19 Search Results for "Schnider, Patrick"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.26
Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice

Authors: Guangya Cai

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We study the problem of finding a fair linear scoring function over (numerical) attributes for top-k selection, ensuring fairness through a proportional representation constraint on the protected group. Existing algorithms do not scale efficiently, particularly in higher dimensions. Our hardness analysis shows that in more than two dimensions, no algorithm is likely to scale efficiently with respect to dataset size, and the computational complexity is likely to grow rapidly with dimensionality. However, the hardness results also provide key insights guiding algorithm design, leading to our two-pronged solution: (1) For small k, our analysis reveals a gap in the hardness barrier. By addressing various engineering challenges, including achieving efficient parallelism, we turn this potential of efficiency into an optimized geometry-based algorithm delivering substantial performance gains. (2) For large k, where the hardness is robust, we employ a practically efficient optimization-based algorithm which, despite being theoretically worse, achieves superior real-world performance. Experimental evaluations on real-world datasets then explore scenarios where worst-case behavior does not manifest, identifying areas critical to practical performance. Our solution achieves speedups of up to several orders of magnitude compared to the state of the art, an efficiency made possible through a tight integration of hardness analysis, algorithm design, practical engineering, and empirical evaluation.
Cite as

Guangya Cai. Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai:LIPIcs.SoCG.2026.26,
  author =	{Cai, Guangya},
  title =	{{Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.26},
  URN =		{urn:nbn:de:0030-drops-258320},
  doi =		{10.4230/LIPIcs.SoCG.2026.26},
  annote =	{Keywords: Fairness, Top-k, Integration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.19
Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

Authors: Michaela Borzechowski, Sebastian Haslebacher, Hung P. Hoang, Patrick Schnider, and Simon Weber

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The famous Ham-Sandwich theorem states that any d point sets in ℝ^d can be simultaneously bisected by a single hyperplane. The α-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the α-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is ∃ℝ-complete, which also implies that the realizability problem for grid Unique Sink Orientations is ∃ℝ-complete.
Cite as

Michaela Borzechowski, Sebastian Haslebacher, Hung P. Hoang, Patrick Schnider, and Simon Weber. Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{borzechowski_et_al:LIPIcs.SoCG.2026.19,
  author =	{Borzechowski, Michaela and Haslebacher, Sebastian and Hoang, Hung P. and Schnider, Patrick and Weber, Simon},
  title =	{{Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.19},
  URN =		{urn:nbn:de:0030-drops-258250},
  doi =		{10.4230/LIPIcs.SoCG.2026.19},
  annote =	{Keywords: \alpha-Ham-Sandwich Theorem, Pseudo-Hyperplanes, Arrangements, Unique Sink Orientations, Oriented Matroids}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.33
Higher Hardness Results for the Reconfiguration of Odd Matchings

Authors: Joseph Dorfer

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We study the reconfiguration of odd matchings of combinatorial graphs. Odd matchings are matchings that cover all but one vertex of a graph. A reconfiguration step, or flip, is an operation that matches the isolated vertex and, consequently, isolates another vertex. The flip graph of odd matchings is a graph that has all odd matchings of a graph as vertices and an edge between two vertices if their corresponding matchings can be transformed into one another via a single flip. We show that computing the diameter of the flip graph of odd matchings is Π₂^p-hard. This complements a recent result by Wulf [FOCS25] that it is Π₂^p-hard to compute the diameter of the flip graph of perfect matchings where a flip swaps matching edges along a single cycle of unbounded size. Further, we show that computing the radius of the flip graph of odd matchings is Σ₃^p-hard. The respective decision problems for the diameter and the radius are also complete in the respective level of the polynomial hierarchy. This shows that computing the radius of the flip graph of odd matchings is provably harder than computing its diameter, unless the polynomial hierarchy collapses. Finally, we reduce set cover to the problem of finding shortest flip sequences. As a consequence, we show APX-hardness and that the problem cannot be approximated by a sublogarithmic factor. By doing so, we answer a question asked by Aichholzer, Brenner, Dorfer, Hoang, Perz, Rieck, and Verciani [GD25].
Cite as

