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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Temporal graphs are dynamic graphs where the edge set can change in each time step, while the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been widely studied. In this paper, we extend the concept of graph automorphisms from static graphs to temporal graphs and show for the first time that symmetries enable faster exploration: We prove that a connected temporal graph with n vertices and orbit number r (i.e., r is the number of automorphism orbits) can be explored in O(r n^{1+ε}) time steps, for any fixed ε > 0. For r = O(n^c) for constant c < 1, this is a significant improvement over the known tight worst-case bound of Θ(n²) time steps for arbitrary connected temporal graphs. We also give two lower bounds for temporal exploration, showing that Ω(n log n) time steps are required for some inputs with r = O(1) and that Ω(rn) time steps are required for some inputs for any r with 1 ≤ r ≤ n.
Moreover, we show that the techniques we develop for fast exploration can be used to derive the following result for rendezvous: Two agents with different programs and without communication ability are placed by an adversary at arbitrary vertices and given full information about the connected temporal graph, except that they do not have consistent vertex labels. Then the two agents can meet at a common vertex after O(n^{1+ε}) time steps, for any constant ε > 0. For some connected temporal graphs with the orbit number being a constant, we also present a complementary lower bound of Ω(nlog n) time steps.

Konstantinos Dogeas, Thomas Erlebach, Frank Kammer, Johannes Meintrup, and William K. Moses Jr.. Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dogeas_et_al:LIPIcs.ICALP.2024.55, author = {Dogeas, Konstantinos and Erlebach, Thomas and Kammer, Frank and Meintrup, Johannes and Moses Jr., William K.}, title = {{Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {55:1--55:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.55}, URN = {urn:nbn:de:0030-drops-201989}, doi = {10.4230/LIPIcs.ICALP.2024.55}, annote = {Keywords: dynamic graphs, parameterized algorithms, algorithmic graph theory, graph automorphism, orbit number} }

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Invited Talk

**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Given a temporal graph 𝒢 and a start vertex v in 𝒢, the temporal exploration problem (TEXP) is the problem of determining a temporal walk that starts at v and visits all vertices of 𝒢, with the objective of minimizing the time when the last unvisited vertex is reached. Studies have investigated the (parameterized) complexity and approximability of TEXP and the worst-case number of time steps required to complete an exploration. While many upper and lower bounds have been obtained for different settings, there are still some large gaps that pose interesting open problems. In this talk, we will give an overview of known results and techniques as well as open problems. Furthermore, we will discuss recent results (from joint work with Konstantinos Dogeas, Frank Kammer, Johannes Meintrup, and William K. Moses Jr) about exploiting symmetries in temporal graphs to get faster exploration. We view the number of automorphism orbits of the temporal graph as a new parameter, termed the orbit number, that may also be useful in other contexts. Finally, we show how a subroutine for quickly exploring a single orbit of the graph can be exploited to solve a certain rendezvous problem with two agents using a near-linear number of time steps in every always-connected temporal graph.

Thomas Erlebach. Exploration and Rendezvous in Temporal Graphs (Invited Talk). In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{erlebach:LIPIcs.SAND.2024.1, author = {Erlebach, Thomas}, title = {{Exploration and Rendezvous in Temporal Graphs}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {1:1--1:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.1}, URN = {urn:nbn:de:0030-drops-198797}, doi = {10.4230/LIPIcs.SAND.2024.1}, annote = {Keywords: Dynamic graphs, parameterized algorithms, orbit number} }

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**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n× n matrix D of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between any pair of vertices matches the one specified by D. We generalize the problem from simple temporal graphs to temporal graphs where each edge can appear up to 𝓁 times in each period, for some given integer 𝓁. For the resulting problem Multi-Label Periodic TGR, we show that it is fixed-parameter tractable for parameter n and for parameter vc+Δ, where vc is the vertex cover number of the underlying graph. We also show the existence of a polynomial kernel for parameter nu+d_max, where nu is the number of non-universal vertices of the underlying graph and d_max is the largest entry of D. Furthermore, we show that the problem is NP-hard for each 𝓁 ≥ 5, even if the underlying graph is a tree, a case that was known to be solvable in polynomial time if the task is to construct a simple periodic temporal graph, that is, if 𝓁 = 1.

