23 Search Results for "Miltzow, Tillmann"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.8
Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects

Authors: Pritam Acharya, Sujoy Bhore, Aaryan Gupta, Arindam Khan, Bratin Mondal, and Andreas Wiese

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

Abstract
We study the geometric knapsack problem in which we are given a set of d-dimensional objects (each with associated profits) and the goal is to find the maximum profit subset that can be packed non-overlappingly into a given d-dimensional (unit hypercube) knapsack. Even if d = 2 and all input objects are disks, this problem is known to be NP-hard [Demaine, Fekete, Lang, 2010]. In this paper, we give polynomial time (1+ε)-approximation algorithms for the following types of input objects in any constant dimension d: - disks and hyperspheres, - a class of fat convex polygons that generalizes regular k-gons for k ≥ 5 (formally, polygons with a constant number of edges, whose lengths are in a bounded range, and in which each angle is strictly larger than π/2), - arbitrary fat convex objects that are sufficiently small compared to the knapsack. We remark that in our PTAS for disks and hyperspheres, we output the computed set of objects, but for a O_ε(1) of them we determine their coordinates only up to an exponentially small error. However, it is not clear whether there always exists a (1+ε)-approximate solution that uses only rational coordinates for the disks' centers. We leave this as an open problem which is related to well-studied geometric questions in the realm of circle packing.
Cite as

Pritam Acharya, Sujoy Bhore, Aaryan Gupta, Arindam Khan, Bratin Mondal, and Andreas Wiese. Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{acharya_et_al:LIPIcs.ICALP.2024.8,
  author =	{Acharya, Pritam and Bhore, Sujoy and Gupta, Aaryan and Khan, Arindam and Mondal, Bratin and Wiese, Andreas},
  title =	{{Approximation Schemes for Geometric Knapsack for Packing Spheres and Fat Objects}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{8:1--8:20},
  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.2024.8},
  URN =		{urn:nbn:de:0030-drops-201511},
  doi =		{10.4230/LIPIcs.ICALP.2024.8},
  annote =	{Keywords: Approximation Algorithms, Polygon Packing, Circle Packing, Sphere Packing, Geometric Knapsack, Resource Augmentation}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.99
Constrained Level Planarity Is FPT with Respect to the Vertex Cover Number

Authors: Boris Klemz and Marie Diana Sieper

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

Abstract
The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity, each level y is equipped with a partial order ≺_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of ≺_y. Constrained Level Planarity is known to be a remarkably difficult problem: previous results by Klemz and Rote [ACM Trans. Alg.'19] and by Brückner and Rutter [SODA'17] imply that it remains NP-hard even when restricted to graphs whose tree-depth and feedback vertex set number are bounded by a constant and even when the instances are additionally required to be either proper, meaning that each edge spans two consecutive levels, or ordered, meaning that all given partial orders are total orders. In particular, these results rule out the existence of FPT-time (even XP-time) algorithms with respect to these and related graph parameters (unless P=NP). However, the parameterized complexity of Constrained Level Planarity with respect to the vertex cover number of the input graph remained open. In this paper, we show that Constrained Level Planarity can be solved in FPT-time when parameterized by the vertex cover number. In view of the previous intractability statements, our result is best-possible in several regards: a speed-up to polynomial time or a generalization to the aforementioned smaller graph parameters is not possible, even if restricting to proper or ordered instances.
Cite as

Boris Klemz and Marie Diana Sieper. Constrained Level Planarity Is FPT with Respect to the Vertex Cover Number. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 99:1-99:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{klemz_et_al:LIPIcs.ICALP.2024.99,
  author =	{Klemz, Boris and Sieper, Marie Diana},
  title =	{{Constrained Level Planarity Is FPT with Respect to the Vertex Cover Number}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{99:1--99:17},
  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.2024.99},
  URN =		{urn:nbn:de:0030-drops-202428},
  doi =		{10.4230/LIPIcs.ICALP.2024.99},
  annote =	{Keywords: Parameterized Complexity, Graph Drawing, Planar Poset Diagram, Level Planarity, Constrained Level Planarity, Vertex Cover, FPT, Computational Geometry}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.129
The Complexity of Computing in Continuous Time: Space Complexity Is Precision

