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

A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a (2e+ε)-approximation for any ε > 0. For the case that all vertices have unit weight, we provide a 2e-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an 8-approximation was known.

Svenja M. Griesbach, Felix Hommelsheim, Max Klimm, and Kevin Schewior. Improved Approximation Algorithms for the Expanding Search Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 54:1-54:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{griesbach_et_al:LIPIcs.ESA.2023.54, author = {Griesbach, Svenja M. and Hommelsheim, Felix and Klimm, Max and Schewior, Kevin}, title = {{Improved Approximation Algorithms for the Expanding Search Problem}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {54:1--54:15}, 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.2023.54}, URN = {urn:nbn:de:0030-drops-187073}, doi = {10.4230/LIPIcs.ESA.2023.54}, annote = {Keywords: Approximation Algorithm, Expanding Search, Search Problem, Graph Exploration, Traveling Repairperson Problem} }

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Track A: Algorithms, Complexity and Games

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

We consider the problem of finding an incremental solution to a cardinality-constrained maximization problem that not only captures the solution for a fixed cardinality, but also describes how to gradually grow the solution as the cardinality bound increases. The goal is to find an incremental solution that guarantees a good competitive ratio against the optimum solution for all cardinalities simultaneously. The central challenge is to characterize maximization problems where this is possible, and to determine the best-possible competitive ratio that can be attained. A lower bound of 2.18 and an upper bound of φ + 1 ≈ 2.618 are known on the competitive ratio for monotone and accountable objectives [Bernstein et al., Math. Prog., 2022], which capture a wide range of maximization problems. We introduce a continuization technique and identify an optimal incremental algorithm that provides strong evidence that φ + 1 is the best-possible competitive ratio. Using this continuization, we obtain an improved lower bound of 2.246 by studying a particular recurrence relation whose characteristic polynomial has complex roots exactly beyond the lower bound. Based on the optimal continuous algorithm combined with a scaling approach, we also provide a 1.772-competitive randomized algorithm. We complement this by a randomized lower bound of 1.447 via Yao’s principle.

Yann Disser, Max Klimm, Kevin Schewior, and David Weckbecker. Incremental Maximization via Continuization. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{disser_et_al:LIPIcs.ICALP.2023.47, author = {Disser, Yann and Klimm, Max and Schewior, Kevin and Weckbecker, David}, title = {{Incremental Maximization via Continuization}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {47:1--47:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.47}, URN = {urn:nbn:de:0030-drops-180992}, doi = {10.4230/LIPIcs.ICALP.2023.47}, annote = {Keywords: incremental optimization, competitive analysis, robust matching, submodular function} }

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APPROX

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop an algorithm with a robustness factor that is decreasing in the curvature c of the submodular function. For the extreme cases c = 0 corresponding to a modular objective, it matches a previously known and best possible robustness factor of 1/2. For the other extreme case of c = 1 it yields a robustness factor of ≈ 0.35 improving over the best previously known robustness factor of ≈ 0.06.

Max Klimm and Martin Knaack. Maximizing a Submodular Function with Bounded Curvature Under an Unknown Knapsack Constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{klimm_et_al:LIPIcs.APPROX/RANDOM.2022.49, author = {Klimm, Max and Knaack, Martin}, title = {{Maximizing a Submodular Function with Bounded Curvature Under an Unknown Knapsack Constraint}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {49:1--49:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.49}, URN = {urn:nbn:de:0030-drops-171711}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.49}, annote = {Keywords: submodular function, knapsack, approximation algorithm, robust optimization} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Wardrop equilibria in nonatomic congestion games are in general inefficient as they do not induce an optimal flow that minimizes the total travel time. Network tolls are a prominent and popular way to induce an optimum flow in equilibrium. The classical approach to find such tolls is marginal cost pricing which requires the exact knowledge of the demand on the network. In this paper, we investigate under which conditions demand-independent optimum tolls exist that induce the system optimum flow for any travel demand in the network. We give several characterizations for the existence of such tolls both in terms of the cost structure and the network structure of the game. Specifically we show that demand-independent optimum tolls exist if and only if the edge cost functions are shifted monomials as used by the Bureau of Public Roads. Moreover, non-negative demand-independent optimum tolls exist when the network is a directed acyclic multi-graph. Finally, we show that any network with a single origin-destination pair admits demand-independent optimum tolls that, although not necessarily non-negative, satisfy a budget constraint.

Riccardo Colini-Baldeschi, Max Klimm, and Marco Scarsini. Demand-Independent Optimal Tolls. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 151:1-151:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{colinibaldeschi_et_al:LIPIcs.ICALP.2018.151, author = {Colini-Baldeschi, Riccardo and Klimm, Max and Scarsini, Marco}, title = {{Demand-Independent Optimal Tolls}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {151:1--151:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.151}, URN = {urn:nbn:de:0030-drops-91553}, doi = {10.4230/LIPIcs.ICALP.2018.151}, annote = {Keywords: nonatomic congestion games, efficiency of equlibria, tolls} }

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

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs.
In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible.
Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance d, the corresponding super-vertices remain at distance at least \varphi(d) in the contracted graph, where \varphi is a tolerance function bounding the permitted distance distortion.
We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions \varphi(x)=x/\alpha-\beta, where \alpha \geq 1 and \beta \geq 0 are arbitrary real-valued parameters.
Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases.
Further we analyze the asymptotic behavior of the size of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.