Thomas Erlebach, Nils Morawietz, and Petra Wolf. Parameterized Algorithms for Multi-Label Periodic Temporal Graph Realization. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{erlebach_et_al:LIPIcs.SAND.2024.12, author = {Erlebach, Thomas and Morawietz, Nils and Wolf, Petra}, title = {{Parameterized Algorithms for Multi-Label Periodic Temporal Graph Realization}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.12}, URN = {urn:nbn:de:0030-drops-198909}, doi = {10.4230/LIPIcs.SAND.2024.12}, annote = {Keywords: Fixed-Parameter Tractability, Almost-Clique, Kernelization, Dynamic Network, Temporal Graph} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

For the problem of delivering a package from a source node to a destination node in a graph using a set of drones, we study the setting where the movements of each drone are restricted to a certain subgraph of the given graph. We consider the objectives of minimizing the delivery time (problem DDT) and of minimizing the total energy consumption (problem DDC). For general graphs, we show a strong inapproximability result and a matching approximation algorithm for DDT as well as NP-hardness and a 2-approximation algorithm for DDC. For the special case of a path, we show that DDT is NP-hard if the drones have different speeds. For trees, we give optimal algorithms under the assumption that all drones have the same speed or the same energy consumption rate. The results for trees extend to arbitrary graphs if the subgraph of each drone is isometric.

Thomas Erlebach, Kelin Luo, and Frits C.R. Spieksma. Package Delivery Using Drones with Restricted Movement Areas. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{erlebach_et_al:LIPIcs.ISAAC.2022.49, author = {Erlebach, Thomas and Luo, Kelin and Spieksma, Frits C.R.}, title = {{Package Delivery Using Drones with Restricted Movement Areas}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {49:1--49:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.49}, URN = {urn:nbn:de:0030-drops-173343}, doi = {10.4230/LIPIcs.ISAAC.2022.49}, annote = {Keywords: Mobile agents, approximation algorithm, inapproximability} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral γ ≥ 2, we present algorithms that are γ-robust and (1+1/γ)-consistent, meaning that they use at most γOPT queries if the predictions are arbitrarily wrong and at most (1+1/γ)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those.

Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{erlebach_et_al:LIPIcs.ESA.2022.49, author = {Erlebach, Thomas and de Lima, Murilo Santos and Megow, Nicole and Schl\"{o}ter, Jens}, title = {{Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {49:1--49:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.49}, URN = {urn:nbn:de:0030-drops-169872}, doi = {10.4230/LIPIcs.ESA.2022.49}, annote = {Keywords: explorable uncertainty, queries, untrusted predictions} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

In this paper we study the fixed-parameter tractability of the problem of deciding whether a given temporal graph 𝒢 admits a temporal walk that visits all vertices (temporal exploration) or, in some problem variants, a certain subset of the vertices. Formally, a temporal graph is a sequence 𝒢 = ⟨ G₁,..., G_L⟩ of graphs with V(G_t) = V(G) and E(G_t) ⊆ E(G) for all t ∈ [L] and some underlying graph G, and a temporal walk is a time-respecting sequence of edge-traversals. For the strict variant, in which edges must be traversed in strictly increasing timesteps, we give FPT algorithms for the problem of finding a temporal walk that visits a given set X of vertices, parameterized by |X|, and for the problem of finding a temporal walk that visits at least k distinct vertices in V, parameterized by k. For the non-strict variant, in which an arbitrary number of edges can be traversed in each timestep, we parameterize by the lifetime L of the input graph and provide an FPT algorithm for the temporal exploration problem. We also give additional FPT or W[2]-hardness results for further problem variants.