Authors: Manon Blanc and Olivier Bournez

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

Abstract
Models of computations over the integers are equivalent from a computability and complexity theory point of view by the (effective) Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but simpler for continuous-time models, as there is a unifying mathematical model, provided by ordinary differential equations (ODEs). Each model corresponds to a particular class of ODEs. For example, the General Purpose Analog Computer model of Claude Shannon, introduced as a mathematical model of analogue machines (Differential Analyzers), is known to correspond to polynomial ODEs. However, the question of a robust complexity theory for such models and its relations to classical (discrete) computation theory is an old problem. There was some recent significant progress: it has been proved that (classical) time complexity corresponds to the length of the involved curves, i.e. to the length of the solutions of the corresponding polynomial ODEs. The question of whether there is a simple and robust way to measure space complexity remains. We argue that space complexity corresponds to precision and conversely. Concretely, we propose and prove an algebraic characterisation of FPSPACE, using continuous ODEs. Recent papers proposed algebraic characterisations of polynomial-time and polynomial-space complexity classes over the reals, but with a discrete-time: those algebras rely on discrete ODE schemes. Here, we use classical (continuous) ODEs, with the classic definition of derivation and hence with the more natural context of continuous-time associated with ODEs. We characterise both the case of polynomial space functions over the integers and the reals. This is done by proving two inclusions. The first is obtained using some original polynomial space method for solving ODEs. For the other, we prove that Turing machines, with a proper representation of real numbers, can be simulated by continuous ODEs and not just discrete ODEs. A major consequence is that the associated space complexity is provably related to the numerical stability of involved schemas and the associated required precision. We obtain that a problem can be solved in polynomial space if and only if it can be simulated by some numerically stable ODE, using a polynomial precision.
Cite as

Manon Blanc and Olivier Bournez. The Complexity of Computing in Continuous Time: Space Complexity Is Precision. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 129:1-129:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blanc_et_al:LIPIcs.ICALP.2024.129,
  author =	{Blanc, Manon and Bournez, Olivier},
  title =	{{The Complexity of Computing in Continuous Time: Space Complexity Is Precision}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{129:1--129:22},
  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.2024.129},
  URN =		{urn:nbn:de:0030-drops-202722},
  doi =		{10.4230/LIPIcs.ICALP.2024.129},
  annote =	{Keywords: Models of computation, Ordinary differential equations, Real computations, Analog computations, Complexity theory, Implicit complexity, Recursion scheme}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.17
Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain

Authors: Sarita de Berg, Tillmann Miltzow, and Frank Staals

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
We devise a data structure that can answer shortest path queries for two query points in a polygonal domain P on n vertices. For any ε > 0, the space complexity of the data structure is O(n^{10+ε}) and queries can be answered in O(log n) time. Alternatively, we can achieve a space complexity of O(n^{9+ε}) by relaxing the query time to O(log² n). This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They presented a data structure with O(n^{11}) space complexity and O(log n) query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with O(n^{9+ε}/𝓁^{4+O(ε)}) space complexity and O(𝓁 log² n) query time, for any integer 1 ≤ 𝓁 ≤ n. Furthermore, we present improved data structures for the special case where we restrict one (or both) of the query points to lie on the boundary of P. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes O(n^{6+ε}) and the query time O(log²n). When both query points are on the boundary, the space complexity is decreased further to O(n^{4+ε}) and the query time to O(log n), thereby improving an earlier result of Bae and Okamoto.
Cite as

Sarita de Berg, Tillmann Miltzow, and Frank Staals. Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deberg_et_al:LIPIcs.SoCG.2024.17,
  author =	{de Berg, Sarita and Miltzow, Tillmann and Staals, Frank},
  title =	{{Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{17:1--17:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.17},
  URN =		{urn:nbn:de:0030-drops-199628},
  doi =		{10.4230/LIPIcs.SoCG.2024.17},
  annote =	{Keywords: data structure, polygonal domain, geodesic distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.77
A Topological Version of Schaefer’s Dichotomy Theorem

