AI 中文总结
本文研究精确最优传输方法，证明在中等规模密集实例上精确匹配（Jonker-Volgenant）比Sinkhorn类近似方法更快更准，并提出kNN池间隙证明和机器人任务分配验证，展示其实际优势。
AI 中文摘要
在n个源和n个单位质量的汇之间的平衡离散最优传输问题，恰好是最小成本分配问题——即二分完美匹配——因此可以通过工业级匹配引擎在毫秒到秒内精确求解。我们探讨了精确方法何时优于标准的近似替代方法（熵正则化的Sinkhorn算法及其加速变体Greenkhorn），并通过教科书式的LP对偶可行性裁剪使稀疏精确侧得到认证。本文有三项贡献。(i) 测量：在密集二维实例上，精确匹配（Jonker-Volgenant算法）在整个中等规模范围内（n=500时0.01秒到n=8000时11.5秒）比两种近似方法更快且严格更准确；在同一硬件上达到1%质量目标，Greenkhorn需要约10-80分钟（比精确方法慢4e2-6e4倍；普通Sinkhorn还要慢20-650倍），这是超出测量范围的粗略幂律投影。Greenkhorn相对于普通Sinkhorn的实测加速比在大多数收敛单元上仅为1.0-1.5倍。(ii) 一个简单的kNN池间隙证明：给定一个池匹配及其Blossom对偶，一次O(n^2)的裁剪产生一个密集可行的下界；结合Sinkhorn对偶势（在每次迭代中有效，不仅限于收敛时），该下界在所有45个测量配置上有效，并随k单调收紧。(iii) 一个多机器人任务分配合理性检查，其中离散计划是交付物：每轮精确分配成本为0.1-68毫秒，而Sinkhorn加硬化流程成本为0.12-15.9秒，并在15轮中累积6-27%的额外行程。Sinkhorn家族的大规模密集场景被承认并保持原样。代码、数据和结果在MIT许可下提供：此https URL。
英文摘要
Balanced discrete optimal transport between n sources and n targets of unit mass is exactly the minimum-cost assignment problem-a bipartite perfect matching-and is therefore solvable exactly by industrial matching engines in milliseconds to seconds. We ask when the exact approach beats the standard approximate alternatives, entropic Sinkhorn and its accelerated variant Greenkhorn, and make the sparse-exact side certified by a textbook LP dual-feasibility clip. Three contributions. (i) Measurement: on dense 2-D instances, exact matching (Jonker-Volgenant) is faster and strictly more accurate than either approximate method throughout the moderate-n regime (0.01 s at n=500 to 11.5 s at n=8000); reaching a 1% quality target on the same hardware requires roughly 10-80 min for Greenkhorn (factors 4e2-6e4 over exact; plain Sinkhorn is 20-650x slower still), a rough power-law projection beyond the measured range. Greenkhorn's measured speedup over plain Sinkhorn is only 1.0-1.5x on most converged cells. (ii) A simple kNN-pool gap certificate: given a pool matching and its Blossom dual, a one-pass O(n^2) clip produces a dense-feasible lower bound; combined with the Sinkhorn dual potential (valid at every iterate, not just at convergence), the bound is valid on all 45 measured configurations and tightens monotonically with k. (iii) A multi-robot task-allocation sanity check where the discrete plan is the deliverable: per-round exact assignment costs 0.1-68 ms, while a Sinkhorn-plus-hardening pipeline costs 0.12-15.9 s and accumulates 6-27% extra travel over 15 rounds. The Sinkhorn family's large-n dense regime is acknowledged and left untouched. Code, data, and results under MIT: https://github.com/dimkadimon/OT-Blossom.
Comments21 pages, 3 figures, 8 tables
