing search strategies under predictions; namely, we give positive and negative results on the consistency-robustness tradeoff, where the performance of the strategy is evaluated at extreme situations in which the prediction is either error-free, or adversarially generated, respectively. For the line, we show tight bounds concerning this tradeoff, under the untrusted advice model, in which the prediction is in the form of a k-bit string which encodes the responses to k binary queries. For the star, we give tight, and near-tight tradeoffs in the positional and the directional models, in which the prediction is related to the position of the target within the star, and to the ray on which the target hides, respectively. Last, for all three prediction models, we show how to generalize our study to a setting in which the performance of the strategy is evaluated as a function of the searcher’s desired tolerance to prediction errors, both in terms of positive and inapproximability results.

Spyros Angelopoulos. Competitive Search in the Line and the Star with Predictions. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{angelopoulos:LIPIcs.MFCS.2023.12, author = {Angelopoulos, Spyros}, title = {{Competitive Search in the Line and the Star with Predictions}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.12}, URN = {urn:nbn:de:0030-drops-185464}, doi = {10.4230/LIPIcs.MFCS.2023.12}, annote = {Keywords: Search problems, line and star search, competitive ratio, predictions, consistency and robustness} }

Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We study the nascent setting of online computation with imperfect advice, in which the online algorithm is enhanced by some prediction encoded in the form of an imperfect, and possibly erroneous binary string. The algorithm is oblivious to the advice error, but defines a desired tolerance, namely an upper bound on the number of erroneous advice bits it can tolerate. This is a model that generalizes the Pareto-based advice model, in which the performance of the algorithm is only evaluated at the extreme values of error (namely, if the advice has either no errors, or if it is generated adversarially). It also subsumes the model in which the algorithm elicits a prediction on the online sequence, via imperfect responses to a number of binary queries.
In this work, we establish connections between games with a lying responder, also known as Rényi-Ulam games, and the design and analysis of online algorithms with imperfect advice. Specifically, we demonstrate how to obtain upper and lower bounds on the competitive ratio for important online problems such as time-series search, online bidding, and fractional knapsack. Our techniques provide the first lower bounds for online problems in this model. We also highlight and exploit connections between competitive analysis with imperfect advice and fault-tolerance in multiprocessor systems. Last, we show how to waive the dependence on the tolerance parameter, by means of resource augmentation and robustification.

Spyros Angelopoulos and Shahin Kamali. Rényi-Ulam Games and Online Computation with Imperfect Advice. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{angelopoulos_et_al:LIPIcs.MFCS.2023.13, author = {Angelopoulos, Spyros and Kamali, Shahin}, title = {{R\'{e}nyi-Ulam Games and Online Computation with Imperfect Advice}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.13}, URN = {urn:nbn:de:0030-drops-185474}, doi = {10.4230/LIPIcs.MFCS.2023.13}, annote = {Keywords: Online computation, R\'{e}nyi-Ulam games, query models, beyond worst-case analysis} }

Document

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

The linear search problem, informally known as the cow path problem, is one of the fundamental problems in search theory. In this problem, an immobile target is hidden at some unknown position on an unbounded line, and a mobile searcher, initially positioned at some specific point of the line called the root, must traverse the line so as to locate the target. The objective is to minimize the worst-case ratio of the distance traversed by the searcher to the distance of the target from the root, which is known as the competitive ratio of the search.
In this work we study this problem in a setting in which the searcher has a hint concerning the target. We consider three settings in regards to the nature of the hint: i) the hint suggests the exact position of the target on the line; ii) the hint suggests the direction of the optimal search (i.e., to the left or the right of the root); and iii) the hint is a general k-bit string that encodes some information concerning the target. Our objective is to study the Pareto-efficiency of strategies in this model. Namely, we seek optimal, or near-optimal tradeoffs between the searcher’s performance if the hint is correct (i.e., provided by a trusted source) and if the hint is incorrect (i.e., provided by an adversary).

