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：我们设计了一种加法近似方案，用于估计最小权重二分匹配问题的代价：给定一个具有非负边代价的二部图以及$\varepsilon > 0$，我们的算法在真正次二次时间$O(n^{2-\delta(\varepsilon)})$内估计出匹配除$O(\varepsilon)$部分顶点之外的所有顶点的代价。该算法在计算地球移动距离（EMD）时具有自然解释，可实现$\varepsilon$加法近似。值得注意的是，我们未对底层度量空间做任何假设（更一般地，代价不必满足三角不等式）。与实例规模（任意$n \times n$代价矩阵）相比，我们的算法运行在{\em 次线性}时间内。该算法还可近似一个更一般的问题：具有背包约束的最大基数二分匹配，其目标是在总代价不超过$B$的条件下最大化可匹配的顶点数量。