  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
$B$.
