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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

In real world applications, important resources like energy are saved by deliberately using so-called low-cost operations that are less reliable. Some of these approaches are based on a dual mode technology where it is possible to choose between high-energy operations (always correct) and low-energy operations (prone to errors), and thus enable to trade energy for correctness.
In this work we initiate the study of algorithms for solving optimization problems that in their computation are allowed to choose between two types of operations: high-energy comparisons (always correct but expensive) and low-energy comparisons (cheaper but prone to errors). For the errors in low-energy comparisons, we assume the persistent setting, which usually makes it impossible to achieve optimal solutions without high-energy comparisons. We propose to study a natural complexity measure which accounts for the number of operations of either type separately.
We provide a new family of algorithms which, for a fairly large class of maximization problems, return a constant approximation using only polylogarithmic many high-energy comparisons and only O(n log n) low-energy comparisons. This result applies to the class of p-extendible system s [Mestre, 2006], which includes several NP-hard problems and matroids as a special case (p=1).
These algorithmic solutions relate to some fundamental aspects studied earlier in different contexts: (i) the approximation guarantee when only ordinal information is available to the algorithm; (ii) the fact that even such ordinal information may be erroneous because of low-energy comparisons and (iii) the ability to approximately sort a sequence of elements when comparisons are subject to persistent errors. Finally, our main result is quite general and can be parametrized and adapted to other error models.

Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, Paolo Penna, and Guido Proietti. Dual-Mode Greedy Algorithms Can Save Energy. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 64:1-64:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{geissmann_et_al:LIPIcs.ISAAC.2019.64, author = {Geissmann, Barbara and Leucci, Stefano and Liu, Chih-Hung and Penna, Paolo and Proietti, Guido}, title = {{Dual-Mode Greedy Algorithms Can Save Energy}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {64:1--64:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.64}, URN = {urn:nbn:de:0030-drops-115604}, doi = {10.4230/LIPIcs.ISAAC.2019.64}, annote = {Keywords: matroids, p-extendible systems, greedy algorithm, approximation algorithms, high-low energy} }

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**Published in:** OASIcs, Volume 75, 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)

We present robust, adaptive routing policies for time-varying networks (temporal graphs) in the presence of random edge-failures. Such a policy answers the following question: How can a traveler navigate a time-varying network where edges fail randomly in order to maximize the traveler’s preference with respect to the arrival time? Our routing policy is computable in near-linear time in the number of edges in the network (for the case when the edges fail independently of each other).
Using our robust routing policy, we show how to travel in a public transit network where the vehicles experience delays. To validate our approach, we present experiments using real-world delay data from the public transit network of the city of Zurich. Our experiments show that we obtain significantly improved outcomes compared to a purely schedule-based policy: The traveler is on time 5-11 percentage points more often for most destinations and 20-40 percentage points more often for certain remote destinations. Our implementation shows that the approach is fast enough for real-time usage. It computes a policy for 1-hour long journeys in around 0.1 seconds.