Joseph Dorfer. Higher Hardness Results for the Reconfiguration of Odd Matchings. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dorfer:LIPIcs.STACS.2026.33,
  author =	{Dorfer, Joseph},
  title =	{{Higher Hardness Results for the Reconfiguration of Odd Matchings}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{33:1--33:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle 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.2026.33},
  URN =		{urn:nbn:de:0030-drops-255222},
  doi =		{10.4230/LIPIcs.STACS.2026.33},
  annote =	{Keywords: Graph Reconfiguration Problems, Flip Graphs, Polynomial Hierarchy, APX-hardness}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.95
ARRIVAL: Recursive Framework & 𝓁₁-Contraction

Authors: Sebastian Haslebacher

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
ARRIVAL is the problem of deciding which out of two possible destinations will be reached first by a token that moves deterministically along the edges of a directed graph, according to so-called switching rules. It is known to lie in NP ∩ CoNP, but not known to lie in 𝖯. The state-of-the-art algorithm due to Gärtner et al. (ICALP `21) runs in time 2^{𝒪(√n log n)} on an n-vertex graph. We prove that ARRIVAL can be solved in time 2^{𝒪(k log² n)} on n-vertex graphs of treewidth k. Our algorithm is derived by adapting a simple recursive algorithm for a generalization of ARRIVAL called G-ARRIVAL. This simple recursive algorithm acts as a framework from which we can also rederive the subexponential upper bound of Gärtner et al. Our second result is a reduction from G-ARRIVAL to the problem of finding an approximate fixed point of an 𝓁₁-contracting function f : [0, 1]ⁿ → [0, 1]ⁿ. Finding such fixed points is a well-studied problem in the case of the 𝓁₂-metric and the 𝓁_∞-metric, but little is known about the 𝓁₁-case. Both of our results highlight parallels between ARRIVAL and the Simple Stochastic Games (SSG) problem. Concretely, Chatterjee et al. (SODA `23) gave an algorithm for SSG parameterized by treewidth that achieves a similar bound as we do for ARRIVAL, and SSG is known to reduce to 𝓁_∞-contraction.
Cite as

Sebastian Haslebacher. ARRIVAL: Recursive Framework & 𝓁₁-Contraction. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 95:1-95:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{haslebacher:LIPIcs.ICALP.2025.95,
  author =	{Haslebacher, Sebastian},
  title =	{{ARRIVAL: Recursive Framework \& 𝓁₁-Contraction}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{95:1--95:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.95},
  URN =		{urn:nbn:de:0030-drops-234723},
  doi =		{10.4230/LIPIcs.ICALP.2025.95},
  annote =	{Keywords: ARRIVAL, G-ARRIVAL, Deterministic Random Walk, Rotor-Routing, 𝓁₁-Contraction, Banach Fixed Point}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.74
The Point-Boundary Art Gallery Problem Is ∃ℝ-Hard

Authors: Jack Stade

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We resolve the complexity of the point-boundary variant of the art gallery problem, showing that it is ∃ℝ-complete, meaning that it is equivalent under polynomial time reductions to deciding whether a system of polynomial equations has a real solution. The art gallery problem asks whether there is a configuration of guards that together can see every point inside of an art gallery modeled by a simple polygon. The original version of this problem (which we call the point-point variant) was shown to be ∃ℝ-hard [Abrahamsen, Adamaszek, and Miltzow, JACM 2021], but the complexity of the variant where guards only need to guard the walls of the art gallery was left as an open problem. We show that this variant is also ∃ℝ-hard. Our techniques can also be used to greatly simplify the proof of ∃ℝ-hardness of the point-point art gallery problem. The gadgets in previous work could only be constructed by using a computer to find complicated rational coordinates with specific algebraic properties. All of our gadgets can be constructed by hand and can be verified with simple geometric arguments.
Cite as

