Abstract
A variation on the Edmonds-Karp scaling approach to the minimum cost network flow problem is examined. This algorithm, which scales costs rather than right-hand sides, also runs in polynomial time. Large-scale computational experiments indicate that the computational behavior of such scaling algorithms may be much better than had been presumed. Within several distributions of square, dense, capacitated transportation problems, a cost scaling code, SCALE, exhibits linear growth in average execution time with the number of edges, while two network simplex codes, RNET and GNET, exhibit greater than linear growth.
Our experiments reveal that median and mean execution times are predictable with surprising accuracy for all of the three codes and all three distributions from which test problems were generated. Moreover, for fixed problem size, individual execution times appear to behave as though they are approximately lognormally distributed with constant variance. The experiments also reveal sensitivity of the parameters in the models, and in the models themselves, to variations in the distribution of problems. This argues for caution in the interpretation of such computational studies beyond the realm in which the computations were performed.
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References
R.K. Ahuja, A.V. Goldberg, J.B. Orlin and R.E. Tarjan, “Finding minimum cost flows by double scaling,”Mathematical Programming 53 (1992) 243–266.
R.K. Ahuja, T.L. Magnanti and J.B. Orlin, “Network flows,” in: G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd, eds.,Optimization, Handbooks in Operations Research and Management Science, vol. 1 (North-Holland, Amsterdam 1989).
R.K. Ahuja, J.B. Orlin and R.E. Tarjan, “Improved time bounds for the maximum flow problem,”SIAM Journal of Computing 18 (1989) 939–954.
D.P. Bertsekas, “A distributed algorithm for the assignment problem,” unpublished manuscript (1979).
D.P. Bertsekas “A unified framework for primal-dual methods in minimum cost network flow problems,”Mathematical Programming 32 (1985) 125–145.
D.P. Bertsekas “Distributed asynchronous relaxation methods for linear network flow problems,” Technical Report LIDS-P-1986, Laboratory for Decision Systems, MIT (Cambridge, MA, 1986).
D.P. Bertsekas and J. Eckstein, “Dual coordinate step methods for linear network flow problems,”Mathematical Programming 42 (1988) 203–243.
R.G. Bland, “Complementary orthogonal subspaces of ℝn and orientability of matroids,” dissertation, Cornell University (Ithaca, NY, 1974).
R.G. Bland and J. Edmonds, “Fast primal algorithms for totally unimodular linear programming,” presented at the XI International Symposium on Mathematical Programming (Bonn, Germany, 1982).
R.G. Bland and D.L. Jensen, “A report on the computational behavior of a polynomial-time network flow algorithm,” Cornell University School of OR/IE Technical Report No. 661 (Ithaca, NY, 1985).
G.H. Bradley, G.G. Brown and G.W. Graves, “Design and implementation of large scale primal transshipment algorithms,”Management Science 24 (1977) 1–28.
N.R. Draper and H. Smith,Applied Regression Analysis (Wiley, New York, 1981, 2nd ed.).
J. Edmonds and R.M. Karp, “Theoretical improvements in algorithmic efficiency for network flow problems,”Journal of the ACM 19 (1972) 248–264.
L.R. Ford, Jr. and D.R. Fulkerson,Flows in Networks (Princeton University Press, Princeton, NJ 1962).
M.L. Fredman and R.E. Tarjan, “Fibonacci heaps and their uses in improved network optimization algorithms,”Journal of the ACM 34 (1987) 596–615.
D.R. Fulkerson, “An out-of-kilter algorithm for minimal cost flow problems,”Journal of the SIAM 9 (1961) 18–27.
D.R. Fulkerson, “Networks, frames, blocking systems,” in: G.B. Dantzig and A.F. Veinott, Jr., eds.,Mathematics of the Decision Sciences, Lectures in Applied Math., vol. 11, (American Mathematical Society, Providence, RI 1968) pp. 303–334.
H.N. Gabow, “Scaling algorithms for network problems,”Journal of Computer and Systems Science 31 (1985) 148–168.
