순열 불변 다중 로봇 경로 계획을 위한 헝가리안 기반 휴리스틱

초록

The Hungarian algorithm is a well-known cubic-time algorithm for finding minimum-cost matchings in weighted bipartite graphs. While utilizing it for multi-agent path planning yields the minimum-total-length set of paths for n agents, these paths may intersect. We propose a quadratic-time heuristic that computes a close-to-minimum-total-length set of agent paths and enables highly efficient collision-free modification of these paths. We employ several techniques to optimize both time efficiency and total path length, including grouping nearby points and exploiting the geometric properties of these groups. First, we utilize k-d trees to group nearby starting/destination points. We then use the Hungarian algorithm to match the centroids of these groups, followed by a second-round Hungarian matching of the individual points within each paired group. This hierarchical approach effectively reduces the time complexity from cubic to quadratic. Second, we accelerate the collision-free refinement of the computed paths by identifying disjoint group pairs through a simple geometric test and excluding them from collision-checking. Based on experimental results, our heuristic algorithm is highly effective in terms of both time efficiency and total path length; it achieves 1.001-optimal solutions for 4K agents in 0.4 s and for 250K agents in 875 s, whereas the Hungarian spends 192 s on computing only the minimum-total-length paths (before collision-checking) for 4K agents. We compare our method against four algorithms: Hungarian and greedy matchings, both with brute-force collision-checking and modification, and two previous algorithms developed by ourselves.

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