Approximate Euclidean shortest path in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
d1-optimal motion for a rod (extended abstract)
Proceedings of the twelfth annual symposium on Computational geometry
Handbook of discrete and computational geometry
Precision-Sensitive Euclidean Shortest Path in 3-Space
SIAM Journal on Computing
Introduction to Algorithms
Retraction: A new approach to motion-planning
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Touring a sequence of polygons
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
ESA'05 Proceedings of the 13th annual European conference on Algorithms
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(MATH) We introduce a technique for computing approximate solutions to optimization problems. If X is the set of feasible solutions, the standard goal of approximation algorithms is to compute &khgr; &egr; X that is an &egr;-approximate solution in the following sense: d(&khgr;)&xie;(1+&egr;)d(&khgr;*) where &khgr;* &Egr; X is an optimal solution, d : X → 0 is the optimization function to be minimized, and $\vareps0 is an input parameter. Our approach is to first devise algorithms that compute pseudo &egr;-approximate solutions satisfying the bound d(&khgr;) &xie; d(&khgr;R *) + &egr;R where R0 is a new input parameter. Here &khgr;* R denotes an optimal solution in the space X R of R-constrained feasible solutions. The parameterization provides a stratification of X in the sense that (1) XR ⊆ XR' , for R R' and (2) XR = X for R sufficiently large.We first describe a highly efficient scheme for converting a pseudo &egr;-approximation algorithm into a true &egr;-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than &egr;-approximation algorithms.We then apply our technique to two problems in robotics: (A) Euclidean Shortest Path (3ESP), namely the shortest path for a point robot amidst polyhedral obstacles in 3D, and (B) d 1-optimal motion for a rod moving amidst polygonal obstacles in 2D. Previously, no true &egr;-approximation algorithm for (B) was known. For (A), our new solution is not only simpler than two previous solutions but also has a lower complexity (in the algebraic model) measured in terms of the input precision. Note that (A) and (B) are the simplest NP-hard motion planning problems in 3-D and 2-D respectively.