Is there fast algorithm for finding min vertex-disjoint path cover of DAG graph of poset of pairs (x, y) where (x, y) < (u, v) iff x < u and y < v
So if I'm interpreting your question you have a set of points in the Euclidean plane and you want to find a decomposition of the points into as few monotone chains as possible? This is called the "layers of maxima" problem and it is easily solved in time O(n log n) by a plane sweep algorithm that sweeps over the points from left to right, using a binary search tree to maintain the set of intersection points of the monotone chains with the sweep line and assigning each point in sweep order greedily to the highest monotone chain that it can be assigned to. See e.g. Buchsbaum and Goodrich, "Three-Dimensional Layers of Maxima", Algorithmica 2004, for a three-dimensional version that can still be solved in O(n log n) and references to earlier work on the problem.