Inputs:

• Set of flights to cover
• Pairs of flights that can use the same plane
• Set of k planes

Output:

• Schedule showing how to use our k planes to cover the flights

Flow graph G:

    vertices = source, sink

    foreach flight i, start u_i & dest v_i
        1) foreach flight i, an edge (u_i, v_i) with capacity 1 and lower bound 1. We want to cover each flight.
        2) foreach flight i and j such that j is “reachable” from i, we have an edge (u_i, v_i) with capacity 1 and lower bound 0. We want to reuse a plane if possible.
        3) foreach flight i, an edge (s, u_i) with capacity 1 and lower bound 0, so that a plane can start its day on any flight.
        4) foreach flight j, and edge (v_j, t) with capacity 1 and lower bound 0, so that a plane can end its day anywhere.
        5) an edge (s,t) with capacity k and lower bound 0, so that we don’t have to use all the planes.

Algorithm:
    Construct G
    Find circulation

“There exists a feasible circulation iff there exists a valid schedule”

Pf(=>): Assume a feasible circulation, meaning each flight is covered by lowe bound on 1. We can read off the schedule by finding edges with flow 1.

Pf(=<): Assume a valid schedule. The schedule can be used to define a circulation. A valid schedule implies all flights are covered which implies that all lower bounds are satisfied.