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ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs

Authors Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi



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Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Fahad Panolan
  • Department of Computer Science and Engineering, IIT Hyderabad, India
Saket Saurabh
  • Department of Informatics, University of Bergen, Norway
  • The Institute of Mathematical Sciences, HBNI and IRL 2000 ReLaX, Chennai, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

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Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 44:1-44:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.44

Abstract

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{𝒪(√k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√k)}(n+m)^𝒪(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{𝒪(√k)}(n+m)^𝒪(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{𝒪(√klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width 𝒪(√k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
Keywords
  • Optimality Program
  • ETH
  • Unit Disk Graphs
  • Parameterized Complexity
  • Long Path
  • Long Cycle

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