We design an additive approximation scheme for estimating the cost of the min-weight bipartite matching problem: given a bipartite graph with non-negative edge costs and $\varepsilon > 0$, our algorithm estimates the cost of matching all but $O(\varepsilon)$-fraction of the vertices in truly subquadratic time $O(n^{2-\delta(\varepsilon)})$. Our algorithm has a natural interpretation for computing the Earth Mover's Distance (EMD), up to a $\varepsilon$-additive approximation. Notably, we make no assumptions about the underlying metric (more generally, the costs do not have to satisfy triangle inequality). Note that compared to the size of the instance (an arbitrary $n \times n$ cost matrix), our algorithm runs in {\em sublinear} time. Our algorithm can approximate a slightly more general problem: max-cardinality bipartite matching with a knapsack constraint, where the goal is to maximize the number of vertices that can be matched up to a total cost $B$.

翻译：我们设计了一种用于估计最小权重二分匹配问题成本的加性近似方案：给定一个具有非负边成本的二分图以及$\varepsilon > 0$，我们的算法能够在真正次二次时间$O(n^{2-\delta(\varepsilon)})$内估计匹配除$O(\varepsilon)$比例顶点之外的所有顶点的成本。该算法在计算推土机距离（EMD）时具有直观的解释，可实现$\varepsilon$-加性近似。值得注意的是，我们不对底层度量空间做任何假设（更一般地，成本不必满足三角不等式）。需要指出的是，与实例规模（任意$n \times n$成本矩阵）相比，我们的算法运行在{\em 次线性}时间内。我们的算法还可近似一个更一般的问题：带背包约束的最大基数二分匹配，其目标是在总成本不超过$B$的条件下最大化可匹配的顶点数量。