Spyros Angelopoulos. Online Search with a Hint. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{angelopoulos:LIPIcs.ITCS.2021.51, author = {Angelopoulos, Spyros}, title = {{Online Search with a Hint}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {51:1--51:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.51}, URN = {urn:nbn:de:0030-drops-135902}, doi = {10.4230/LIPIcs.ITCS.2021.51}, annote = {Keywords: Search problems, searching on the line, competitive analysis, predictions} }

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

The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner. To this end, we focus on well- studied online problems such as ski rental, online bidding, bin packing, and list update. For ski-rental and online bidding, we show how to obtain algorithms that are Pareto-optimal with respect to the competitive ratios achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in which Pareto-optimality is not necessarily guaranteed. For bin packing and list update, we give online algorithms with worst-case tradeoffs in their competitiveness, depending on whether the advice is trusted or not; this is motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging problem, but in which the competitiveness depends on the reliability of the advice. Furthermore, we demonstrate how to prove lower bounds, within this model, on the tradeoff between the number of advice bits and the competitiveness of any online algorithm. Last, we study the effect of randomization: here we show that for ski-rental there is a randomized algorithm that Pareto-dominates any deterministic algorithm with advice of any size. We also show that a single random bit is not always inferior to a single advice bit, as it happens in the standard model.

Spyros Angelopoulos, Christoph Dürr, Shendan Jin, Shahin Kamali, and Marc Renault. Online Computation with Untrusted Advice. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{angelopoulos_et_al:LIPIcs.ITCS.2020.52, author = {Angelopoulos, Spyros and D\"{u}rr, Christoph and Jin, Shendan and Kamali, Shahin and Renault, Marc}, title = {{Online Computation with Untrusted Advice}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {52:1--52:15}, 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.2020.52}, URN = {urn:nbn:de:0030-drops-117372}, doi = {10.4230/LIPIcs.ITCS.2020.52}, annote = {Keywords: Online computation, competitive analysis, advice complexity, robust algorithms, untrusted advice} }

Document

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

In search problems, a mobile searcher seeks to locate a target that hides in some unknown position of the environment. Such problems are typically considered to be of an on-line nature, in that the input is unknown to the searcher, and the performance of a search strategy is usually analyzed by means of the standard framework of the competitive ratio, which compares the cost incurred by the searcher to an optimal strategy that knows the location of the target. However, one can argue that even for simple search problems, competitive analysis fails to distinguish between strategies which, intuitively, should have different performance in practice.
Motivated by the above, in this work we introduce and study measures supplementary to competitive analysis in the context of search problems. In particular, we focus on the well-known problem of linear search, informally known as the cow-path problem, for which there is an infinite number of strategies that achieve an optimal competitive ratio equal to 9. We propose a measure that reflects the rate at which the line is being explored by the searcher, and which can be seen as an extension of the bijective ratio over an uncountable set of requests. Using this measure we show that a natural strategy that explores the line aggressively is optimal among all 9-competitive strategies. This provides, in particular, a strict separation from the competitively optimal doubling strategy, which is much more conservative in terms of exploration. We also provide evidence that this aggressiveness is requisite for optimality, by showing that any optimal strategy must mimic the aggressive strategy in its first few explorations.

Spyros Angelopoulos, Christoph Dürr, and Shendan Jin. Best-Of-Two-Worlds Analysis of Online Search. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{angelopoulos_et_al:LIPIcs.STACS.2019.7, author = {Angelopoulos, Spyros and D\"{u}rr, Christoph and Jin, Shendan}, title = {{Best-Of-Two-Worlds Analysis of Online Search}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.7}, URN = {urn:nbn:de:0030-drops-102467}, doi = {10.4230/LIPIcs.STACS.2019.7}, annote = {Keywords: Online computation, search problems, linear search, performance measures} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter k. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most k such actions per edge take place, where k is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. [Avitabile et al., 2013], whereas the special case k=2 was studied by Boyar et al. [Boyar et al., 2017].
In the first part of this paper, we consider the edge arrival model, in which an arriving edge never disappears from the graph. Here, we first show an improved analysis on the performance of the algorithm AMP given in [Avitabile et al., 2013], by exploiting the structure of the matching problem. In addition, we extend the result of [Boyar et al., 2017] and show that the greedy algorithm has competitive ratio 3/2 for every even k and ratio 2 for every odd k. Moreover, we present and analyze an improvement of the greedy algorithm which we call L-Greedy, and we show that for small values of k it outperforms the algorithm of [Avitabile et al., 2013]. In terms of lower bounds, we show that no deterministic algorithm better than 1+1/(k-1) exists, improving upon the lower bound of 1+1/k shown in [Avitabile et al., 2013].
The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of online matching with recourse. The analysis of L-Greedy and AMP carry through in this model; moreover we show a lower bound of (k^2-3k+6)/(k^2-4k+7) for all even k >= 4. For k in {2,3}, the competitive ratio is 3/2.