Thomas Erlebach and Jakob T. Spooner. Parameterized Temporal Exploration Problems. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{erlebach_et_al:LIPIcs.SAND.2022.15, author = {Erlebach, Thomas and Spooner, Jakob T.}, title = {{Parameterized Temporal Exploration Problems}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {15:1--15:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.15}, URN = {urn:nbn:de:0030-drops-159570}, doi = {10.4230/LIPIcs.SAND.2022.15}, annote = {Keywords: Temporal graphs, fixed-parameter tractability, parameterized complexity} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distribution. Using competitive analysis, we compare the expected query cost of an algorithm with the expected cost of an optimal query set for the given instance. For the general case, we give a polynomial-time f(α)-competitive algorithm, where f(α) ∈ [1.618+ε,2] depends on the approximation ratio α for an underlying vertex cover problem. We also show that no algorithm using a similar approach can be better than 1.5-competitive. Furthermore, we give polynomial-time 4/3-competitive algorithms for bipartite graphs with arbitrary query costs and for hypergraphs with a single hyperedge and uniform query costs, with matching lower bounds.

Evripidis Bampis, Christoph Dürr, Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bampis_et_al:LIPIcs.ESA.2021.10, author = {Bampis, Evripidis and D\"{u}rr, Christoph and Erlebach, Thomas and de Lima, Murilo Santos and Megow, Nicole and Schl\"{o}ter, Jens}, title = {{Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.10}, URN = {urn:nbn:de:0030-drops-145910}, doi = {10.4230/LIPIcs.ESA.2021.10}, annote = {Keywords: Explorable uncertainty, queries, stochastic optimization, graph orientation, selection problems} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

The area of computing with uncertainty considers problems where some information about the input elements is uncertain, but can be obtained using queries. For example, instead of the weight of an element, we may be given an interval that is guaranteed to contain the weight, and a query can be performed to reveal the weight. While previous work has considered models where queries are asked either sequentially (adaptive model) or all at once (non-adaptive model), and the goal is to minimize the number of queries that are needed to solve the given problem, we propose and study a new model where k queries can be made in parallel in each round, and the goal is to minimize the number of query rounds. We use competitive analysis and present upper and lower bounds on the number of query rounds required by any algorithm in comparison with the optimal number of query rounds. Given a set of uncertain elements and a family of m subsets of that set, we present an algorithm for determining the value of the minimum of each of the subsets that requires at most (2+ε) ⋅ opt_k+O(1/(ε) ⋅ lg m) rounds for every 0 < ε < 1, where opt_k is the optimal number of rounds, as well as nearly matching lower bounds. For the problem of determining the i-th smallest value and identifying all elements with that value in a set of uncertain elements, we give a 2-round-competitive algorithm. We also show that the problem of sorting a family of sets of uncertain elements admits a 2-round-competitive algorithm and this is the best possible.

Thomas Erlebach, Michael Hoffmann, and Murilo Santos de Lima. Round-Competitive Algorithms for Uncertainty Problems with Parallel Queries. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{erlebach_et_al:LIPIcs.STACS.2021.27, author = {Erlebach, Thomas and Hoffmann, Michael and de Lima, Murilo Santos}, title = {{Round-Competitive Algorithms for Uncertainty Problems with Parallel Queries}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {27:1--27:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.27}, URN = {urn:nbn:de:0030-drops-136728}, doi = {10.4230/LIPIcs.STACS.2021.27}, annote = {Keywords: online algorithms, competitive analysis, explorable uncertainty, parallel algorithms, minimum problem, selection problem} }

Document

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)

A temporal graph is a graph whose edge set can change over time. We only require that the edge set in each time step forms a connected graph. The temporal exploration problem asks for a temporal walk that starts at a given vertex, moves over at most one edge in each time step, visits all vertices, and reaches the last unvisited vertex as early as possible. We show in this paper that every temporal graph with n vertices can be explored in O(n^{1.75}) time steps provided that either the degree of the graph is bounded in each step or the temporal walk is allowed to make two moves per step. This result is interesting because it breaks the lower bound of Omega(n^2) steps that holds for the worst-case exploration time if only one move per time step is allowed and the graph in each step can have arbitrary degree. We complement this main result by a logarithmic inapproximability result and a proof that for sparse temporal graphs (i.e., temporal graphs with O(n) edges in the underlying graph) making O(1) moves per time step can improve the worst-case exploration time at most by a constant factor.

Thomas Erlebach, Frank Kammer, Kelin Luo, Andrej Sajenko, and Jakob T. Spooner. Two Moves per Time Step Make a Difference. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 141:1-141:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{erlebach_et_al:LIPIcs.ICALP.2019.141, author = {Erlebach, Thomas and Kammer, Frank and Luo, Kelin and Sajenko, Andrej and Spooner, Jakob T.}, title = {{Two Moves per Time Step Make a Difference}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {141:1--141:14}, 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.2019.141}, URN = {urn:nbn:de:0030-drops-107176}, doi = {10.4230/LIPIcs.ICALP.2019.141}, annote = {Keywords: Temporal Graph Exploration, Algorithmic Graph Theory, NP-Complete Problem} }

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

We study an on-line scheduling problem that is motivated by applications such as car-sharing for trips between an airport and a group of hotels. Users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time, the pick-up location, and the drop-off location, where one of the two locations must be the airport. A request must be submitted a fixed amount of time before the pick-up time. The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). In the unit travel time variant, the travel time between the airport and any hotel is a fixed value t. We give a 2-competitive algorithm for the case in which the booking interval (pick-up time minus booking time) is at least t and the number of servers is even. In the arbitrary travel time variant, the travel time between the airport and a hotel may have arbitrary length between t and L t for some L >= 1. We give an algorithm with competitive ratio O(log L) if the number of servers is at least ceil[log L]. For both variants, we prove matching lower bounds on the competitive ratio of any deterministic on-line algorithm.

Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Car-Sharing on a Star Network: On-Line Scheduling with k Servers. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:LIPIcs.STACS.2019.51, author = {Luo, Kelin and Erlebach, Thomas and Xu, Yinfeng}, title = {{Car-Sharing on a Star Network: On-Line Scheduling with k Servers}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {51:1--51:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.51}, URN = {urn:nbn:de:0030-drops-102907}, doi = {10.4230/LIPIcs.STACS.2019.51}, annote = {Keywords: Car-Sharing System, On-Line Scheduling, Competitive Analysis, Star Network} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm and prove that this algorithm is a 3/2-approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem and prove that, in one particular case, the algorithm is a 5/4-approximation algorithm. Our analysis is tight in all cases except one.

Annette M. C. Ficker, Thomas Erlebach, Matús Mihalák, and Frits C. R. Spieksma. Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 45:1-45:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ficker_et_al:LIPIcs.ISAAC.2018.45, author = {Ficker, Annette M. C. and Erlebach, Thomas and Mihal\'{a}k, Mat\'{u}s and Spieksma, Frits C. R.}, title = {{Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {45:1--45:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.45}, URN = {urn:nbn:de:0030-drops-99933}, doi = {10.4230/LIPIcs.ISAAC.2018.45}, annote = {Keywords: approximation algorithm, matching, clustering problem} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study an on-line scheduling problem that is motivated by applications such as car-sharing, in which users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destination). The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). We consider two variants of the problem with respect to constraints on the booking time: In the fixed booking time variant, a request must be submitted a fixed amount of time before the pick-up time. In the variable booking time variant, a request can be submitted at any time during a certain time interval (called the booking horizon) that precedes the pick-up time. We present lower bounds on the competitive ratio for both variants and propose a balanced greedy algorithm (BGA) that achieves the best possible competitive ratio. We prove that, for the fixed booking time variant, BGA is 1.5-competitive if k=3i ( i in N) and the fixed booking length is not less than the travel time between the two locations; for the variable booking time variant, BGA is 1.5-competitive if k=3i ( i in N) and the length of the booking horizon is less than the travel time between the two locations, and BGA is 5/3-competitive if k=5i ( i in N) and the length of the booking horizon is not less than the travel time between the two locations.

Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Online Scheduling of Car-Sharing Requests Between Two Locations with Many Cars and Flexible Advance Bookings. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luo_et_al:LIPIcs.ISAAC.2018.64, author = {Luo, Kelin and Erlebach, Thomas and Xu, Yinfeng}, title = {{Online Scheduling of Car-Sharing Requests Between Two Locations with Many Cars and Flexible Advance Bookings}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {64:1--64:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.64}, URN = {urn:nbn:de:0030-drops-100122}, doi = {10.4230/LIPIcs.ISAAC.2018.64}, annote = {Keywords: Car-sharing system, Competitive analysis, On-line scheduling} }

Document

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

A temporal graph can be viewed as a sequence of static graphs indexed by discrete time steps. The vertex set of each graph in the sequence remains the same; however, the edge sets are allowed to differ. A natural problem on temporal graphs is the Temporal Exploration problem (TEXP): given, as input, a temporal graph G of order n, we are tasked with computing an exploration schedule (i.e., a temporal walk that visits all vertices in G), such that the time step at which the walk arrives at the last unvisited vertex is minimised (we refer to this time step as the arrival time). It can be easily shown that general temporal graphs admit exploration schedules with arrival time no greater than O(n^2). Moreover, it has been shown previously that there exists an infinite family of temporal graphs for which any exploration schedule has arrival time Omega(n^2), making these bounds tight for general TEXP instances. We consider restricted instances of TEXP, in which the temporal graph given as input is, in every time step, of maximum degree d; we show an O(n^2/log n) bound on the arrival time when d is constant, and an O(d log d * n^2/log n) bound when d is given as some function of n.

Thomas Erlebach and Jakob T. Spooner. Faster Exploration of Degree-Bounded Temporal Graphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{erlebach_et_al:LIPIcs.MFCS.2018.36, author = {Erlebach, Thomas and Spooner, Jakob T.}, title = {{Faster Exploration of Degree-Bounded Temporal Graphs}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {36:1--36:13}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.36}, URN = {urn:nbn:de:0030-drops-96181}, doi = {10.4230/LIPIcs.MFCS.2018.36}, annote = {Keywords: temporal graph exploration, algorithmic graph theory, NP-complete problem} }

Document

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

In this paper, we consider an on-line scheduling problem that is motivated by applications such as car sharing, in which users submit ride requests, and the scheduler aims to accept requests of maximum total profit using two servers (cars). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destination). The length of the time interval between the submission of a request (booking time) and the pick-up time is fixed. The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted. We present lower bounds on the competitive ratio for this problem and propose a smart greedy algorithm that achieves the best possible competitive ratio.

Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Car-Sharing between Two Locations: Online Scheduling with Two Servers. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luo_et_al:LIPIcs.MFCS.2018.50, author = {Luo, Kelin and Erlebach, Thomas and Xu, Yinfeng}, title = {{Car-Sharing between Two Locations: Online Scheduling with Two Servers}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {50:1--50:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.50}, URN = {urn:nbn:de:0030-drops-96325}, doi = {10.4230/LIPIcs.MFCS.2018.50}, annote = {Keywords: Car-sharing system, Competitive analysis, On-line scheduling} }

Document

**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We introduce a novel model for scheduling with explorable uncertainty. In this model, the processing time of a job can potentially be reduced (by an a priori unknown amount) by testing the job. Testing a job j takes one unit of time and may reduce its processing time from the given upper limit p'_j (which is the time taken to execute the job if it is not tested) to any value between 0 and p'_j. This setting is motivated e.g. by applications where a code optimizer can be run on a job before executing it. We consider the objective of minimizing the sum of completion times on a single machine. All jobs are available from the start, but the reduction in their processing times as a result of testing is unknown, making this an online problem that is amenable to competitive analysis. The need to balance the time spent on tests and the time spent on job executions adds a novel flavor to the problem. We give the first and nearly tight lower and upper bounds on the competitive ratio for deterministic and randomized algorithms. We also show that minimizing the makespan is a considerably easier problem for which we give optimal deterministic and randomized online algorithms.

