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

We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line [Gouleakis et al., 2023], in which the predictions correspond to the locations of the requests and extend it to more general metric spaces.
We first propose an oracle-based algorithmic framework, inspired by previous work [Bampis et al., 2023]. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness).
Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results.

Evripidis Bampis, Bruno Escoffier, Themis Gouleakis, Niklas Hahn, Kostas Lakis, Golnoosh Shahkarami, and Michalis Xefteris. Learning-Augmented Online TSP on Rings, Trees, Flowers and (Almost) Everywhere Else. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bampis_et_al:LIPIcs.ESA.2023.12, author = {Bampis, Evripidis and Escoffier, Bruno and Gouleakis, Themis and Hahn, Niklas and Lakis, Kostas and Shahkarami, Golnoosh and Xefteris, Michalis}, title = {{Learning-Augmented Online TSP on Rings, Trees, Flowers and (Almost) Everywhere Else}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {12:1--12:17}, 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.2023.12}, URN = {urn:nbn:de:0030-drops-186659}, doi = {10.4230/LIPIcs.ESA.2023.12}, annote = {Keywords: TSP, Online algorithms, Learning-augmented algorithms, Algorithms with predictions, Competitive analysis} }

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**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

A temporal graph is a sequence of graphs, indexed by discrete time steps, with a fixed vertex set but with an edge set that is able to change over time. In the temporal graph exploration problem, an agent wants to visit all the vertices of a given temporal graph. In the classical model, at each time step the agent can either stay where they are, or move along one edge. In this work we add a constraint called restlessness that forces the agent to move along one edge at each time step. We mainly focus on (infinite) periodical temporal graphs. We show that if the period is 2 one can decide in polynomial time whether exploring the whole graph is possible or not, while this problem turns out to be NP-hard for any period p ≥ 3. We also show some time bounds on the explorations of such graphs when the exploration is possible.

Thomas Bellitto, Cyril Conchon-Kerjan, and Bruno Escoffier. Restless Exploration of Periodic Temporal Graphs. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bellitto_et_al:LIPIcs.SAND.2023.13, author = {Bellitto, Thomas and Conchon-Kerjan, Cyril and Escoffier, Bruno}, title = {{Restless Exploration of Periodic Temporal Graphs}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.13}, URN = {urn:nbn:de:0030-drops-179497}, doi = {10.4230/LIPIcs.SAND.2023.13}, annote = {Keywords: Temporal graphs, Graph exploration, NP-completeness} }

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

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,...,n}, a collection F subseteq 2^N of subsets thereof such that the empty set is in F, and an objective (profit) function p: F -> R_+. The task is to choose a set S in F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions F_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors.
We study multistage subset maximization problems in the online setting, that is, p_t (along with possibly F_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future.
We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one.

Evripidis Bampis, Bruno Escoffier, Kevin Schewior, and Alexandre Teiller. Online Multistage Subset Maximization Problems. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bampis_et_al:LIPIcs.ESA.2019.11, author = {Bampis, Evripidis and Escoffier, Bruno and Schewior, Kevin and Teiller, Alexandre}, title = {{Online Multistage Subset Maximization Problems}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {11:1--11:14}, 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.2019.11}, URN = {urn:nbn:de:0030-drops-111320}, doi = {10.4230/LIPIcs.ESA.2019.11}, annote = {Keywords: Multistage optimization, Online algorithms} }

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

Many systems have to be maintained while the underlying constraints, costs and/or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incured for transitioning from one state to another. In order to model such situations, Gupta et al. (ICALP 2014) and Eisenstat et al. (ICALP 2014) introduced a multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that simultaneously (i) have good quality on the time steps and (ii) as stable as possible. We focus on the multistage version of the Knapsack problem where we are given a time horizon t=1,2,...,T, and a sequence of knapsack instances I_1,I_2,...,I_T, one for each time step, defined on a set of n objects. In every time step t we have to choose a feasible knapsack S_t of I_t, which gives a knapsack profit. To measure the stability/similarity of two consecutive solutions S_t and S_{t+1}, we identify the objects for which the decision, to be picked or not, remains the same in S_t and S_{t+1}, giving a transition profit. We are asked to produce a sequence of solutions S_1,S_2,...,S_T so that the total knapsack profit plus the overall transition profit is maximized.
We propose a PTAS for the Multistage Knapsack problem. This is the first approximation scheme for a combinatorial optimization problem in the considered multistage setting, and its existence contrasts with the inapproximability results for other combinatorial optimization problems that are even polynomial-time solvable in the static case (e.g.multistage Spanning Tree, or multistage Bipartite Perfect Matching). Then, we prove that there is no FPTAS for the problem even in the case where T=2, unless P=NP. Furthermore, we give a pseudopolynomial time algorithm for the case where the number of steps is bounded by a fixed constant and we show that otherwise the problem remains NP-hard even in the case where all the weights, profits and capacities are 0 or 1.

Evripidis Bampis, Bruno Escoffier, and Alexandre Teiller. Multistage Knapsack. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bampis_et_al:LIPIcs.MFCS.2019.22, author = {Bampis, Evripidis and Escoffier, Bruno and Teiller, Alexandre}, title = {{Multistage Knapsack}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {22:1--22:14}, 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.2019.22}, URN = {urn:nbn:de:0030-drops-109664}, doi = {10.4230/LIPIcs.MFCS.2019.22}, annote = {Keywords: Knapsack, Approximation Algorithms, Multistage Optimization} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

We consider a multistage version of the Perfect Matching problem which models the scenario where the costs of edges change over time and we seek to obtain a solution that achieves low total cost, while minimizing the number of changes from one instance to the next. Formally, we are given a sequence of edge-weighted graphs on the same set of vertices V, and are asked to produce a perfect matching in each instance so that the total edge cost plus the transition cost (the cost of exchanging edges), is minimized. This model was introduced by Gupta et al. (ICALP 2014), who posed as an open problem its approximability for bipartite instances. We completely resolve this question by showing that Minimum Multistage Perfect Matching (Min-MPM) does not admit an n^{1-epsilon}-approximation, even on bipartite instances with only two time steps.
Motivated by this negative result, we go on to consider two variations of the problem. In Metric Minimum Multistage Perfect Matching problem (Metric-Min-MPM) we are promised that edge weights in each time step satisfy the triangle inequality. We show that this problem admits a 3-approximation when the number of time steps is 2 or 3. On the other hand, we show that even the metric case is APX-hard already for 2 time steps. We then consider the complementary maximization version of the problem, Maximum Multistage Perfect Matching problem (Max-MPM), where we seek to maximize the total profit of all selected edges plus the total number of non-exchanged edges. We show that Max-MPM is also APX-hard, but admits a constant factor approximation algorithm for any number of time steps.

Evripidis Bampis, Bruno Escoffier, Michael Lampis, and Vangelis Th. Paschos. Multistage Matchings. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bampis_et_al:LIPIcs.SWAT.2018.7, author = {Bampis, Evripidis and Escoffier, Bruno and Lampis, Michael and Paschos, Vangelis Th.}, title = {{Multistage Matchings}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.7}, URN = {urn:nbn:de:0030-drops-88338}, doi = {10.4230/LIPIcs.SWAT.2018.7}, annote = {Keywords: Perfect Matching, Temporal Optimization, Multistage Optimization} }

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