Approximations for Maximum Transportation Problem with Permutable Supply Vector and Other Capacitated Star Packing Problems

The input to the transportation problem consists of a complete weighted bipartite graph G = (V 1,V 2,w), integer supplies a i ≥ 0, i ∈ V 1, and integer demands b j ≥ 0, j ∈ V 2, where w.l.o.g. \( \sum\nolimits_{i \in V_1 } {a_i } = \sum\nolimits_{j \in V_2 } {b_j } \) . The problem is to compute flows x ij i ∈ V 1, j ∈ V 2 such that \( \sum\nolimits_j {x_{ij} = a_i } \) for every i ∈ V 1, \( \sum\nolimits_j {x_{ij} = b_j } \) for every j ∈ V 2, and \( \sum\nolimits_E {w_{ij} x_{ij} } \) is maximized (or minimized). The transportation problem is polynomially solvable even when the flows are required to be integers.