# Universally Maximum Flow with Piecewise-Constant Capacities

## Abstract

The *maximum dynamic flow problem* generalizes the standard maximum flow problem by introducing time. The object is to send as much flow from source to sink in *T* time units as possible, where capacities are interpreted as an upper bound on the rate of flow entering an arc. A related problem is the *universally maximum flow*, which is to send a flow from source to sink that maximizes the amount of flow arriving at the sink by time *t* simultaneously for all *t* ≤ *T*. We consider a further generalization of this problem that allows arc and node capacities to change over time. In particular, given a network with arc and node capacities that are piecewise constant functions of time with at most *k* breakpoints, and a time bound *T*, we show how to compute a flow that maximizes the amount of flow reaching the sink in all time intervals (0, *t*] simultaneously for all 0 < *t* ≤ *T*, in *O*(*k* ^{2} *mn*log(*kn* ^{2}/*m*)) time. The best previous algorithm requires *O*(*nk*) maximum flow computations on a network with (*m*+ *n*)*k* arcs and *nk* nodes.

## Keywords

Dynamic Network Sink Node Polynomial Time Algorithm Capacity Function Node Capacity## Preview

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## References

- 1.E. J. Anderson and P. Nash.
*Linear Programming in Infinite-Dimensional Spaces*. John Wiley & Sons, 1987.Google Scholar - 2.E. J. Anderson, P. Nash, and A. B. Philpott. A class of continuous network flow problems.
*Mathematics of Operations Research*, 7:501–14, 1982.zbMATHMathSciNetGoogle Scholar - 3.E. J. Anderson and A. B. Philpott. A continuous-time network simplex algorithm.
*Networks*, 19:395–425, 1989.zbMATHMathSciNetCrossRefGoogle Scholar - 4.J. E. Aronson. A survey of dynamic network flows.
*Annals of Operations Research*, 20:1–66, 1989.zbMATHCrossRefMathSciNetGoogle Scholar - 5.R. E. Burkard, K. Dlaska, and B. Klinz. The quickest flow problem.
*ZOR Methods and Models of Operations Research*, 37(1):31–58, 1993.zbMATHCrossRefMathSciNetGoogle Scholar - 6.L. Fleischer. Faster algorithms for the quickest transshipment problem with zero transit times. In
*Proceedings of the Ninth Annual ACM/SIAM Symposium on Discrete Algorithms*, pages 147–156, 1998. Submitted to SIAM Journal on Optimization.Google Scholar - 7.L. R. Ford and D. R. Fulkerson.
*Flows in Networks*. Princeton University Press, 1962.Google Scholar - 8.G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications.
*SIAM J. Comput.*, 18(1):30–55, 1989.zbMATHCrossRefMathSciNetGoogle Scholar - 9.A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem.
*Journal of ACM*, 35:921–940, 1988.zbMATHCrossRefMathSciNetGoogle Scholar - 10.B. Hajek and R. G. Ogier. Optimal dynamic routing in communication networks with continuous traffic.
*Networks*, 14:457–487, 1984.zbMATHMathSciNetCrossRefGoogle Scholar - 11.B. Hoppe.
*Efficient Dynamic Network Flow Algorithms*. PhD thesis, Cornell University, June 1995. Department of Computer Science Technical Report TR95-1524.Google Scholar - 12.B. Hoppe and É. Tardos. Polynomial time algorithms for some evacuation problems. In
*Proc. of 5th Annual ACM-SIAM Symp. on Discrete Algorithms*, pages 433–441, 1994.Google Scholar - 13.B. Hoppe and É. Tardos. The quickest transshipment problem. In
*Proc. of 6th Annual ACM-SIAM Symp. on Discrete Algorithms*, pages 512–521, 1995.Google Scholar - 14.E. Minieka. Maximal, lexicographic, and dynamic network flows.
*Operations Research*, 21:517–527, 1973.zbMATHMathSciNetGoogle Scholar - 15.F. H. Moss and A. Segall. An optimal control approach to dynamic routing in networks.
*IEEE Transactions on Automatic Control*, 27(2):329–339, 1982.zbMATHCrossRefMathSciNetGoogle Scholar - 16.R. G. Ogier. Minimum-delay routing in continuous-time dynamic networks with piecewise-constant capacities.
*Networks*, 18:303–318, 1988.zbMATHMathSciNetCrossRefGoogle Scholar - 17.J. B. Orlin. Minimum convex cost dynamic network flows.
*Mathematics of Operations Research*, 9(2):190–207, 1984.zbMATHMathSciNetCrossRefGoogle Scholar - 18.A. B. Philpott. Continuous-time flows in networks.
*Mathematics of Operations Research*, 15(4):640–661, November 1990.zbMATHMathSciNetGoogle Scholar - 19.A. B. Philpott and M. Craddock. An adaptive discretization method for a class of continuous network programs.
*Networks*, 26:1–11, 1995.zbMATHMathSciNetCrossRefGoogle Scholar - 20.W. B. Powell, P. Jaillet, and A. Odoni. Stochastic and dynamic networks and routing. In M. O. Ball, T. L. Magnanti, C. L. Monma, and G. L. Nemhauser, editors,
*Handbooks in Operations Research and Management Science: Networks*. Elsevier Science Publishers B. V., 1995.Google Scholar - 21.M. C. Pullan. An algorithm for a class of continuous linear programs.
*SIAM J. Control and Optimization*, 31(6):1558–1577, November 1993.zbMATHCrossRefMathSciNetGoogle Scholar - 22.M. C. Pullan. A study of general dynamic network programs with arc time-delays.
*SIAM Journal on Optimization*, 7:889–912, 1997.zbMATHCrossRefMathSciNetGoogle Scholar - 23.G. I. Stassinopoulos and P. Konstantopoulos. Optimal congestion control in single destination networks.
*IEEE transactions on communications*, 33(8):792–800, 1985.zbMATHCrossRefGoogle Scholar - 24.W. L. Wilkinson. An algorithm for universal maximal dynamic flows in a network.
*Operations Research*, 19:1602–1612, 1971.zbMATHMathSciNetGoogle Scholar