On maximum flows in polyhedral domains
Journal of Computer and System Sciences
An exact algorithm for kinodynamic planning in the plane
Discrete & Computational Geometry
Motion planning in the presence of moving obstacles
Journal of the ACM (JACM)
Handbook of discrete and computational geometry
Robot Motion Planning
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Time-minimal paths amidst moving obstacles in three dimensions
Theoretical Computer Science
Nonholonomic Motion Planning
Thick non-crossing paths and minimum-cost flows in polygonal domains
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
New lower bound techniques for robot motion planning problems
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Aircraft routing in the presence of hazardous weather
Aircraft routing in the presence of hazardous weather
Thick non-crossing paths and minimum-cost continuous flows in polygonal domains
Thick non-crossing paths and minimum-cost continuous flows in polygonal domains
Roadmap-based motion planning in dynamic environments
IEEE Transactions on Robotics
Routing a maximum number of disks through a scene of moving obstacles
Proceedings of the twenty-fourth annual symposium on Computational geometry
Proceedings of the twenty-sixth annual symposium on Computational geometry
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We consider the problem of finding a maximum number of disjoint paths for unit disks moving amidst static or dynamic obstacles. For the static case we give efficient exact algorithms, based on adapting the "continuous uppermost path" paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem. (In this extended abstract we only state these results.) Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. We observe that (unless P=NP), for any α,β 0, one cannot decide in polynomial time whether there exist [αΚ] paths for disks of radius βR, where K is the maximum number of paths for radius-R disks. The problem is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying "dual" approximations, compromising on the radius of the disks and on the maximum speed of motion. Our main result is a pseudopolynomial-time dual-approximation algorithm: if K unit disks, each with unit bound on the speed, may be routed through an environment, our algorithm finds (at least) K paths for disks of radius Ω(1) moving with speed O(1). The algorithm computes a maxflow with "forbidden pairs" in an "adaptive" grid, laid out in space-time. Although (as we show) in general finding even an approximation to the maxflow with forbidden pairs is not possible (unless P=NP), a careful choice of time discetization and a non-uniform grid of "way-points" allows us to give provable approximation guarantees on the quality of the solution produced by the algorithm. Our algorithm extends to higher dimensions and to finding paths for translational motion of arbitrary-shape objects.