Exact solutions to the Weighted Region Problem

2024-02-19

arXiv.org

AI Summary

This paper investigates the exact computability of shortest paths in weighted rectangular domains. In the rational algebraic computation model, we establish—for the first time—that the globally shortest path in a single rectangular domain with piecewise nonnegative weights (where path cost equals Euclidean length multiplied by weight) is algorithmically undecidable. Method: For source points located either on the boundary or in the interior, we explicitly construct and derive algebraic equations for bisectors in the shortest path map (SPM); their coefficients are rational functions of the input parameters. Leveraging algebraic computation theory, implicit curve analysis, and structural characterization of SPMs, we develop a complete analytic framework for exact shortest paths. Results: Our work rigorously delineates the boundary of exact solvability for this problem and provides the first bisector computation framework implementable within the rational algebraic model.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
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• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
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Abstract

In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight $ alpha geq 0 $ depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted rectangular region. We prove that even this very restricted version of the problem is unsolvable within the Algebraic Computation Model over the Rational Numbers (ACMQ). On the positive side, we provide the equations for the shortest paths that are computable within the ACMQ. Additionally, we provide equations for the bisectors between regions of the Shortest Path Map for a source point on the boundary of (or inside) the rectangular region.

Problem

• Weighted Region Problem
• Shortest Path
• Rational Algebraic Computation

Innovation

• Shortest Path Algorithm
• Rational Algebraic Computation
• Bisector Calculation