Jack Stade. The Point-Boundary Art Gallery Problem Is ∃ℝ-Hard. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 74:1-74:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{stade:LIPIcs.SoCG.2025.74,
  author =	{Stade, Jack},
  title =	{{The Point-Boundary Art Gallery Problem Is \exists\mathbb{R}-Hard}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{74:1--74:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.74},
  URN =		{urn:nbn:de:0030-drops-232269},
  doi =		{10.4230/LIPIcs.SoCG.2025.74},
  annote =	{Keywords: Art Gallery Problem, Complexity, ETR, Polygon}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.65
Recognizing 2-Layer and Outer k-Planar Graphs

Authors: Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. In both cases, edges are drawn as straight-line segments. Both variants are NP-hard, but admit FPT-algorithms with respect to the natural parameter. In recent years, in the context of beyond-planar graphs, a local version of the crossing number has also received considerable attention. A graph is k-planar if it admits a drawing with at most k crossings per edge. In contrast to the crossing number, recognizing k-planar graphs is NP-hard even if k = 1 and hence not likely to be FPT with respect to the natural parameter k. In this paper, we consider the two above variants in the local setting. The k-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer k-planar and outer k-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to the natural parameter k. For k = 0, the two classes of graphs are exactly the caterpillars and outerplanar graphs, respectively, which can be recognized in linear time. Two groups of researchers independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked explicitly whether outer 2-planar graphs can be recognized in polynomial time. Our main contribution consists of XP-algorithms for recognizing 2-layer k-planar graphs and outer k-planar graphs, which implies that both recognition problems can be solved in polynomial time for every fixed k. We complement these results by showing that recognizing 2-layer k-planar graphs is XNLP-complete and that recognizing outer k-planar graphs is XNLP-hard. This implies that both problems are W[t]-hard for every t and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer k-planar graphs where the order of the vertices on one layer is specified.
Cite as

Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff. Recognizing 2-Layer and Outer k-Planar Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kobayashi_et_al:LIPIcs.SoCG.2025.65,
  author =	{Kobayashi, Yasuaki and Okada, Yuto and Wolff, Alexander},
  title =	{{Recognizing 2-Layer and Outer k-Planar Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{65:1--65:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.65},
  URN =		{urn:nbn:de:0030-drops-232170},
  doi =		{10.4230/LIPIcs.SoCG.2025.65},
  annote =	{Keywords: 2-layer k-planar graphs, outer k-planar graphs, recognition algorithms, local crossing number, bandwidth, FPT, XNLP, XP, W\lbrackt\rbrack}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.71
Unfairly Splitting Separable Necklaces

Authors: Patrick Schnider, Linus Stalder, and Simon Weber

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

Abstract
The Necklace Splitting problem is a classical problem in combinatorics that has been intensively studied both from a combinatorial and a computational point of view. It is well-known that the Necklace Splitting problem reduces to the discrete Ham Sandwich problem. This reduction was crucial in the proof of PPA-completeness of the Ham Sandwich problem. Recently, Borzechowski, Schnider and Weber [ISAAC'23] introduced a variant of Necklace Splitting that similarly reduces to the α-Ham Sandwich problem, which lies in the complexity class UEOPL but is not known to be complete. To make this reduction work, the input necklace is guaranteed to be n-separable. They showed that these necklaces can be fairly split in polynomial time and thus this subproblem cannot be used to prove UEOPL-hardness for α-Ham Sandwich. We consider the more general unfair necklace splitting problem on n-separable necklaces, i.e., the problem of splitting these necklaces such that each thief gets a desired fraction of each type of jewels. This more general problem is the natural necklace-splitting-type version of α-Ham Sandwich, and its complexity status is one of the main open questions posed by Borzechowski, Schnider and Weber. We show that the unfair splitting problem is also polynomial-time solvable, and can thus also not be used to show UEOPL-hardness for α-Ham Sandwich.
Cite as

Patrick Schnider, Linus Stalder, and Simon Weber. Unfairly Splitting Separable Necklaces. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 71:1-71:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{schnider_et_al:LIPIcs.STACS.2025.71,
  author =	{Schnider, Patrick and Stalder, Linus and Weber, Simon},
  title =	{{Unfairly Splitting Separable Necklaces}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{71:1--71:19},
  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.2025.71},
  URN =		{urn:nbn:de:0030-drops-228963},
  doi =		{10.4230/LIPIcs.STACS.2025.71},
  annote =	{Keywords: Necklace splitting, n-separability, well-separation, Ham Sandwich, alpha-Ham Sandwich, unfair splitting, fair division}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.32
Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents

