Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles

Authors Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri, Jie Xue

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Author Details

Neeraj Kumar
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Daniel Lokshtanov
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
  • University of Bergen, Norway
Subhash Suri
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Jie Xue
  • New York University Shanghai, China

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Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri, and Jie Xue. Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Suppose we are given a pair of points s, t and a set 𝒮 of n geometric objects in the plane, called obstacles. We show that in polynomial time one can construct an auxiliary (multi-)graph G with vertex set 𝒮 and every edge labeled from {0, 1}, such that a set 𝒮_d ⊆ 𝒮 of obstacles separates s from t if and only if G[𝒮_d] contains a cycle whose sum of labels is odd. Using this structural characterization of separating sets of obstacles we obtain the following algorithmic results. In the Obstacle-removal problem the task is to find a curve in the plane connecting s to t intersecting at most q obstacles. We give a 2.3146^q n^{O(1)} algorithm for Obstacle-removal, significantly improving upon the previously best known q^{O(q³)} n^{O(1)} algorithm of Eiben and Lokshtanov (SoCG'20). We also obtain an alternative proof of a constant factor approximation algorithm for Obstacle-removal, substantially simplifying the arguments of Kumar et al. (SODA'21). In the Generalized Points-separation problem input consists of the set 𝒮 of obstacles, a point set A of k points and p pairs (s₁, t₁), … (s_p, t_p) of points from A. The task is to find a minimum subset 𝒮_r ⊆ 𝒮 such that for every i, every curve from s_i to t_i intersects at least one obstacle in 𝒮_r. We obtain 2^{O(p)} n^{O(k)}-time algorithm for Generalized Points-separation. This resolves an open problem of Cabello and Giannopoulos (SoCG'13), who asked about the existence of such an algorithm for the special case where (s₁, t₁), … (s_p, t_p) contains all the pairs of points in A. Finally, we improve the running time of our algorithm to f(p,k) ⋅ n^{O(√k)} when the obstacles are unit disks, where f(p,k) = 2^{O(p)} k^{O(k)}, and show that, assuming the Exponential Time Hypothesis (ETH), the running time dependence on k of our algorithms is essentially optimal.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • points-separation
  • min color path
  • constraint removal
  • barrier resillience


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  1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(sqrt(log n)) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22-24, 2005, pages 573-581, 2005. Google Scholar
  2. Paul Balister, Zizhan Zheng, Santosh Kumar, and Prasun Sinha. Trap coverage: Allowing coverage holes of bounded diameter in wireless sensor networks. In IEEE INFOCOM 2009, pages 136-144. IEEE, 2009. Google Scholar
  3. Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi Varadarajan. Improved approximation bounds for the minimum constraint removal problem. Computational Geometry, 90:101650, 2020. Google Scholar
  4. Sergey Bereg and David G. Kirkpatrick. Approximating barrier resilience in wireless sensor networks. In Proc. of 5th ALGOSENSORS, volume 5804, pages 29-40, 2009. Google Scholar
  5. S. Cabello and P. Giannopoulos. The complexity of separating points in the plane. Algorithmica, 74(2):643-663, 2016. Google Scholar
  6. David Yu Cheng Chan and David G. Kirkpatrick. Approximating barrier resilience for arrangements of non-identical disk sensors. In Proc. of 8th ALGOSENSORS, pages 42-53, 2012. Google Scholar
  7. David Yu Cheng Chan and David G. Kirkpatrick. Multi-path algorithms for minimum-colour path problems with applications to approximating barrier resilience. Theor. Comput. Sci., 553:74-90, 2014. Google Scholar
  8. E. Eiben and I. Kanj. How to navigate through obstacles? In Proc. of 45th ICALP, 2018. Google Scholar
  9. Eduard Eiben, Jonathan Gemmell, Iyad A. Kanj, and Andrew Youngdahl. Improved results for minimum constraint removal. In Proc. of 32nd AAAI, pages 6477-6484, 2018. Google Scholar
  10. Eduard Eiben and Iyad Kanj. A colored path problem and its applications. ACM Trans. Algorithms, 16(4):47:1-47:48, 2020. Google Scholar
  11. Eduard Eiben and Daniel Lokshtanov. Removing connected obstacles in the plane is FPT. In Proc. of 36th SoCG, volume 164, pages 39:1-39:14, 2020. Google Scholar
  12. Lawrence H. Erickson and Steven M. LaValle. A simple, but NP-Hard, motion planning problem. In Proc. of 27th AAAI, 2013. Google Scholar
  13. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  14. Matias Korman, Maarten Löffler, Rodrigo I. Silveira, and Darren Strash. On the complexity of barrier resilience for fat regions and bounded ply. Comput. Geom., 72:34-51, 2018. Google Scholar
  15. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. Journal of the ACM (JACM), 67(3):1-50, 2020. Google Scholar
  16. Santosh Kumar, Ten-Hwang Lai, and Anish Arora. Barrier coverage with wireless sensors. Wirel. Networks, 13(6):817-834, 2007. Google Scholar
  17. James R. Lee. Separators in region intersection graphs. In Proc. of 8th ITCS, volume 67, pages 1-8, 2017. Google Scholar
  18. Daniel Lokshtanov, NS Narayanaswamy, Venkatesh Raman, MS Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms (TALG), 11(2):1-31, 2014. Google Scholar
  19. Dániel Marx. Can you beat treewidth? In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pages 169-179. IEEE, 2007. Google Scholar
  20. Dániel Marx. A tight lower bound for planar multiway cut with fixed number of terminals. In International Colloquium on Automata, Languages, and Programming, pages 677-688. Springer, 2012. Google Scholar
  21. Saket Saurabh Neeraj Kumar, Daniel Lokshtanov and Subhash Suri. A constant factor approximation for navigating through connected obstacles in the plane. In Proc. 32nd SODA, 2021. Google Scholar
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