Christoph Dürr, Thomas Erlebach, Nicole Megow, and Julie Meißner. Scheduling with Explorable Uncertainty. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{durr_et_al:LIPIcs.ITCS.2018.30, author = {D\"{u}rr, Christoph and Erlebach, Thomas and Megow, Nicole and Mei{\ss}ner, Julie}, title = {{Scheduling with Explorable Uncertainty}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {30:1--30:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.30}, URN = {urn:nbn:de:0030-drops-83360}, doi = {10.4230/LIPIcs.ITCS.2018.30}, annote = {Keywords: online scheduling, explorable uncertainty, competitive ratio, makespan, sum of completion times} }

Document

Complete Volume

**Published in:** OASIcs, Volume 14, 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10) (2010)

OASIcs, Volume 14, ATMOS'10, Complete Volume

10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Open Access Series in Informatics (OASIcs), Volume 14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@Proceedings{erlebach_et_al:OASIcs.ATMOS.2010, title = {{OASIcs, Volume 14, ATMOS'10, Complete Volume}}, booktitle = {10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10)}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-20-0}, ISSN = {2190-6807}, year = {2012}, volume = {14}, editor = {Erlebach, Thomas and L\"{u}bbecke, Marco}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2010}, URN = {urn:nbn:de:0030-drops-35767}, doi = {10.4230/OASIcs.ATMOS.2010}, annote = {Keywords: Analysis of Algorithms and Problem Complexity, Optimization, Graph Theory, Applications} }

Document

Front Matter

**Published in:** OASIcs, Volume 14, 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10) (2010)

Titlepage, Table of Contents, Preface, Organization.

10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Open Access Series in Informatics (OASIcs), Volume 14, pp. i-ix, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{erlebach_et_al:OASIcs.ATMOS.2010.i, author = {Erlebach, Thomas and L\"{u}bbecke, Marco}, title = {{Frontmatter, Table of Contents, Preface, Organization}}, booktitle = {10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10)}, pages = {i--ix}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-20-0}, ISSN = {2190-6807}, year = {2010}, volume = {14}, editor = {Erlebach, Thomas and L\"{u}bbecke, Marco}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2010.i}, URN = {urn:nbn:de:0030-drops-27584}, doi = {10.4230/OASIcs.ATMOS.2010.i}, annote = {Keywords: Titlepage, Table of Contents, Preface, Organization} }

Document

**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

For $t,g>0$, a vertex-weighted graph of total weight $W$ is
$(t,g)$-trimmable if it contains a vertex-induced subgraph of total
weight at least $(1-1/t)W$ and with no simple path of more than $g$
edges. A family of graphs is trimmable if for each constant $t>0$,
there is a constant $g=g(t)$ such that every vertex-weighted graph
in the family is $(t,g)$-trimmable. We show that every family of
graphs of bounded domino treewidth is trimmable. This implies that
every family of graphs of bounded degree is trimmable if the graphs
in the family have bounded treewidth or are planar. Based on this
result, we derive a polynomial-time approximation scheme for the
problem of labeling weighted points with nonoverlapping sliding
labels of unit height and given lengths so as to maximize the total
weight of the labeled points. This settles one of the last major
open questions in the theory of map labeling.