Authors: Patrick Schnider and Simon Weber

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Schaefer’s dichotomy theorem states that a Boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of four given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, we analyze Boolean CSPs in terms of their topological complexity, instead of their computational complexity. Motivated by complexity and topological universality results in computational geometry, we attach a natural topological space to the set of solutions of a Boolean CSP and introduce the notion of projection-universality. We prove that a Boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer’s dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality.
Cite as

Patrick Schnider and Simon Weber. A Topological Version of Schaefer’s Dichotomy Theorem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 77:1-77:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{schnider_et_al:LIPIcs.SoCG.2024.77,
  author =	{Schnider, Patrick and Weber, Simon},
  title =	{{A Topological Version of Schaefer’s Dichotomy Theorem}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{77:1--77:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.77},
  URN =		{urn:nbn:de:0030-drops-200220},
  doi =		{10.4230/LIPIcs.SoCG.2024.77},
  annote =	{Keywords: Computational topology, Boolean CSP, satisfiability, computational complexity, solution space, homotopy universality, homological connectivity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.1
Geometric Embeddability of Complexes Is ∃ℝ-Complete

Authors: Mikkel Abrahamsen, Linda Kleist, and Tillmann Miltzow

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ℝ^d is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability.
Cite as

Mikkel Abrahamsen, Linda Kleist, and Tillmann Miltzow. Geometric Embeddability of Complexes Is ∃ℝ-Complete. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.1,
  author =	{Abrahamsen, Mikkel and Kleist, Linda and Miltzow, Tillmann},
  title =	{{Geometric Embeddability of Complexes Is \exists\mathbb{R}-Complete}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{1:1--1:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.1},
  URN =		{urn:nbn:de:0030-drops-178518},
  doi =		{10.4230/LIPIcs.SoCG.2023.1},
  annote =	{Keywords: simplicial complex, geometric embedding, linear embedding, hypergraph, recognition, existential theory of the reals}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.58
Topological Universality of the Art Gallery Problem

Authors: Jack Stade and Jamie Tucker-Foltz

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We prove that any compact semi-algebraic set is homeomorphic to the solution space of some art gallery problem. Previous works have established similar universality theorems, but holding only up to homotopy equivalence, rather than homeomorphism, and prior to this work, the existence of art galleries even for simple spaces such as the Möbius strip or the three-holed torus were unknown. Our construction relies on an elegant and versatile gadget to copy guard positions with minimal overhead. It is simpler than previous constructions, consisting of a single rectangular room with convex slits cut out from the edges. We show that both the orientable and non-orientable surfaces of genus n admit galleries with only O(n) vertices.
Cite as

Jack Stade and Jamie Tucker-Foltz. Topological Universality of the Art Gallery Problem. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{stade_et_al:LIPIcs.SoCG.2023.58,
  author =	{Stade, Jack and Tucker-Foltz, Jamie},
  title =	{{Topological Universality of the Art Gallery Problem}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{58:1--58:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.58},
  URN =		{urn:nbn:de:0030-drops-179082},
  doi =		{10.4230/LIPIcs.SoCG.2023.58},
  annote =	{Keywords: Art gallery, Homeomorphism, Exists-R, ETR, Semi-algebraic sets, Universality}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.48
The Complexity of the Hausdorff Distance

Authors: Paul Jungeblut, Linda Kleist, and Tillmann Miltzow

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀∃_<ℝ. This implies that the problem is NP-, co-NP-, ∃ℝ- and ∀ℝ-hard.
Cite as

Paul Jungeblut, Linda Kleist, and Tillmann Miltzow. The Complexity of the Hausdorff Distance. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jungeblut_et_al:LIPIcs.SoCG.2022.48,
  author =	{Jungeblut, Paul and Kleist, Linda and Miltzow, Tillmann},
  title =	{{The Complexity of the Hausdorff Distance}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{48:1--48:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.48},
  URN =		{urn:nbn:de:0030-drops-160567},
  doi =		{10.4230/LIPIcs.SoCG.2022.48},
  annote =	{Keywords: Hausdorff Distance, Semi-Algebraic Set, Existential Theory of the Reals, Universal Existential Theory of the Reals, Complexity Theory}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.3
Classifying Convex Bodies by Their Contact and Intersection Graphs

