Bipartite Minimum Path Cover

Problem

Compute the minimum number of vertex-disjoint paths needed to cover every node of a DAG.

Build a bipartite graph with a left and a right copy of each node and an edge from left u to right v for every DAG edge u to v. By Dilworth's theorem the answer is n minus the maximum matching. edges is a list of [u, v] pairs.

Hint

Each matched pair joins two path segments; unmatched right nodes are path starts.

Test cases

case 1

in: [4,[[0,1],[1,2],[2,3]]]

expected: 1

case 2

in: [4,[[0,1],[0,2],[1,3],[2,3]]]

expected: 2

case 3

in: [3,[]]

expected: 3

case 4

in: [2,[[0,1]]]

expected: 1

case 5

in: [5,[[0,1],[1,2],[3,4]]]

expected: 2

Starter codePython

def min_path_cover(n, edges):
    # Your code here
    pass

5 test cases · run your code against them in the editor