A Note on the Unsolvability of the Weighted Region Shortest Path Problem

TL;DR

This paper proves that the weighted region shortest path problem cannot be solved exactly within the Algebraic Computation Model over the Rational Numbers, using Galois theory and Bajaj's technique.

Contribution

It establishes the unsolvability of the weighted region shortest path problem in the ACMQ, highlighting fundamental computational limitations.

Findings

The problem cannot be solved exactly in ACMQ.

The proof employs Galois theory and Bajaj's technique.

It delineates the boundaries of algebraic solvability for geometric problems.

Abstract

Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s, t in R^2, where the distances are measured according to the weighted Euclidean metric-the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's technique.