Authors: Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber

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

Abstract
We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the α-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our α-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.
Cite as

Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber. Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{borzechowski_et_al:LIPIcs.ICALP.2024.32,
  author =	{Borzechowski, Michaela and Fearnley, John and Gordon, Spencer and Savani, Rahul and Schnider, Patrick and Weber, Simon},
  title =	{{Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.32},
  URN =		{urn:nbn:de:0030-drops-201751},
  doi =		{10.4230/LIPIcs.ICALP.2024.32},
  annote =	{Keywords: P-LCP, Unique Sink Orientation, \alpha-Ham Sandwich, search complexity, TFNP, UEOPL}
}
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.ISAAC.2023.15
An FPT Algorithm for Splitting a Necklace Among Two Thieves

Authors: Michaela Borzechowski, Patrick Schnider, and Simon Weber

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
It is well-known that the 2-Thief-Necklace-Splitting problem reduces to the discrete Ham Sandwich problem. In fact, this reduction was crucial in the proof of the PPA-completeness of the Ham Sandwich problem [Filos-Ratsikas and Goldberg, STOC'19]. Recently, a variant of the Ham Sandwich problem called α-Ham Sandwich has been studied, in which the point sets are guaranteed to be well-separated [Steiger and Zhao, DCG'10]. The complexity of this search problem remains unknown, but it is known to lie in the complexity class UEOPL [Chiu, Choudhary and Mulzer, ICALP'20]. We define the analogue of this well-separation condition in the necklace splitting problem - a necklace is n-separable, if every subset A of the n types of jewels can be separated from the types [n]⧵A by at most n separator points. Since this version of necklace splitting reduces to α-Ham Sandwich in a solution-preserving way it follows that instances of this version always have unique solutions. We furthermore provide two FPT algorithms: The first FPT algorithm solves 2-Thief-Necklace-Splitting on (n-1+𝓁)-separable necklaces with n types of jewels and m total jewels in time 2^O(𝓁log𝓁) + O(m²). In particular, this shows that 2-Thief-Necklace-Splitting is polynomial-time solvable on n-separable necklaces. Thus, attempts to show hardness of α-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting problem cannot work. The second FPT algorithm tests (n-1+𝓁)-separability of a given necklace with n types of jewels in time 2^O(𝓁²) ⋅ n⁴. In particular, n-separability can thus be tested in polynomial time, even though testing well-separation of point sets is co-NP-complete [Bergold et al., SWAT'22].
Cite as

Michaela Borzechowski, Patrick Schnider, and Simon Weber. An FPT Algorithm for Splitting a Necklace Among Two Thieves. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{borzechowski_et_al:LIPIcs.ISAAC.2023.15,
  author =	{Borzechowski, Michaela and Schnider, Patrick and Weber, Simon},
  title =	{{An FPT Algorithm for Splitting a Necklace Among Two Thieves}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.15},
  URN =		{urn:nbn:de:0030-drops-193178},
  doi =		{10.4230/LIPIcs.ISAAC.2023.15},
  annote =	{Keywords: Necklace splitting, n-separability, well-separation, ham sandwich, FPT}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.30
On Connectivity in Random Graph Models with Limited Dependencies

Authors: Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
For any positive edge density p, a random graph in the Erdős-Rényi G_{n,p} model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability ρ(n), such that for any distribution 𝒢 (in this model) on graphs with n vertices in which each potential edge has a marginal probability of being present at least ρ(n), a graph drawn from 𝒢 is connected with non-zero probability? As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold ρ(n). For each condition, we provide upper and lower bounds for ρ(n). In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that ρ(n) → 2-ϕ ≈ 0.38 for n → ∞, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that ρ(n) > 0.5-o(n). In stark contrast to the coloring model, for our weakest independence condition - pairwise independence of non-adjacent edges - we show that ρ(n) lies within O(1/n²) of the threshold 1-2/n for completely arbitrary distributions.
Cite as

Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl. On Connectivity in Random Graph Models with Limited Dependencies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lengler_et_al:LIPIcs.APPROX/RANDOM.2023.30,
  author =	{Lengler, Johannes and Martinsson, Anders and Petrova, Kalina and Schnider, Patrick and Steiner, Raphael and Weber, Simon and Welzl, Emo},
  title =	{{On Connectivity in Random Graph Models with Limited Dependencies}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{30:1--30:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.30},
  URN =		{urn:nbn:de:0030-drops-188556},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.30},
  annote =	{Keywords: Random Graphs, Independence, Dependency, Connectivity, Threshold, Probabilistic Method}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.55
Combinatorial Depth Measures for Hyperplane Arrangements

Authors: Patrick Schnider and Pablo Soberón

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

Abstract
Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth.
Cite as

Patrick Schnider and Pablo Soberón. Combinatorial Depth Measures for Hyperplane Arrangements. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{schnider_et_al:LIPIcs.SoCG.2023.55,
  author =	{Schnider, Patrick and Sober\'{o}n, Pablo},
  title =	{{Combinatorial Depth Measures for Hyperplane Arrangements}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{55:1--55:14},
  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.2023.55},
  URN =		{urn:nbn:de:0030-drops-179055},
  doi =		{10.4230/LIPIcs.SoCG.2023.55},
  annote =	{Keywords: Depth measures, Hyperplane arrangements, Regression depth, Tverberg theorem}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2022.16
Well-Separation and Hyperplane Transversals in High Dimensions

Authors: Helena Bergold, Daniel Bertschinger, Nicolas Grelier, Wolfgang Mulzer, and Patrick Schnider

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ℝ^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an Ω((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete. Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ℝ^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ≥ d+1 finite point sets in ℝ^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d.
Cite as

Helena Bergold, Daniel Bertschinger, Nicolas Grelier, Wolfgang Mulzer, and Patrick Schnider. Well-Separation and Hyperplane Transversals in High Dimensions. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bergold_et_al:LIPIcs.SWAT.2022.16,
  author =	{Bergold, Helena and Bertschinger, Daniel and Grelier, Nicolas and Mulzer, Wolfgang and Schnider, Patrick},
  title =	{{Well-Separation and Hyperplane Transversals in High Dimensions}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.16},
  URN =		{urn:nbn:de:0030-drops-161766},
  doi =		{10.4230/LIPIcs.SWAT.2022.16},
  annote =	{Keywords: hyperplane transversal, high-dimension, hardness}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.6
Edge Partitions of Complete Geometric Graphs

Authors: Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber

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

Abstract
In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.
Cite as

Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber. Edge Partitions of Complete Geometric Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2022.6,
  author =	{Aichholzer, Oswin and Obenaus, Johannes and Orthaber, Joachim and Paul, Rosna and Schnider, Patrick and Steiner, Raphael and Taubner, Tim and Vogtenhuber, Birgit},
  title =	{{Edge Partitions of Complete Geometric Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{6:1--6:16},
  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.2022.6},
  URN =		{urn:nbn:de:0030-drops-160141},
  doi =		{10.4230/LIPIcs.SoCG.2022.6},
  annote =	{Keywords: edge partition, complete geometric graph, plane spanning tree, wheel set}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.10
Enclosing Depth and Other Depth Measures

Authors: Patrick Schnider

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.
Cite as

Patrick Schnider. Enclosing Depth and Other Depth Measures. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{schnider:LIPIcs.ISAAC.2021.10,
  author =	{Schnider, Patrick},
  title =	{{Enclosing Depth and Other Depth Measures}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.10},
  URN =		{urn:nbn:de:0030-drops-154431},
  doi =		{10.4230/LIPIcs.ISAAC.2021.10},
  annote =	{Keywords: Depth measures, Tukey depth, Tverberg theorem, Combinatorial Geometry}
}
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