Motion planning seeks a collision-free path in a configuration space (C-space), representing all possible robot configurations in the environment. As it is challenging to construct a C-space explicitly for a high-dimensional robot, we generally build a graph structure called a roadmap, a discrete approximation of a complex continuous C-space, to reason about connectivity. Checking collision-free connectivity in the roadmap requires expensive edge-evaluation computations, and thus, reducing the number of evaluations has become a significant research objective. However, in practice, we often face infeasible problems: those in which there is no collision-free path in the roadmap between the start and the goal locations. Existing studies often overlook the possibility of infeasibility, becoming highly inefficient by performing many edge evaluations. In this work, we address this oversight in scenarios where a prior roadmap is available; that is, the edges of the roadmap contain the probability of being a collision-free edge learned from past experience. To this end, we propose an algorithm called iterative path and cut finding (IPC) that iteratively searches for a path and a cut in a prior roadmap to detect infeasibility while reducing expensive edge evaluations as much as possible. We further improve the efficiency of IPC by introducing a second algorithm, iterative decomposition and path and cut finding (IDPC), that leverages the fact that cut-finding algorithms partition the roadmap into smaller subgraphs. We analyze the theoretical properties of IPC and IDPC, such as completeness and computational complexity, and evaluate their performance in terms of completion time and the number of edge evaluations in large-scale simulations.

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