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Documents authored by Kaul, Matthias

Document
DOI: 10.4230/LIPIcs.STACS.2025.57
Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously

Authors: Matthias Kaul, Kelin Luo, Matthias Mnich, and Heiko Röglin

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We study the problem of partitioning a set of n objects in a metric space into k clusters V₁,...,V_k. The quality of the clustering is measured by considering the vector of cluster costs and then minimizing some monotone symmetric norm of that vector (in particular, this includes the 𝓁_p-norms). For the costs of the clusters we take the weight of a minimum-weight spanning tree on the objects in V_i, which may serve as a proxy for the cost of traversing all objects in the cluster, for example in the context of Multirobot Coverage as studied by Zheng, Koenig, Kempe, Jain (IROS 2005), but also as a shape-invariant measure of cluster density similar to Single-Linkage Clustering. This problem has been studied by Even, Garg, Könemann, Ravi, Sinha (Oper. Res. Lett., 2004) for the setting of minimizing the weight of the largest cluster (i.e., using 𝓁_∞) as Min-Max Tree Cover, for which they gave a constant-factor approximation algorithm. We provide a careful adaptation of their algorithm to compute solutions which are approximately optimal with respect to all monotone symmetric norms simultaneously, and show how to find them in polynomial time. In fact, our algorithm is purely combinatorial and can process metric spaces with 10,000 points in less than a second. As an extension, we also consider the case where instead of a target number of clusters we are provided with a set of depots in the space such that every cluster should contain at least one such depot. One can consider these as the fixed starting points of some agents that will traverse all points of a cluster. For this setting also we are able to give a polynomial-time algorithm computing a constant-factor approximation with respect to all monotone symmetric norms simultaneously. To show that the algorithmic results are tight up to the precise constant of approximation attainable, we also prove that such clustering problems are already APX-hard when considering only one single 𝓁_p norm for the objective.
Cite as

Matthias Kaul, Kelin Luo, Matthias Mnich, and Heiko Röglin. Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kaul_et_al:LIPIcs.STACS.2025.57,
  author =	{Kaul, Matthias and Luo, Kelin and Mnich, Matthias and R\"{o}glin, Heiko},
  title =	{{Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{57:1--57:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.57},
  URN =		{urn:nbn:de:0030-drops-228821},
  doi =		{10.4230/LIPIcs.STACS.2025.57},
  annote =	{Keywords: Clustering, spanning trees, all-norm approximation}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2024.19
Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates

Authors: Matthias Kaul, Matthias Mnich, and Hendrik Molter

Published in: LIPIcs, Volume 321, 19th International Symposium on Parameterized and Exact Computation (IPEC 2024)

Abstract
We study the fundamental scheduling problem 1|r_j|∑ w_j U_j: schedule a set of n jobs with weights, processing times, release dates, and due dates on a single machine, such that each job starts after its release date and we maximize the weighted number of jobs that complete execution before their due date. Problem 1|r_j|∑ w_j U_j generalizes both Knapsack and Partition, and the simplified setting without release dates was studied by Hermelin et al. [Annals of Operations Research, 2021] from a parameterized complexity viewpoint. Our main contribution is a thorough complexity analysis of 1|r_j|∑ w_j U_j in terms of four key problem parameters: the number p_# of processing times, the number w_# of weights, the number d_# of due dates, and the number r_# of release dates of the jobs. 1|r_j|∑ w_j U_j is known to be weakly para-NP-hard even if w_#+d_#+r_# is constant, and Heeger and Hermelin [ESA, 2024] recently showed (weak) 𝖶[1]-hardness parameterized by p_# or w_# even if r_# is constant. Algorithmically, we show that 1|r_j|∑ w_j U_j is fixed-parameter tractable parameterized by p_# combined with any two of the remaining three parameters w_#, d_#, and r_#. We further provide pseudo-polynomial XP-time algorithms for parameter r_# and d_#. To complement these algorithms, we show that 1|r_j|∑ w_j U_j is (strongly) 𝖶[1]-hard when parameterized by d_#+r_# even if w_# is constant. Our results provide a nearly complete picture of the complexity of 1|r_j|∑ w_j U_j for p_#, w_#, d_#, and r_# as parameters, and extend those of Hermelin et al. [Annals of Operations Research, 2021] for the problem 1||∑ w_j U_j without release dates.
Cite as

Matthias Kaul, Matthias Mnich, and Hendrik Molter. Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kaul_et_al:LIPIcs.IPEC.2024.19,
  author =	{Kaul, Matthias and Mnich, Matthias and Molter, Hendrik},
  title =	{{Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates}},
  booktitle =	{19th International Symposium on Parameterized and Exact Computation (IPEC 2024)},
  pages =	{19:1--19:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-353-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{321},
  editor =	{Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.19},
  URN =		{urn:nbn:de:0030-drops-222450},
  doi =		{10.4230/LIPIcs.IPEC.2024.19},
  annote =	{Keywords: Scheduling, Release Dates, Fixed-Parameter Tractability}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.39
A (3/2 + ε)-Approximation for Multiple TSP with a Variable Number of Depots

Authors: Max Deppert, Matthias Kaul, and Matthias Mnich

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

Abstract
One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the Multiple TSP: a set of m ≥ 1 salespersons collectively traverses a set of n cities by m non-trivial tours, to minimize the total length of their tours. This problem can also be considered to be a variant of Uncapacitated Vehicle Routing, where the objective is to minimize the sum of all tour lengths. When all m tours start from and end at a single common depot v₀, then the metric Multiple TSP can be approximated equally well as the standard metric TSP, as shown by Frieze (1983). The metric Multiple TSP becomes significantly harder to approximate when there is a set D of d ≥ 1 depots that form the starting and end points of the m tours. For this case, only a (2-1/d)-approximation in polynomial time is known, as well as a 3/2-approximation for constant d which requires a prohibitive run time of n^Θ(d) (Xu and Rodrigues, INFORMS J. Comput., 2015). A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another approximation algorithm for metric Multiple TSP with run time n^Θ(d), which reduces the problem to approximating metric TSP. In this paper we overcome the n^Θ(d) time barrier: we give the first efficient approximation algorithm for Multiple TSP with a variable number d of depots that yields a better-than-2 approximation. Our algorithm runs in time (1/ε)^O(dlog d) ⋅ n^O(1), and produces a (3/2+ε)-approximation with constant probability. For the graphic case, we obtain a deterministic 3/2-approximation in time 2^d ⋅ n^O(1).
Cite as

Max Deppert, Matthias Kaul, and Matthias Mnich. A (3/2 + ε)-Approximation for Multiple TSP with a Variable Number of Depots. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{deppert_et_al:LIPIcs.ESA.2023.39,
  author =	{Deppert, Max and Kaul, Matthias and Mnich, Matthias},
  title =	{{A (3/2 + \epsilon)-Approximation for Multiple TSP with a Variable Number of Depots}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{39:1--39:15},
  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.2023.39},
  URN =		{urn:nbn:de:0030-drops-186925},
  doi =		{10.4230/LIPIcs.ESA.2023.39},
  annote =	{Keywords: Traveling salesperson problem, rural postperson problem, multiple TSP, vehicle routing}
}
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