H.N. Gabow, “On the theoretic and practical efficiency of scaling algorithms for network problems,” presented at the Fall 1984 ORSA/TIMS meeting (Dallas, TX, 1984).
A.V. Goldberg, “A new max-flow algorithm,” Technical Report MIT/LCS/TM-291, Laboratory for Computer Science, MIT (Cambridge, MA, 1985).
A.V. Goldberg, “Efficient graph algorithms for sequential and parallel computers”, Dissertation, MIT (Cambridge, MA., 1987).
A.V. Goldberg, E. Tardos and R.E. Tarjan, “Network flow algorithms,” in: B. Korte, L. Lovász, H.J. Prömel and A. Schrÿver, eds.,Flows, Paths, and VLSI Layout (Springer, Berlin, 1990) pp 101–164.
A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”Proceedings of the 18th ACM Symposium on Theory of Computing (1986) 136–146.
A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”Journal of the ACM 35 (1988) 921–940.
A.V. Goldberg and R.E. Tarjan, “Finding minimum cost circulations by successive approximations,”Mathematics of OR 15 (1990) 430–466.
M.D. Grigoriadis, “An efficient implementation of the network simplex method,”Mathematical Programming Study 26 (1986) 83–111.
M.D. Grigoriadis, private communication.
M.D. Grigoriadis and T. Hsu, “RNET — The Rutgers minimum-cost network flow subroutines — user documentation,” Computer Science Technical Report, Rutgers University (New Brunswick, NY, 1979).
D.L. Jensen, “Coloring and duality: combinatorial augmentation methods,” Dissertation, Cornell University (Ithaca, NY, 1985).
D. Klingman, A. Napier and J. Stutz, “NETGEN: a program for generating large-scale assignment, transportation and minimum cost network flow problems,”Management Science 20 (1974) 814–821.
L.G. Khachiyan, “A polynomial algorithm in linear programming,”Doklady Akademiia Nauk SSSR 224 (1979) 1093–1096 (in Russian). [English translation in:Soviet Mathematics Doklady 20 (1979) 191–194.]
V.M. Malhotra, M. Pramodh Kumar and S.N. Maheshwari, “An O(|V 3|) algorithm for finding maximum flows in networks,”Information Processing Letters 7 (1978) 277–278.
G.J. Minty, “Monotone networks,”Proceedings of the Royal Society, London, Series A 257 (1960) 194–212.
J.B. Orlin, “A faster polynomial minimum cost flow algorithm,”Proceedings of the 20th ACM Symposium on Theory of Computing (1988) 377–387.
H. Röck, “Scaling techniques for minimal cost network flows,” in: V. Page, ed.,Discrete structures and algorithms (Carl Hansen, Munich, 1980).
P.D. Seymour, “Decomposition of regular matroids,”Journal of Combinatorial Theory, Series B 28 (1980) 305–359.
D.D. Sleator, “An O(nm logn) algorithm for maximum network flow,” Dissertation, Technical Report STAN-CS-80-831, Stanford University (Stanford, CA, 1980).
D.D. Sleator and R.E. Tarjan, “A data structure for dynamic trees,”Journal of Computer and System Sciences 26 (1983) 362–391.
É. Tardos, “A strongly polynomial minimum cost circulation algorithm,”Combinatorica 5 (1985) 247–255.
É. Tardos, “A strongly polynomial algorithm for solving combinatorial linear programs,”Operations Research 34 (1986) 250–256.
R.E. Tarjan,Data Structures and Network Algorithms (SIAM, Philadelphia, PA, 1983).
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This work has been supported in part by NSF grants ENG-7910807, ECS-8313853, DMS-8706133, and DDM-8813054, and by AFOSR, NSF, and ONR under NSF grant DMS-8920550 to Cornell University, and by a Sloan Foundation research fellowship held by the first author.
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Bland, R.G., Jensen, D.L. On the computational behavior of a polynomial-time network flow algorithm. Mathematical Programming 54, 1–39 (1992). https://doi.org/10.1007/BF01586039
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DOI: https://doi.org/10.1007/BF01586039