Spyros Angelopoulos, Christoph Dürr, and Shendan Jin. Online Maximum Matching with Recourse. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{angelopoulos_et_al:LIPIcs.MFCS.2018.8, author = {Angelopoulos, Spyros and D\"{u}rr, Christoph and Jin, Shendan}, title = {{Online Maximum Matching with Recourse}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.8}, URN = {urn:nbn:de:0030-drops-95908}, doi = {10.4230/LIPIcs.MFCS.2018.8}, annote = {Keywords: Competitive ratio, maximum cardinality matching, recourse} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We study the problem of searching for a hidden target in an environment that is modeled by an edge-weighted graph. Most of the previous work on this problem considers the pathwise cost formulation, in which the cost incurred by the searcher is the overall time to locate the target, assuming that the searcher moves at unit speed. More recent work introduced the setting of expanding search in which the searcher incurs cost only upon visiting previously unexplored areas of the graph. Such a paradigm is useful in modeling problems in which the cost of re-exploration is negligible (such as coal mining).
In our work we study algorithmic and computational issues of expanding search, for a variety of search environments including general graphs, trees and star-like graphs. In particular, we rely on the deterministic and randomized search ratio as the performance measures of search strategies, which were originally introduced by Koutsoupias and Papadimitriou [ICALP 1996] in the context of pathwise search. The search ratio is essentially the best competitive ratio among all possible strategies. Our main objective is to explore how the transition from pathwise to expanding search affects the competitive analysis, which has applications to optimization problems beyond the strict boundaries of search problems.

Spyros Angelopoulos, Christoph Dürr, and Thomas Lidbetter. The Expanding Search Ratio of a Graph. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{angelopoulos_et_al:LIPIcs.STACS.2016.9, author = {Angelopoulos, Spyros and D\"{u}rr, Christoph and Lidbetter, Thomas}, title = {{The Expanding Search Ratio of a Graph}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.9}, URN = {urn:nbn:de:0030-drops-57109}, doi = {10.4230/LIPIcs.STACS.2016.9}, annote = {Keywords: Search games, randomized algorithms, competitive analysis, game theory} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 9171, Adaptive, Output Sensitive, Online and Parameterized Algorithms (2009)

Steiner tree problems occupy a central place in both areas of approximation and on-line algorithms. Many variants have been studied from the point of view of competitive analysis, and for several of these variants tight bounds are known. However, in several cases, worst-case analysis is overly pessimistic, which fails to explain the relative performance of algorithms. We show how adaptive analysis can help resolve this problem. As case studies, we consider the Steiner tree problem in directed graphs, and the Priority Steiner tree problem.

Spyros Angelopoulos. Parameterized Analysis of Online Steiner Tree Problems. In Adaptive, Output Sensitive, Online and Parameterized Algorithms. Dagstuhl Seminar Proceedings, Volume 9171, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{angelopoulos:DagSemProc.09171.3, author = {Angelopoulos, Spyros}, title = {{Parameterized Analysis of Online Steiner Tree Problems}}, booktitle = {Adaptive, Output Sensitive, Online and Parameterized Algorithms}, pages = {1--11}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {9171}, editor = {J\'{e}r\'{e}my Barbay and Rolf Klein and Alejandro Ortiz-L\'{o}pez and Rolf Niedermeier}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09171.3}, URN = {urn:nbn:de:0030-drops-21210}, doi = {10.4230/DagSemProc.09171.3}, annote = {Keywords: Online algorithms, Steiner tree problems, adaptive and parameteried analysis} }

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