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Computing Shortest Paths in the Plane with Removable Obstacles

Authors Pankaj K. Agarwal, Neeraj Kumar, Stavros Sintos, Subhash Suri

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ACM Subject Classification
  • Theory of computation → Shortest paths
Keywords
  • Euclidean shortest paths
  • Removable polygonal obstacles
  • Stochastic shortest paths
  • L_1 shortest paths

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Abstract

We consider the problem of computing a Euclidean shortest path in the presence of removable obstacles in the plane. In particular, we have a collection of pairwise-disjoint polygonal obstacles, each of which may be removed at some cost c_i > 0. Given a cost budget C > 0, and a pair of points s, t, which obstacles should be removed to minimize the path length from s to t in the remaining workspace? We show that this problem is NP-hard even if the obstacles are vertical line segments. Our main result is a fully-polynomial time approximation scheme (FPTAS) for the case of convex polygons. Specifically, we compute an (1 + epsilon)-approximate shortest path in time O({nh}/{epsilon^2} log n log n/epsilon) with removal cost at most (1+epsilon)C, where h is the number of obstacles, n is the total number of obstacle vertices, and epsilon in (0, 1) is a user-specified parameter. Our approximation scheme also solves a shortest path problem for a stochastic model of obstacles, where each obstacle's presence is an independent event with a known probability. Finally, we also present a data structure that can answer s-t path queries in polylogarithmic time, for any pair of points s, t in the plane.

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Pankaj K. Agarwal, Neeraj Kumar, Stavros Sintos, and Subhash Suri. Computing Shortest Paths in the Plane with Removable Obstacles. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SWAT.2018.5

Author Details

Pankaj K. Agarwal
  • Duke University, Durham, NC, USA
Neeraj Kumar
  • University of California, Santa Barbara, CA, USA
Stavros Sintos
  • Duke University, Durham, NC, USA
Subhash Suri
  • University of California, Santa Barbara, CA, USA

Funding

Work by Agarwal and Sintos is supported by NSF under grants CCF-15-13816, CCF-15-46392, and IIS-14-08846, by ARO grant W911NF-15-1-0408, and by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Work by Suri and Kumar is supported by NSF under grant CCF-1525817.

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