Othar Hansson

Criticizing solutions to relaxed models yields powerful admissible heuristics

Abstract Branch-and-bound techniques allow intractable problems to be solved by using heuristics to bound the cost of partial solutions. The use of admissible heuristics can guarantee that the solutions found are optimal. This paper examines one paradigm—problem relaxation by constraint deletion—which has been used to develop many admissible heuristics. The paradigm suggests three steps: simplify (or relax) a problem, solve the simplified problem, and use that solution to guide the search for a solution to the original problem. We introduce the following extension to this methodology: by criticizing the feasibility of a relaxed solution, we arrive at a closer approximation of the solution to the original problem. We apply this methodology to two well-studied problems in operations research and artificial intelligence. For the traveling-salesman problem, iteration of our technique yields a series of novel heuristics, culminating in Held and Karps minimum-spanning-tree heuristic. For the eight puzzle, it yields a heretofore undiscovered heuristic which is shown to perform significantly better than any previously known.

Probabilistic heuristic estimates

Though they constitute the major knowledge source in problem-solving systems, no unified theory of heuristics has emerged. Pearl [15] defines heuristics as “criteria, methods, or principles for deciding which among several alternative courses of action promises to be the most effective in order to achieve some goal”. The absence of a more precise definition has impeded our efforts to understand, utilize, and discover heuristics. Another consequence is that problem-solving techniques which rely on heuristic knowledge cannot be relied upon to act rationally — in the sense of the normative theory of rationality.To provide a sound basis for BPS, the Bayesian Problem-Solver, we have developed a simple formal theory of heuristics, which is general enough to subsume traditional heuristic functions as well as other forms of problem-solving knowledge, and to straddle disparate problem domains. Probabilistic heuristic estimates represent a probabilistic association of sensations with prior experience — specifically, a mapping from observations directly to subjective probabilities which enables the use of theoretically principled mechanisms for coherent inference and decision making during problem-solving. This paper discusses some of the implications of this theory, and describes its successful application in BPS.

A new result on the complexity of heuristic estimates for the A★ algorithm

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.

Generating Admissible Heuristics by Criticizing Solutions to Relaxed Models

CUCS-119-85 This paper examines one paradigm used to develop admissible heuristics: problem relaxation [10, 11,32]. This consists of three steps: simplify (or relax) a problem, solve the simplified problem, and use that solution as advice to guide the search for a solution to the original problem. We introduce an extension to this methodology which exploits the simplicity of relaxed models. By criticizing the feasibility of a relaxed solution, we arrive at a closer approximation of the solution to the original problem. This solution-criticism process recovers some of the information lost by relaxation. and yields more powerful admissible heuristics than by relaxation alone. We apply our methodology to the TravelingSalesman problem and the N Puzzle. For the Traveling-Salesman Problem, it yields the well known. admissible minimum spanning tree heuristic. For the Eight and Fifteen Puzzles (in general the N puzzle), it yields a new heuristic which performs significantly better than all previously known heuristics.