Figure 4

In Fig. 4(a), an input DAG G with two subpath constraints P ₁ and P ₂ ; we take. $V^{\prime }=V\left(G\right)$, $E^{\prime }=\mathrm{0̸}$, S = {a, d, e} and T = {f, c}; weights are not drawn. In Fig. 4(b), the graph transformed by Steps 1-6; the vertices still in $V^{\prime }$ are drawn as circles, other vertices as squares. In Fig. 4(c), the reduction to a min-cost circulation problem; the edges with flow lower bound 1 are labeled as '1'; other edges have flow lower bound 0. In a min-cost circulation of value 3, all highlighted edges have flow value 1, except for (f, t) with flow value 2, and (t, s) with value 3. Any decomposition of the min-cost circulation into 3 paths gives the solution for Problem MW-MPC-SC.