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DOI: 10.4230/LIPIcs.ESA.2024.51

Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs

Authors: Lech Duraj, Filip Konieczny, and Krzysztof Potępa

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract

We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in 𝒪{(k ⋅ n^{1-1/d} ⋅ m ⋅ polylog(n))} time complexity for the graph on n vertices and m edges, where k is the diameter and d is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an 𝒪{(n^{7/4} ⋅ polylog(n))} parameterized diameter algorithm for unit square intersection graph of size n, as well as a more general algorithm for convex polygon intersection graphs.

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Lech Duraj, Filip Konieczny, and Krzysztof Potępa. Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{duraj_et_al:LIPIcs.ESA.2024.51,
  author =	{Duraj, Lech and Konieczny, Filip and Pot\k{e}pa, Krzysztof},
  title =	{{Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.51},
  URN =		{urn:nbn:de:0030-drops-211229},
  doi =		{10.4230/LIPIcs.ESA.2024.51},
  annote =	{Keywords: Graph Diameter, Geometric Intersection Graphs, Vapnik-Chervonenkis Dimension}
}

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DOI: 10.4230/LIPIcs.ESA.2021.76

Faster Deterministic Modular Subset Sum

Authors: Krzysztof Potępa

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract

We consider the Modular Subset Sum problem: given a multiset X of integers from ℤ_m and a target integer t, decide if there exists a subset of X with a sum equal to t (mod m). Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in 𝒪(m log m) time, while Axiotis et al. gave a deterministic algorithm that runs in 𝒪(m polylog m) time. Both results work by reduction to a text problem, which is solved using a dynamic strings data structure. In this work, we develop a simple data structure, designed specifically to handle the text problem that arises in the algorithms for Modular Subset Sum. Our data structure, which we call the shift-tree, is a simple variant of a segment tree. We provide both a hashing-based and a deterministic variant of the shift-trees. We then apply our data structure to the Modular Subset Sum problem and obtain two algorithms. The first algorithm is Monte-Carlo randomized and matches the 𝒪(m log m) runtime of the Las-Vegas algorithm by Cardinal and Iacono. The second algorithm is fully deterministic and runs in 𝒪(m log m ⋅ α(m)) time, where α is the inverse Ackermann function.

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Krzysztof Potępa. Faster Deterministic Modular Subset Sum. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{potepa:LIPIcs.ESA.2021.76,
  author =	{Pot\k{e}pa, Krzysztof},
  title =	{{Faster Deterministic Modular Subset Sum}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{76:1--76:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.76},
  URN =		{urn:nbn:de:0030-drops-146574},
  doi =		{10.4230/LIPIcs.ESA.2021.76},
  annote =	{Keywords: Modular Subset Sum, String Problem, Segment Tree, Data Structure}
}

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Documents authored by Potępa, Krzysztof

Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs

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Faster Deterministic Modular Subset Sum

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