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In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is an approximation framework that gives an O(√nα(n))-approximation for polygonal obstacles, where α(n) denotes the inverse Ackermann’s function. For pseudodisks and rectilinear polygons, the same technique achieves an O(√n)-approximation. The technique also gives O (√n)-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.
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A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane
Given two points s and t in the plane and a set of obstacles defined by closed curves, what is the minimum number of obstacles touched by a path connecting s and t? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory (under the names Min-Color Path and Minimum Label Path), wireless sensor networks (Barrier Resilience) and motion planning (Minimum Constraint Removal). It remains NP-hard even for very simple-shaped obstacles such as unit-length line segments. In this paper we give the first constant factor approximation algorithm for this problem, resolving an open problem of [Chan and Kirkpatrick, TCS, 2014] and [Bandyapadhyay et al., CGTA, 2020]. We also obtain a constant factor approximation for the Minimum Color Prize Collecting Steiner Forest where the goal is to connect multiple request pairs (s1, t1), . . . , (sk, tk) while minimizing the number of obstacles touched by any (si, ti) path plus a fixed cost of wi for each pair (si, ti) left disconnected. This generalizes the classic Steiner Forest and Prize-Collecting Steiner Forest problems on planar graphs, for which intricate PTASes are known. In contrast, no PTAS is possible for Min-Color Path even on planar graphs since the problem is known to be APX- hard [Eiben and Kanj, TALG, 2020]. Additionally, we show that generalizations of the problem to disconnected obstacles in the plane or connected obstacles in higher dimensions are strongly inapproximable assuming some well-known hardness conjectures.
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- Award ID(s):
- 1814172
- PAR ID:
- 10284625
- Date Published:
- Journal Name:
- Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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