Abstract:The problem we consider is a stochastic shortest path problem in the presence of a dynamic learning capability. Specifically, a spatial arrangement of possible obstacles needs to be traversed as swiftly as possible, and the status of the obstacles may be disambiguated (at a cost) en route. No efficiently computable optimal policy is known, and many similar problems have been proven intractable. In this article, we adapt a policy which is optimal for a related problem and prove that this policy is indeed also optimal for a restricted class of instances of our problem. Otherwise, this policy is generally suboptimal but, nonetheless, it is both effective and efficiently computable. Examples/simulations are provided in a mine countermeasures application. Of central use is the Tangent Arc Graph, a polynomially sized topological superimposition of exponentially many visibility graphs. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 389-399, 2011 Keywords: mine countermeasures; probabilistic path planning; random disambiguation path; tangent arc graph; markov decision process THE DISAMBIGUATION PROBLEMA disambiguation problem instance is a tuple (s, t, A, ρ, c), where s and t are points in R 2 , A is a finite set of open discs in R 2 , ρ is a function A → (0, 1], and c is a function A → R ≥0 . An agent wants to traverse from s to t through R 2 , along a continuous curve which is as short as possible in the sense of arclength. However, the discs of A are potential obstacles; for each A ∈ A, the probability that A is an obstacle is ρ(A), independently from the other discs in A. If ρ(A) < 1 then we say A is ambiguous and if ρ(A) = 1 then A is definitely an obstacle. The traversing agent cannot enter discs which are obstacles or ambiguous but, if and when the agent is located at the boundary ∂A for any A ∈ A, the agent has the option to disambiguate the disc A at a cost c(A) added to the traversal arclength, and the agent will learn whether or not A is actually an obstacle. The status of a disc will never change; if A is revealed to be an obstacle then the traversing agent may never enter A, and if A is not an obstacle then A may be entered anytime thereafter. The central issue is how to direct the agent's traversal to optimally utilize this disambiguation capability; that is, to find a policy for the agent which minimizes the expected length of the agent's s, t traversal.Correspondence to: C.E. Priebe (cep@jhu.edu) An example of a disambiguation problem instance is shown in Fig. 1; suppose the values of ρ(A i ), for i = 1, 2, 3, 4, 5 are 0.6, 0.4, 0.9, 0.8, 0.7, and suppose c(A i ) = 1.1 for all i. One particular traversal policy is illustrated in Fig. 1; from s the agent proceeds to the red bullet labeled 1, at which point A 1 is disambiguated. If A 1 is traversable then the agent is to continue till the red bullet labeled 2, at which point A 2 is disambiguated. Then the agent is to proceed to t through A 2 or clockwise around A 2 , according as A 2 is traversable or not. If A 1 was not traversa...
The authors consider the problem of navigating an agent to safely and swiftly traverse a two dimensional terrain populated with possible hazards. Each potential hazard is marked with a probabilistic estimate of whether it is indeed true. In proximity to any of these potential hazards, the agent is able to disambiguate, at a cost, whether it is indeed true or false. The method presented in this paper is to discretize the terrain using a two dimensional grid with 8-adjacency and approximately solve the problem by dynamically searching for shortest paths using the A* algorithm in the positively weighted grid graph with changing weight function.
The primal-dual algorithm for linear programming is very effective for solving network flow problems. For the method to work, an initial feasible solution to the dual is required. In this article, we show that, for the shortest path problem in a positively weighted graph equipped with a consistent heuristic function, the primal-dual algorithm will become the well-known A* algorithm if a special initial feasible solution to the dual is chosen. We also show how the improvements of the dual objective are related to the A* iterations.
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