标题： 关于参数化运动规划算法的复杂性 显示英文标题

标题： On the complexity of parametrized motion planning algorithms

摘要： 我们研究了r次顺序参数化拓扑复杂性的概率变体，它从下方界定了这个经典不变量，并衡量了构建宽容的参数化运动规划算法的难度。 一方面，我们使用上同调来证明这个新不变量在Fadell-Neuwirth纤维丛和定向球面丛上与经典不变量表现出相似的行为；另一方面，我们使用等变同伦理论来证明，在纤维为实射影空间且结构群为特殊正交群的纤维丛上，它的行为截然不同。 我们还探讨了该不变量的其他几个特性及其与由拓扑机器人学引发的其他不变量之间的关系。 显示英文摘要

摘要： We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.