Pseudo Approximation Algorithms with Applications to Optimal Motion Planning

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) 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.