Thomas Erlebach, Torben Hagerup, Klaus Jansen, Moritz Minzlaff, and Alexander Wolff. Trimming of Graphs, with Application to Point Labeling. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 265-276, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{erlebach_et_al:LIPIcs.STACS.2008.1350, author = {Erlebach, Thomas and Hagerup, Torben and Jansen, Klaus and Minzlaff, Moritz and Wolff, Alexander}, title = {{Trimming of Graphs, with Application to Point Labeling}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {265--276}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1350}, URN = {urn:nbn:de:0030-drops-13509}, doi = {10.4230/LIPIcs.STACS.2008.1350}, annote = {Keywords: Trimming weighted graphs, domino treewidth, planar graphs, point-feature label placement, map labeling, polynomial-time approximation schemes} }

Document

**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We consider the minimum spanning tree problem in a setting where
information about the edge weights of the given graph is uncertain.
Initially, for each edge $e$ of the graph only a set $A_e$, called
an uncertainty area, that contains the actual edge weight
$w_e$ is known. The algorithm can `update' $e$ to obtain the edge
weight $w_e in A_e$. The task is to output the edge set of a
minimum spanning tree after a minimum number of updates. An
algorithm is $k$-update competitive if it makes at most $k$ times
as many updates as the optimum. We present a $2$-update
competitive algorithm if all areas $A_e$ are open or trivial, which
is the best possible among deterministic algorithms. The condition
on the areas $A_e$ is to exclude degenerate inputs for which no
constant update competitive algorithm can exist.
Next, we consider a setting where the vertices of the graph
correspond to points in Euclidean space and the weight of an edge
is equal to the distance of its endpoints. The location of each
point is initially given as an uncertainty area, and an update
reveals the exact location of the point. We give a general
relation between the edge uncertainty and the vertex uncertainty
versions of a problem and use it to derive a $4$-update competitive
algorithm for the minimum spanning tree problem in the vertex
uncertainty model. Again, we show that this is best possible among
deterministic algorithms.

Michael Hoffmann, Thomas Erlebach, Danny Krizanc, Matús Mihal'ák, and Rajeev Raman. Computing Minimum Spanning Trees with Uncertainty. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 277-288, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{hoffmann_et_al:LIPIcs.STACS.2008.1358, author = {Hoffmann, Michael and Erlebach, Thomas and Krizanc, Danny and Mihal'\'{a}k, Mat\'{u}s and Raman, Rajeev}, title = {{Computing Minimum Spanning Trees with Uncertainty}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {277--288}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1358}, URN = {urn:nbn:de:0030-drops-13581}, doi = {10.4230/LIPIcs.STACS.2008.1358}, annote = {Keywords: Algorithms and data structures; Current challenges: mobile and net computing} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 5031, Algorithms for Optimization with Incomplete Information (2005)

Consider the problem of discovering (or verifying) the edges and non-edges of a network, modelled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and non-edges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O(sqrt(n*log n)) for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless P=NP. Furthermore, we prove that the optimal number of queries for d-dimensional hypercubes is Theta(d/log d).

Zuzana Beerliova, Felix Eberhard, Thomas Erlebach, Alexander Hall, Michael Hoffmann, Matus Mihalak, and L. Shankar Ram. Network Discovery and Verification. In Algorithms for Optimization with Incomplete Information. Dagstuhl Seminar Proceedings, Volume 5031, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{beerliova_et_al:DagSemProc.05031.17, author = {Beerliova, Zuzana and Eberhard, Felix and Erlebach, Thomas and Hall, Alexander and Hoffmann, Michael and Mihalak, Matus and Ram, L. Shankar}, title = {{Network Discovery and Verification}}, booktitle = {Algorithms for Optimization with Incomplete Information}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5031}, editor = {Susanne Albers and Rolf H. M\"{o}hring and Georg Ch. Pflug and R\"{u}diger Schultz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05031.17}, URN = {urn:nbn:de:0030-drops-594}, doi = {10.4230/DagSemProc.05031.17}, annote = {Keywords: on-line algorithms , set cover , landmarks , metric dimension} }

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