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

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.SoCG.2022.52

Abstract

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
Keywords
  • points-separation
  • min color path
  • constraint removal
  • barrier resillience

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