Network Flow
The network flow problem considers a graph 
with a set of sources 
and sinks 
and for which each edge has an assigned capacity (weight), and then asks to find the maximum flow that can be routed from 
to 
while respecting the given edge capacities. The network flow problem can be solved in time 
(Edmonds and Karp 1972; Skiena 1990, p. 237). It is implemented in the Wolfram Language as FindMaximumFlow[g, source, sink].
REFERENCES
Edmonds, J. and Karp, R. M. "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems." J. ACM 19, 248-264, 1972.
Even, S. and Tarjan, R. E. "Network Flow and Testing Graph Connectivity." SIAM J. Comput. 4, 507-518, 1975.
Ford, L. R. and Fulkerson, D. R. Flows in Networks. Princeton, NJ: Princeton University Press, 1962.
Gonery, R. E. and Hu, T. C. "Multiterminal Network Flows." J. SIAM 9, 551-570, 1961.
Orlin, J. B. "A Faster Strongly Polynomial Minimum Cost Flow Algorithm." Proc. 20th ACM Symposium Theorem of Computing. pp. 377-387, 1988.
Skiena, S. "Network Flow." §6.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 237-239, 1990.
Skiena, S. S. "Network Flow." §8.4.9 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 297-300, 1997.
Tarjan, R. E. Data Structures and Network Algorithms. Philadelphia, PA: SIAM Press, 1983.
