We study the problem of path planning with soft homology constraints on a surface topologically equivalent to a disk with punctures. Specifically, we propose an algorithm, named $\Hstar$, for the efficient computation of a path homologous to a user-provided reference path. We show that the algorithm can generate a suite of paths in distinct homology classes, from the overall shortest path to the shortest path homologous to the reference path, ordered both by path length and similarity to the reference path. Rollout is shown to improve the results produced by the algorithm. Experiments demonstrate that $\Hstar$ can be an efficient alternative to optimal methods, especially for configuration spaces with many obstacles.

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