A New Result on the Complexity of Heuristic Estimates for the A* Algorithm

TL;DR

This paper investigates the computational complexity of heuristic estimates in A* search, demonstrating that hierarchical relaxation does not reduce effort unless specific optimizations are applied, and provides theoretical proofs of dominance relations.

Contribution

It offers a formal analysis showing that searching through relaxed models does not inherently reduce complexity unless certain transformations are used.

Findings

Searching in the hierarchy of relaxed models does not reduce computational effort without optimization.

Heuristic A* search with a heuristic derived from a relaxed model dominates other searches.

The paper provides theoretical proofs of dominance relations between heuristics in hierarchical search.

Abstract

Relaxed models are abstract problem descriptions generated by ignoring constraints that are present in base-level problems. They play an important role in planning and search algorithms, as it has been shown that the length of an optimal solution to a relaxed model yields a monotone heuristic for an A? search of a base-level problem. Optimal solutions to a relaxed model may be computed algorithmically or by search in a further relaxed model, leading to a search that explores a hierarchy of relaxed models. In this paper, we review the traditional definition of problem relaxation and show that searching in the abstraction hierarchy created by problem relaxation will not reduce the computational effort required to find optimal solutions to the base- level problem, unless the relaxed problem found in the hierarchy can be transformed by some optimization (e.g., subproblem factoring). Specifically, we prove that any A* search of the base-level using a heuristic h2 will largely dominate an A* search of the base-level using a heuristic h1, if h1 must be computed by an A* search of the relaxed model using h2.