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| Title: | Pseudo approximation algorithms with applications to optimal motion planning |
| Authors: | Asano, Tetsuo
Kirkpatrick, David
Yap, Chee |
| Issue Date: | 2004-01 |
| Publisher: | Springer |
| Magazine name: | Discrete and Computational Geometry |
| Volume: | 31 |
| Number: | 1 |
| Start page: | 139 |
| End page: | 171 |
| DOI: | 10.1007/s00454-003-2952-3 |
| Abstract: | 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 x ∈ X that is an ε - approximate solution in the following sense: d(x) ≤ - (1+ε)d(x^*), where x^* ∈ X is an optimal solution, d : X → R ≥ 0 is the optimization function to be minimized, andε>0 is an input parameter. Our approach is first to devise algorithms that compute pseudo ε-approximate solutions satisfying the bound d(x) ≤ - d(x^*_R) +εR, where R>0 is a new input parameter. Here x^*_R denotes an optimal solution in the space X_R of R-constrained feasible solutions. The parameter R provides a stratification of X in the sense that (1) X_R ⊆ X_<R’> for R < R’ and (2) X_R = X for R sufficiently large. We first describe a highly efficient scheme for converting a pseudo ε-approximation algorithm into a trueε-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than ε-approximation algorithms. Another benefit is that our algorithm is automatically precision-sensitive. We 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 planar obstacles (1ORM). Previously, no polynomial time ε-approximation algorithm for (B) was known. For (A), our new solution is simpler than previous solutions and has an exponentially smaller complexity in terms of the input precision. |
| Rights: | This is the author-created version of Springer, Tetsuo Asano, David Kirkpatrick and Chee Yap, Discrete and Computational Geometry, 31(1), 2004, 139-171. The original publication is available at www.springerlink.com, http://dx.doi.org/10.1007/s00454-003-2952-3 |
| URI: | https://hdl.handle.net/10119/4919 |
| Material Type: | author |
| Appears in Collections: | b10-1. 雑誌掲載論文 (Journal Articles)
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