Aaron Bernstein, Karl Däubel, Yann Disser, Max Klimm, Torsten Mütze, and Frieder Smolny. Distance-Preserving Graph Contractions. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bernstein_et_al:LIPIcs.ITCS.2018.51, author = {Bernstein, Aaron and D\"{a}ubel, Karl and Disser, Yann and Klimm, Max and M\"{u}tze, Torsten and Smolny, Frieder}, title = {{Distance-Preserving Graph Contractions}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {51:1--51:14}, 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.2018.51}, URN = {urn:nbn:de:0030-drops-83427}, doi = {10.4230/LIPIcs.ITCS.2018.51}, annote = {Keywords: distance oracle, contraction, spanner} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We revisit a classical problem in transportation, known as the continuous (bilevel) network design problem, CNDP for short. Given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed, the goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing cost of the induced Wardrop equilibrium and the investment cost for installing the capacity. While this problem is considered as challenging in the literature, its complexity status was still unknown. We close this gap showing that CNDP is strongly NP-complete and APX-hard, both on directed and undirected networks and even for instances with affine latencies.
As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions (Math. Program., Vol. 34, 1986). We derive a closed form expression of its approximation guarantee for arbitrary sets of latency functions. We then propose a different approximation algorithm and show that it has the same approximation guarantee. However, we show that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we derive a closed form expression. For affine latencies, e.g., this algorithm achieves a 49/41-approximation which improves on the 5/4 that has been shown before by Marcotte. We finally discuss the case of hard budget constraints on the capacity investment.

Martin Gairing, Tobias Harks, and Max Klimm. Complexity and Approximation of the Continuous Network Design Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 226-241, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{gairing_et_al:LIPIcs.APPROX-RANDOM.2014.226, author = {Gairing, Martin and Harks, Tobias and Klimm, Max}, title = {{Complexity and Approximation of the Continuous Network Design Problem}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {226--241}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.226}, URN = {urn:nbn:de:0030-drops-46998}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.226}, annote = {Keywords: Bilevel optimization, Optimization under equilibrium constraints, Network design, Wardrop equilibrium, Computational complexity, Approximation algorit} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We study the existence of approximate pure Nash equilibria in weighted congestion games and develop techniques to obtain approximate potential functions that prove the existence of alpha-approximate pure Nash equilibria and the convergence of alpha-improvement steps. Specifically, we show how to obtain upper bounds for approximation factor alpha for a given class of cost functions. For example for concave cost functions the factor is at most 3/2, for quadratic cost functions it is at most 4/3, and for polynomial cost functions of maximal degree d it is at at most d + 1. For games with two players we obtain tight bounds which are as small as for example 1.054 in the case of quadratic cost functions.

Christoph Hansknecht, Max Klimm, and Alexander Skopalik. Approximate Pure Nash Equilibria in Weighted Congestion Games. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 242-257, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{hansknecht_et_al:LIPIcs.APPROX-RANDOM.2014.242, author = {Hansknecht, Christoph and Klimm, Max and Skopalik, Alexander}, title = {{Approximate Pure Nash Equilibria in Weighted Congestion Games}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {242--257}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.242}, URN = {urn:nbn:de:0030-drops-47005}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.242}, annote = {Keywords: Congestion game, Pure Nash equilibrium, Approximate equilibrium, Existence, Potential function} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio. Both factors are shown to be best possible.
In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items and try to pack the items in this order, independent of the observations made while packing. We give efficient algorithms computing these policies. On the other hand, we show that, for any a>1, the problem of deciding whether a given universal policy achieves a factor of a is coNP-complete. If a is part of the input, the same problem is shown to be coNP-complete for items with unit densities. Finally, we show that it is coNP-hard to decide, for given a, whether a set of items admits a universal policy with factor a, even if all items have unit densities.

Yann Disser, Max Klimm, Nicole Megow, and Sebastian Stiller. Packing a Knapsack of Unknown Capacity. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 276-287, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{disser_et_al:LIPIcs.STACS.2014.276, author = {Disser, Yann and Klimm, Max and Megow, Nicole and Stiller, Sebastian}, title = {{Packing a Knapsack of Unknown Capacity}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {276--287}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.276}, URN = {urn:nbn:de:0030-drops-44642}, doi = {10.4230/LIPIcs.STACS.2014.276}, annote = {Keywords: Knapsack, unknown capacity, robustness, approximation algorithms} }

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