Authors: Anders Aamand, Mikkel Abrahamsen, Jakob Bæk Tejs Knudsen, and Peter Michael Reichstein Rasmussen

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Let A be a convex body in the plane and A₁,…,A_n be translates of A. Such translates give rise to an intersection graph of A, G = (V,E), with vertices V = {1,… ,n} and edges E = {uv∣ A_u ∩ A_v ≠ ∅}. The subgraph G' = (V, E') satisfying that E' ⊂ E is the set of edges uv for which the interiors of A_u and A_v are disjoint is a unit distance graph of A. If furthermore G' = G, i.e., if the interiors of A_u and A_v are disjoint whenever u≠ v, then G is a contact graph of A. In this paper, we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies A and B are equivalent if there exists a linear transformation B' of B such that for any slope, the longest line segments with that slope contained in A and B', respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of A and B are the same if and only if A and B are equivalent. We prove the same statement for unit distance and intersection graphs.
Cite as

Anders Aamand, Mikkel Abrahamsen, Jakob Bæk Tejs Knudsen, and Peter Michael Reichstein Rasmussen. Classifying Convex Bodies by Their Contact and Intersection Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{aamand_et_al:LIPIcs.SoCG.2021.3,
  author =	{Aamand, Anders and Abrahamsen, Mikkel and Knudsen, Jakob B{\ae}k Tejs and Rasmussen, Peter Michael Reichstein},
  title =	{{Classifying Convex Bodies by Their Contact and Intersection Graphs}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.3},
  URN =		{urn:nbn:de:0030-drops-138024},
  doi =		{10.4230/LIPIcs.SoCG.2021.3},
  annote =	{Keywords: convex body, contact graph, intersection graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.5
Chasing Puppies: Mobile Beacon Routing on Closed Curves

Authors: Mikkel Abrahamsen, Jeff Erickson, Irina Kostitsyna, Maarten Löffler, Tillmann Miltzow, Jérôme Urhausen, Jordi Vermeulen, and Giovanni Viglietta

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others’ work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.
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Mikkel Abrahamsen, Jeff Erickson, Irina Kostitsyna, Maarten Löffler, Tillmann Miltzow, Jérôme Urhausen, Jordi Vermeulen, and Giovanni Viglietta. Chasing Puppies: Mobile Beacon Routing on Closed Curves. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2021.5,
  author =	{Abrahamsen, Mikkel and Erickson, Jeff and Kostitsyna, Irina and L\"{o}ffler, Maarten and Miltzow, Tillmann and Urhausen, J\'{e}r\^{o}me and Vermeulen, Jordi and Viglietta, Giovanni},
  title =	{{Chasing Puppies: Mobile Beacon Routing on Closed Curves}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.5},
  URN =		{urn:nbn:de:0030-drops-138046},
  doi =		{10.4230/LIPIcs.SoCG.2021.5},
  annote =	{Keywords: Beacon routing, navigation, generic smooth curves, puppies}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.44
A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Authors: Simon B. Hengeveld and Tillmann Miltzow