Barbara Geissmann and Lukas Gianinazzi. Routing in Stochastic Public Transit Networks. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019). Open Access Series in Informatics (OASIcs), Volume 75, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{geissmann_et_al:OASIcs.ATMOS.2019.4, author = {Geissmann, Barbara and Gianinazzi, Lukas}, title = {{Routing in Stochastic Public Transit Networks}}, booktitle = {19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)}, pages = {4:1--4:18}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-128-3}, ISSN = {2190-6807}, year = {2019}, volume = {75}, editor = {Cacchiani, Valentina and Marchetti-Spaccamela, Alberto}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2019.4}, URN = {urn:nbn:de:0030-drops-114167}, doi = {10.4230/OASIcs.ATMOS.2019.4}, annote = {Keywords: Route Planning, Public Transit Network, Temporal Graphs} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We consider the problem of sorting n elements in the case of persistent comparison errors. In this problem, each comparison between two elements can be wrong with some fixed (small) probability p, and comparisons cannot be repeated (Braverman and Mossel, SODA'08). Sorting perfectly in this model is impossible, and the objective is to minimize the dislocation of each element in the output sequence, that is, the difference between its true rank and its position. Existing lower bounds for this problem show that no algorithm can guarantee, with high probability, maximum dislocation and total dislocation better than Omega(log n) and Omega(n), respectively, regardless of its running time.
In this paper, we present the first O(n log n)-time sorting algorithm that guarantees both O(log n) maximum dislocation and O(n) total dislocation with high probability. This settles the time complexity of this problem and shows that comparison errors do not increase its computational difficulty: a sequence with the best possible dislocation can be obtained in O(n log n) time and, even without comparison errors, Omega(n log n) time is necessary to guarantee such dislocation bounds.
In order to achieve this optimality result, we solve two sub-problems in the persistent error comparisons model, and the respective methods have their own merits for further application. One is how to locate a position in which to insert an element in an almost-sorted sequence having O(log n) maximum dislocation in such a way that the dislocation of the resulting sequence will still be O(log n). The other is how to simultaneously insert m elements into an almost sorted sequence of m different elements, such that the resulting sequence of 2m elements remains almost sorted.

Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal Sorting with Persistent Comparison Errors. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{geissmann_et_al:LIPIcs.ESA.2019.49, author = {Geissmann, Barbara and Leucci, Stefano and Liu, Chih-Hung and Penna, Paolo}, title = {{Optimal Sorting with Persistent Comparison Errors}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.49}, URN = {urn:nbn:de:0030-drops-111706}, doi = {10.4230/LIPIcs.ESA.2019.49}, annote = {Keywords: approximate sorting, comparison errors, persistent errors} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

We study the problem of sorting N elements in presence of persistent errors in comparisons: In this classical model, each comparison between two elements is wrong independently with some probability p, but repeating the same comparison gives always the same result. The best known algorithms for this problem have running time O(N^2) and achieve an optimal maximum dislocation of O(log N) for constant error probability. Note that no algorithm can achieve dislocation o(log N), regardless of its running time.
In this work we present the first subquadratic time algorithm with optimal maximum dislocation: Our algorithm runs in tilde{O}(N^{3/2}) time and guarantees O(log N) maximum dislocation with high probability. Though the first version of our algorithm is randomized, it can be derandomized by extracting the necessary random bits from the results of the comparisons (errors).

Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal Dislocation with Persistent Errors in Subquadratic Time. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{geissmann_et_al:LIPIcs.STACS.2018.36, author = {Geissmann, Barbara and Leucci, Stefano and Liu, Chih-Hung and Penna, Paolo}, title = {{Optimal Dislocation with Persistent Errors in Subquadratic Time}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {36:1--36:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.36}, URN = {urn:nbn:de:0030-drops-85266}, doi = {10.4230/LIPIcs.STACS.2018.36}, annote = {Keywords: sorting, recurrent comparison errors, maximum dislocation} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We present a sorting algorithm for the case of recurrent random comparison errors. The algorithm essentially achieves simultaneously good properties of previous algorithms for sorting n distinct elements in this model. In particular, it runs in O(n^2) time, the maximum dislocation of the elements in the output is O(log n), while the total dislocation is O(n). These guarantees are the best possible since we prove that even randomized algorithms cannot achieve o(log n) maximum dislocation with high probability, or o(n) total dislocation in expectation, regardless of their
running time.

Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Sorting with Recurrent Comparison Errors. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 38:1-38:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{geissmann_et_al:LIPIcs.ISAAC.2017.38, author = {Geissmann, Barbara and Leucci, Stefano and Liu, Chih-Hung and Penna, Paolo}, title = {{Sorting with Recurrent Comparison Errors}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {38:1--38:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.38}, URN = {urn:nbn:de:0030-drops-82652}, doi = {10.4230/LIPIcs.ISAAC.2017.38}, annote = {Keywords: sorting, recurrent comparison error, maximum and total dislocation} }

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