### 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 ε 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, and ε > 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 true ε-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 three dimensions, 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.

Original language | English (US) |
---|---|

Pages (from-to) | 139-171 |

Number of pages | 33 |

Journal | Discrete and Computational Geometry |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - 2004 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Discrete and Computational Geometry*,

*31*(1), 139-171. https://doi.org/10.1007/s00454-003-2952-3

**Pseudo Approximation Algorithms with Applications to Optimal Motion Planning.** / Asano, Tetsuo; Kirkpatrick, David; Yap, Chee.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 31, no. 1, pp. 139-171. https://doi.org/10.1007/s00454-003-2952-3

}

TY - JOUR

T1 - Pseudo Approximation Algorithms with Applications to Optimal Motion Planning

AU - Asano, Tetsuo

AU - Kirkpatrick, David

AU - Yap, Chee

PY - 2004

Y1 - 2004

N2 - 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 ε 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, and ε > 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) XR ⊆ X R′ for R < R′ and (2) XR = X for R sufficiently large. We first describe a highly efficient scheme for converting a pseudo ε-approximation algorithm into a true ε-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 three dimensions, and (B) d1-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.

AB - 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 ε 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, and ε > 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) XR ⊆ X R′ for R < R′ and (2) XR = X for R sufficiently large. We first describe a highly efficient scheme for converting a pseudo ε-approximation algorithm into a true ε-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 three dimensions, and (B) d1-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.

UR - http://www.scopus.com/inward/record.url?scp=1142264267&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1142264267&partnerID=8YFLogxK

U2 - 10.1007/s00454-003-2952-3

DO - 10.1007/s00454-003-2952-3

M3 - Article

VL - 31

SP - 139

EP - 171

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -