### 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) |
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Pages (from-to) | 139-171 |

Number of pages | 33 |

Journal | Discrete and Computational Geometry |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2004 |

### ASJC Scopus subject areas

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

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## Cite this

*Discrete and Computational Geometry*,

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