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Given a closed simple polygon P, we say two points p,q see each other if the segment seg(p,q) is fully contained in P. The art gallery problem seeks a minimum size set G ⊂ P of guards that sees P completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods. As the art gallery problem is ∃ ℝ-complete, it seems unlikely to avoid algebraic methods, for any exact algorithm, without additional assumptions. In this paper, we introduce the notion of vision-stability. In order to describe vision-stability consider an enhanced guard that can see "around the corner" by an angle of δ or a diminished guard whose vision is by an angle of δ "blocked" by reflex vertices. A polygon P has vision-stability δ if the optimal number of enhanced guards to guard P is the same as the optimal number of diminished guards to guard P. We will argue that most relevant polygons are vision-stable. We describe a one-shot vision-stable algorithm that computes an optimal guard set for vision-stable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision-stable polygon. We implemented an iterative vision-stable algorithm and show its practical performance is slower, but comparable with other state-of-the-art algorithms. The practical implementation can be found at: https://github.com/simonheng/AGPIterative. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord c of a polygon, we denote by n(c) the number of vertices visible from c. The chord-visibility width (cw(P)) of a polygon is the maximum n(c) over all possible chords c. The set of vision-stable polygons admit an FPT algorithm when parameterized by the chord-visibility width. Furthermore, the one-shot algorithm runs in FPT time when parameterized by the number of reflex vertices.
Cite as

Simon B. Hengeveld and Tillmann Miltzow. A Practical Algorithm with Performance Guarantees for the Art Gallery Problem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 44:1-44:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hengeveld_et_al:LIPIcs.SoCG.2021.44,
  author =	{Hengeveld, Simon B. and Miltzow, Tillmann},
  title =	{{A Practical Algorithm with Performance Guarantees for the Art Gallery Problem}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{44:1--44:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.44},
  URN =		{urn:nbn:de:0030-drops-138433},
  doi =		{10.4230/LIPIcs.SoCG.2021.44},
  annote =	{Keywords: Art Gallery, Parametrized complexity, Integer Programming, Visibility}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.13
Between Shapes, Using the Hausdorff Distance

Authors: Marc van Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, and Jordi L. Vermeulen

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and show some related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.
Cite as

Marc van Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, and Jordi L. Vermeulen. Between Shapes, Using the Hausdorff Distance. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{vankreveld_et_al:LIPIcs.ISAAC.2020.13,
  author =	{van Kreveld, Marc and Miltzow, Tillmann and Ophelders, Tim and Sonke, Willem and Vermeulen, Jordi L.},
  title =	{{Between Shapes, Using the Hausdorff Distance}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.13},
  URN =		{urn:nbn:de:0030-drops-133572},
  doi =		{10.4230/LIPIcs.ISAAC.2020.13},
  annote =	{Keywords: computational geometry, Hausdorff distance, shape interpolation}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.17
Maximum Clique in Disk-Like Intersection Graphs

Authors: Édouard Bonnet, Nicolas Grelier, and Tillmann Miltzow

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes I follow the same road. They show that, for every graph G of a large-enough class C, the complement of an even subdivision of G belongs to the intersection class I. Then they conclude by invoking the hardness of Maximum Independent Set on the class C, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.
Cite as

Édouard Bonnet, Nicolas Grelier, and Tillmann Miltzow. Maximum Clique in Disk-Like Intersection Graphs. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonnet_et_al:LIPIcs.FSTTCS.2020.17,
  author =	{Bonnet, \'{E}douard and Grelier, Nicolas and Miltzow, Tillmann},
  title =	{{Maximum Clique in Disk-Like Intersection Graphs}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{17:1--17:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.17},
  URN =		{urn:nbn:de:0030-drops-132587},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.17},
  annote =	{Keywords: Disk Graphs, Intersection Graphs, Maximum Clique, Algorithms, NP-hardness, APX-hardness}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2020.78
Hiding Sliding Cubes: Why Reconfiguring Modular Robots Is Not Easy (Media Exposition)

Authors: Tillmann Miltzow, Irene Parada, Willem Sonke, Bettina Speckmann, and Jules Wulms

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Face-connected configurations of cubes are a common model for modular robots in three dimensions. In this abstract and the accompanying video we study reconfigurations of such modular robots using so-called sliding moves. Using sliding moves, it is always possible to reconfigure one face-connected configuration of n cubes into any other, while keeping the robot connected at all stages of the reconfiguration. For certain configurations Ω(n²) sliding moves are necessary. In contrast, the best current upper bound is O(n³). It has been conjectured that there is always a cube on the outside of any face-connected configuration of cubes which can be moved without breaking connectivity. The existence of such a cube would immediately imply a straight-forward O(n²) reconfiguration algorithm. However, we present a configuration of cubes such that no cube on the outside can move without breaking connectivity. In other words, we show that this particular avenue towards an O(n²) reconfiguration algorithm for face-connected cubes is blocked.
Cite as

