Abstract
This paper presents a faster algorithm for the M-convex submodular flow problem, which is a generalization of the minimum-cost flow problem with an M-convex cost function for the flow-boundary, where an M-convex function is a nonlinear nonseparable discrete convex function on integer points. The algorithm extends the capacity scaling approach for the submodular flow problem by Fleischer, Iwata and McCormick (2002) with the aid of a novel technique of changing the potential by solving maximum submodular flow problems.
Supported by the Kayamori Foundation of Informational Science Advancement.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cunningham, W.H., Frank, A.: A primal-dual algorithm for submodular flows. Math. Oper. Res. 10, 251–262 (1985)
Dress, A.W.M., Wenzel, W.: Valuated matroids. Adv. Math. 93, 214–250 (1992)
Edmonds, J., Giles, R.: A min-max relation for submodular functions on graphs. Ann. Discrete Math. 1, 185–204 (1977)
Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)
Fleischer, L., Iwata, S., McCormick, S.T.: A faster capacity scaling algorithm for minimum cost submodular flow. Math. Program., Ser. A 92, 119–139 (2002)
Frank, A.: Finding feasible vectors of Edmonds-Giles Polyhedra. J. Comb. Theory, Ser. B 36, 221–239 (1984)
Fujishige, S.: Submodular Functions and Optimization. North-Holland, Amsterdam (1991)
Fujishige, S., Zhang, X.: New algorithms for the intersection problem of submodular systems. Japan J. Indust. Appl. Math. 9, 369–382 (1992)
Iwata, S.: A capacity scaling algorithm for convex cost submodular flows. Math. Program. 76, 299–308 (1997)
Iwata, S.: A faster scaling algorithm for minimizing submodular functions. SIAM J. Comput. 32, 833–840 (2003)
Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)
Iwata, S., Shigeno, M.: Conjugate scaling algorithm for Fenchel-type duality in discrete convex optimization. SIAM J. Optim. 13, 204–211 (2003)
Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming–The State of the Art, pp. 235–257. Springer, Heidelberg (1983)
Moriguchi, S., Murota, K.: Capacity scaling algorithm for scalable M-convex submodular flow problems. Optim. Methods Softw. 18, 207–218 (2003)
Murota, K.: Valuated matroid intersection, I: optimality criteria. SIAM J. Discrete Math. 9, 545–561 (1996)
Murota, K.: Submodular flow problem with a nonseparable cost function. Combinatorica 19, 87–109 (1999)
Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)
Murota, K., Tamura, A.: Application of M-convex submodular flow problem to mathematical economics. Japan J. Indust. Appl. Math. 20, 257–277 (2003)
Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory, Ser. B 80, 346–355 (2000)
Shioura, A.: Fast scaling algorithms for M-convex function minimization with application to resource allocation problem. Discrete Appl. Math. 134, 303–316 (2003)
Tamura, A.: Coordinatewise domain scaling algorithm for M-convex function minimization. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 21–35. Springer, Heidelberg (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Iwata, S., Moriguchi, S., Murota, K. (2004). A Capacity Scaling Algorithm for M-convex Submodular Flow. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_27
Download citation
DOI: https://doi.org/10.1007/978-3-540-25960-2_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22113-5
Online ISBN: 978-3-540-25960-2
eBook Packages: Springer Book Archive