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**Published in:** Dagstuhl Reports, Volume 13, Issue 7 (2024)

Parameterization and approximation are two established approaches of coping with intractability in combinatorial optimization. In this Dagstuhl Seminar, we studied parameterized approximation as a relatively new algorithmic paradigm that combines these two popular research areas. In particular, we analyzed the solution quality (approximation ratio) as well as the running time of an algorithm in terms of a parameter that captures the "complexity" of a problem instance.
While the field has grown and yielded some promising results, our understanding of the area is rather ad-hoc compared to our knowledge in approximation or parameterized algorithms alone. In this seminar, we brought together researchers from both communities in order to bridge this gap by accommodating the exchange and unification of scientific knowledge.

Karthik C. S., Parinya Chalermsook, Joachim Spoerhase, Meirav Zehavi, and Martin Herold. Parameterized Approximation: Algorithms and Hardness (Dagstuhl Seminar 23291). In Dagstuhl Reports, Volume 13, Issue 7, pp. 96-107, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{c.s._et_al:DagRep.13.7.96, author = {C. S., Karthik and Chalermsook, Parinya and Spoerhase, Joachim and Zehavi, Meirav and Herold, Martin}, title = {{Parameterized Approximation: Algorithms and Hardness (Dagstuhl Seminar 23291)}}, pages = {96--107}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2024}, volume = {13}, number = {7}, editor = {C. S., Karthik and Chalermsook, Parinya and Spoerhase, Joachim and Zehavi, Meirav and Herold, Martin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.7.96}, URN = {urn:nbn:de:0030-drops-197764}, doi = {10.4230/DagRep.13.7.96}, annote = {Keywords: approximation algorithms, Hardness of approximation, Parameterized algorithms} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We propose and study a generalization to the well-known problem of polyline simplification. Instead of a single polyline, we are given a set of l polylines possibly sharing some line segments and bend points. Our goal is to minimize the number of bend points in the simplified bundle with respect to some error tolerance δ (measuring Fréchet distance) but under the additional constraint that shared parts have to be simplified consistently. We show that polyline bundle simplification is NP-hard to approximate within a factor n^(1/3 - ε) for any ε > 0 where n is the number of bend points in the polyline bundle. This inapproximability even applies to instances with only l=2 polylines. However, we identify the sensitivity of the solution to the choice of δ as a reason for this strong inapproximability. In particular, we prove that if we allow δ to be exceeded by a factor of 2 in our solution, we can find a simplified polyline bundle with no more than 𝒪(log (l + n)) ⋅ OPT bend points in polytime, providing us with an efficient bi-criteria approximation. As a further result, we show fixed-parameter tractability in the number of shared bend points.

Joachim Spoerhase, Sabine Storandt, and Johannes Zink. Simplification of Polyline Bundles. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{spoerhase_et_al:LIPIcs.SWAT.2020.35, author = {Spoerhase, Joachim and Storandt, Sabine and Zink, Johannes}, title = {{Simplification of Polyline Bundles}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {35:1--35:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.35}, URN = {urn:nbn:de:0030-drops-122821}, doi = {10.4230/LIPIcs.SWAT.2020.35}, annote = {Keywords: Polyline Simplification, Bi-criteria Approximation, Hardness of Approximation, Geometric Set Cover} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant epsilon >0 achieves a guarantee of 1-(1/e)-epsilon while violating only the covering constraints by a multiplicative factor of 1-epsilon. Our algorithm is based on a novel enumeration method, which unlike previously known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraint as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach. While this approach does not give tight bounds it yields deterministic and in some special cases also considerably faster algorithms. For example, for the well-studied special case of only packing constraints (Kulik et al. [Math. Oper. Res. `13] and Chekuri et al. [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.

Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, and Sumedha Uniyal. A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mizrachi_et_al:LIPIcs.ICALP.2019.85, author = {Mizrachi, Eyal and Schwartz, Roy and Spoerhase, Joachim and Uniyal, Sumedha}, title = {{A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {85:1--85:15}, 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.2019.85}, URN = {urn:nbn:de:0030-drops-106610}, doi = {10.4230/LIPIcs.ICALP.2019.85}, annote = {Keywords: submodular function, approximation algorithm, covering, packing} }

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

We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far.
Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity.
Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.

Timothy M. Chan, Thomas C. van Dijk, Krzysztof Fleszar, Joachim Spoerhase, and Alexander Wolff. Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chan_et_al:LIPIcs.ISAAC.2018.61, author = {Chan, Timothy M. and van Dijk, Thomas C. and Fleszar, Krzysztof and Spoerhase, Joachim and Wolff, Alexander}, title = {{Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {61:1--61:13}, 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.2018.61}, URN = {urn:nbn:de:0030-drops-100094}, doi = {10.4230/LIPIcs.ISAAC.2018.61}, annote = {Keywords: Geometric optimization, NP-hard, approximation, shallow-cell complexity, line stabbing} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

In their seminal work, Mustafa and Ray [Nabil H. Mustafa and Saurabh Ray, 2010] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search - this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [Greg N. Frederickson, 1987]. Obtaining similar results for the related maximum coverage problem (MC) seems non-trivial due to the hard cardinality constraint. In fact, while Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] have shown (via a different analysis) that local search yields a PTAS for two-dimensional real halfspaces, they only conjectured that the same holds true for dimension three. Interestingly, at this point it was already known that local search provides a PTAS for the corresponding set cover case and this followed directly from the approach of Mustafa and Ray.
In this work we provide a way to address the above-mentioned issue. First, we propose a color-balanced version of the planar separator theorem. The resulting subdivision approximates locally in each part the global distribution of the colors. Second, we show how this roughly balanced subdivision can be employed in a more careful analysis to strictly obey the hard cardinality constraint. More specifically, we obtain a PTAS for any "planarizable" instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we confirm the conjecture of Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] regarding real halfspaces in dimension three. We feel that our ideas could also be helpful in other geometric settings involving a cardinality constraint.

Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase. Approximation Schemes for Geometric Coverage Problems. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chaplick_et_al:LIPIcs.ESA.2018.17, author = {Chaplick, Steven and De, Minati and Ravsky, Alexander and Spoerhase, Joachim}, title = {{Approximation Schemes for Geometric Coverage Problems}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.17}, URN = {urn:nbn:de:0030-drops-94809}, doi = {10.4230/LIPIcs.ESA.2018.17}, annote = {Keywords: balanced separators, maximum coverage, local search, approximation scheme, geometric approximation algorithms} }

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Brief Announcement

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

In this announcement, we show that the classical Maximum Coverage problem (MC) admits a PTAS via local search in essentially all cases where the corresponding instances of Set Cover (SC) admit a PTAS via the local search approach by Mustafa and Ray [Nabil H. Mustafa and Saurabh Ray, 2010]. As a corollary, we answer an open question by Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] regarding half-spaces in R^3 thereby settling the existence of PTASs for essentially all natural cases of geometric MC problems. As an intermediate result, we show a color-balanced version of the classical planar subdivision theorem by Frederickson [Greg N. Frederickson, 1987]. We believe that some of our ideas may be useful for analyzing local search in other settings involving a hard cardinality constraint.

Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase. Brief Announcement: Approximation Schemes for Geometric Coverage Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 107:1-107:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chaplick_et_al:LIPIcs.ICALP.2018.107, author = {Chaplick, Steven and De, Minati and Ravsky, Alexander and Spoerhase, Joachim}, title = {{Brief Announcement: Approximation Schemes for Geometric Coverage Problems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {107:1--107:4}, 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.2018.107}, URN = {urn:nbn:de:0030-drops-91113}, doi = {10.4230/LIPIcs.ICALP.2018.107}, annote = {Keywords: balanced separators, maximum coverage, local search, approximation scheme, geometric approximation algorithms} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We study the classical NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is Omega(log^{1/2 - varepsilon} n), assuming NP not subseteq ZPTIME(n^{poly log n}). This constitutes a significant gap to the best known approximation upper bound of O(n^1/2) due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an O(1)-approximation when edges (or nodes) may be used by O(log n/log log n) paths.
In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following.
- For MaxEDP, we give an O(r^0.5 log^1.5 kr)-approximation algorithm. As r<=n, up to logarithmic factors, our result strengthens the best known ratio O(n^0.5) due to Chekuri et al.
- Further, we show how to route Omega(opt) pairs with congestion O(log(kr)/log log(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.
- For MaxNDP, we give an algorithm that gives the optimal answer in time (k+r)^O(r)n. This is a substantial improvement on the run time of 2^kr^O(r)n, which can be obtained via an algorithm by Scheffler.
We complement these positive results by proving that MaxEDP is NP-hard even for r=1, and MaxNDP is W[1]-hard for parameter r. This shows that neither problem is fixed-parameter tractable in r unless FPT = W[1] and that our approximability results are relevant even for very small constant values of r.

Krzysztof Fleszar, Matthias Mnich, and Joachim Spoerhase. New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fleszar_et_al:LIPIcs.ESA.2016.42, author = {Fleszar, Krzysztof and Mnich, Matthias and Spoerhase, Joachim}, title = {{New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {42:1--42:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.42}, URN = {urn:nbn:de:0030-drops-63542}, doi = {10.4230/LIPIcs.ESA.2016.42}, annote = {Keywords: disjoint paths, approximation algorithms, feedback vertex set} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In a directed graph G with non-correlated edge lengths and costs, the network design problem with bounded distances asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria (2+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for this problem. This improves on the currently best known linear approximation bound, at the cost of violating the distance bound by a factor of at most 2+\varepsilon.
In the course of proving this result, the related problem of directed shallow-light Steiner trees arises as a subproblem. In the context of directed graphs, approximations to this problem have been elusive. We present the first non-trivial result by proposing a (1+\varepsilon,O(|R|^{\varepsilon}))-ap\-proximation, where R is the set of terminals.
Finally, we show how to apply our results to obtain an (\alpha+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for light-weight directed \alpha-spanners. For this, no non-trivial approximation algorithm has been known before. All running times depends on n and
\varepsilon and are polynomial in n for any fixed \varepsilon>0.

Markus Chimani and Joachim Spoerhase. Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 238-248, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chimani_et_al:LIPIcs.STACS.2015.238, author = {Chimani, Markus and Spoerhase, Joachim}, title = {{Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {238--248}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.238}, URN = {urn:nbn:de:0030-drops-49170}, doi = {10.4230/LIPIcs.STACS.2015.238}, annote = {Keywords: network design, approximation algorithm, shallow-light spanning trees, spanners} }

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