Tillmann Miltzow, Irene Parada, Willem Sonke, Bettina Speckmann, and Jules Wulms. Hiding Sliding Cubes: Why Reconfiguring Modular Robots Is Not Easy (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 78:1-78:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{miltzow_et_al:LIPIcs.SoCG.2020.78,
  author =	{Miltzow, Tillmann and Parada, Irene and Sonke, Willem and Speckmann, Bettina and Wulms, Jules},
  title =	{{Hiding Sliding Cubes: Why Reconfiguring Modular Robots Is Not Easy}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{78:1--78:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.78},
  URN =		{urn:nbn:de:0030-drops-122363},
  doi =		{10.4230/LIPIcs.SoCG.2020.78},
  annote =	{Keywords: Sliding cubes, Reconfiguration, Modular robots}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.8
Parameterized Streaming Algorithms for Min-Ones d-SAT

Authors: Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
In this work, we initiate the study of the Min-Ones d-SAT problem in the parameterized streaming model. An instance of the problem consists of a d-CNF formula F and an integer k, and the objective is to determine if F has a satisfying assignment which sets at most k variables to 1. In the parameterized streaming model, input is provided as a stream, just as in the usual streaming model. A key difference is that the bound on the read-write memory available to the algorithm is O(f(k) log n) (f: N -> N, a computable function) as opposed to the O(log n) bound of the usual streaming model. The other important difference is that the number of passes the algorithm makes over its input must be a (preferably small) function of k. We design a (k + 1)-pass parameterized streaming algorithm that solves Min-Ones d-SAT (d >= 2) using space O((kd^(ck) + k^d)log n) (c > 0, a constant) and a (d + 1)^k-pass algorithm that uses space O(k log n). We also design a streaming kernelization for Min-Ones 2-SAT that makes (k + 2) passes and uses space O(k^6 log n) to produce a kernel with O(k^6) clauses. To complement these positive results, we show that any k-pass algorithm for or Min-Ones d-SAT (d >= 2) requires space Omega(max{n^(1/k) / 2^k, log(n / k)}) on instances (F, k). This is achieved via a reduction from the streaming problem POT Pointer Chasing (Guha and McGregor [ICALP 2008]), which might be of independent interest. Given this, our (k + 1)-pass parameterized streaming algorithm is the best possible, inasmuch as the number of passes is concerned. In contrast to the results of Fafianie and Kratsch [MFCS 2014] and Chitnis et al. [SODA 2015], who independently showed that there are 1-pass parameterized streaming algorithms for Vertex Cover (a restriction of Min-Ones 2-SAT), we show using lower bounds from Communication Complexity that for any d >= 1, a 1-pass streaming algorithm for Min-Ones d-SAT requires space Omega(n). This excludes the possibility of a 1-pass parameterized streaming algorithm for the problem. Additionally, we show that any p-pass algorithm for the problem requires space Omega(n/p).
Cite as

Akanksha Agrawal, Arindam Biswas, Édouard Bonnet, Nick Brettell, Radu Curticapean, Dániel Marx, Tillmann Miltzow, Venkatesh Raman, and Saket Saurabh. Parameterized Streaming Algorithms for Min-Ones d-SAT. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.FSTTCS.2019.8,
  author =	{Agrawal, Akanksha and Biswas, Arindam and Bonnet, \'{E}douard and Brettell, Nick and Curticapean, Radu and Marx, D\'{a}niel and Miltzow, Tillmann and Raman, Venkatesh and Saurabh, Saket},
  title =	{{Parameterized Streaming Algorithms for Min-Ones d-SAT}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.8},
  URN =		{urn:nbn:de:0030-drops-115708},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.8},
  annote =	{Keywords: min, ones, sat, d-sat, parameterized, kernelization, streaming, space, efficient, algorithm, parameter}
}
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