Pseudo Approximation Algorithms with Applications to Optimal Motion Planning

Tetsuo Asano, David Kirkpatrick, Chee Yap

Research output: Contribution to journalArticle

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) 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.

Original languageEnglish (US)
Pages (from-to)139-171
Number of pages33
JournalDiscrete and Computational Geometry
Volume31
Issue number1
DOIs
StatePublished - 2004

Fingerprint

Motion Planning
Approximation algorithms
Motion planning
Approximation Algorithms
Approximate Solution
Shortest path
Optimal Solution
Function Optimization
Stratification
Polynomial-time Algorithm
Three-dimension
Robotics
Euclidean
Robot
Polynomials
Robots
Optimization Problem
Denote
Motion
Computing

ASJC Scopus subject areas

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

Cite this

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

In: Discrete and Computational Geometry, Vol. 31, No. 1, 2004, p. 139-171.

Research output: Contribution to journalArticle

@article{cdec3c54e4ca4329afe9466bd7bc7859,
title = "Pseudo Approximation Algorithms with Applications to Optimal Motion Planning",
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) 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.",
author = "Tetsuo Asano and David Kirkpatrick and Chee Yap",
year = "2004",
doi = "10.1007/s00454-003-2952-3",
language = "English (US)",
volume = "31",
pages = "139--171",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "1",

}

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 -