Abstract
Max-flow in planar graphs has always been studied with the assumption that there are capacities only on the edges. Here we consider a more general version of the problem when the vertices as well as edges have capacity constraints. In the context of general graphs considering only edge capacities is not restrictive, since the vertex-capacity problem can be reduced to the edge-capacity problem. However, in the case of planar graphs this reduction does not maintainplanarity and cannot be used. We study different versions of the planar flow problem (all of which have been extensively investigated in the context of edge capacities).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Birk and J. B. Lotspiech, A fast algorithm for connecting grid points to the boundary with nonintersecting straight lines,Proceedings of the second Symposium on Discrete Algorithms, pp. 465–474 (1991).
J. A. Bondy and U. S. R. Murty,Graph Theory with Applications, North-Holland, Amsterdam (1977).
T. H. Cormen, C. E. Leiserson, and R. L. Rivest,Introduction to Algorithms, MIT Press, Cambridge, MA (1990).
B. Codenetti and R. Tamassia, A network flow approach to reconfiguration of VLSI arrays,IEEE Transactions on Computers, Vol. 40, No. 1, pp. 118–121 (1991).
G. N. Frederickson, Fast algorithms for shortest paths in planar graphs, with applications,SIAM Journal on Computing, Vol. 16, pp. 1004–1022 (1987).
L. R. Ford and D. R. Fulkerson, Maximal flow through a network,Canadian Journal of Mathematics, Vol. 8, pp. 399–404 (1956).
M. L. Fredman and D. E. Willard, Blasting through the information theoretic barrier with fusion trees,Proceedings of the 22nd Annual Symposium on Theory of Computing, pp. 1–7 (1990).
J. W. Greene and A. El-Gamal, Configuration of VLSI arrays in the presence of defects,Journal of the ACM, Vol. 31, No. 4, pp. 694–717 (1984).
R. E. Gomory and T. C. Hu, Multi-terminal network flows,SIAM Journal on Applied Mathematics, Vol. 9, pp. 551–570 (1961).
F. Granot and R. Hassin, Multi-terminal maximum flows in node-capacitated networks,Discrete Applied Mathematics, Vol. 13, pp. 157–163 (1986).
L. Goldschlager, R. Shaw, and J. Staples, The maximum flow problem is log space complete forP, Theoretical Computer Science, Vol. 21, pp. 105–111 (1982).
A. V. Goldberg, E. Tardos, and R. E. Tarjan, Network flow algorithms, inPaths, Flows and VLSI-Layout (B. Korte, ed.), Springer-Verlag, New York, pp. 101–164 (1990).
R. Hassin, Maximum flows in (s, t) planar networks,Information Processing Letters, Vol. 13, page 107 (1981).
R. Hassin and D. B. Johnson, An O(n log2 n) algorithm for maximum flow in undirected planar networks,SIAM Journal on Computing, Vol. 14, pp. 612–624 (1985).
A. Itai and Y. Shiloach, Maximum flow in planar networks,SIAM Journal on Computing, Vol. 8, pp. 135–150 (1979).
D. B. Johnson, Parallel algorithms for minimum cuts and maximum flows in planar networks,Journal of the ACM, Vol. 34, No. 4, pp. 950–967 (1987).
D. B. Johnson and S. Venkatesan, Using divide and conquer to find flows in directed planar networks in O(n1.5 logn) time,Proceedings of the 20th Annual Allerton Conference on Communication, Control and Computing, University of Illinois, Urbana-Champaign, IL, pp. 898–905 (1982).
S. Khuller and B. Schieber, Efficient parallel algorithms for testingk-connectivity and finding disjoints-t paths in graphs,SIAM Journal on Computing, Vol. 20, No. 2, pp. 352–375 (1991).
R. J. Lipton, D. J. Rose, and R. E. Tarjan, Generalized nested dissection,SIAM Journal on Numerical Analysis, Vol. 16, pp. 346–358 (1979).
G. L. Miller, Finding small simple separators for 2-connected planar graphs,Journal of Computer and System Sciences, Vol. 32, pp. 265–279 (1986).
G. L. Miller and J. Naor, Flow in planar graphs with multiple sources and sinks,Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pp. 112–117 (1989).
T. Nishizeki and N. Chiba,Planar Graphs: Theory and Algorithms, Annals of Discrete Mathematics, Vol. 32, North-Holland, Amsterdam (1988).
V. Pan and J. H. Reif, Fast and efficient solution of path algebra problems,Journal of Computer and System Sciences, Vol. 38, No. 3, pp. 494–510 (1989).
J. H. Reif, Minimums-t cut of a planar undirected network in O(n log2 n) time,SIAM Journal on Computing, Vol. 12, No. 1, pp. 71–81 (1983).
V. P. Roychowdhury and J. Bruck, On finding non-intersecting paths in a grid and its application in reconfiguration of VLSI/WSI arrays,Proceedings of the First Symposium on Discrete Algorithms, pp. 454–464 (1990).
V. P. Roychowdhury, J. Bruck, and T. Kailath, Efficient algorithms for reconfiguration in VLSI/WSI arrays,IEEE Transactions on Computers (Special Issue on Fault Tolerant Computing), Vol. 39, No. 4, pp. 480–489 (1990).
Author information
Authors and Affiliations
Additional information
Communicated by Harold N. Gabow.
A preliminary version of this paper appeared in theProceedings of the First Integer Programming and Combinatorial Optimization Conference, Waterloo, Canada, May 1990, pp. 367–383. Samir Khuller is currently supported by NSF Grant CCR-8906949. Part of this research was done while he was visiting the IBM Thomas J. Watson Research Center and was supported by an IBM Graduate Fellowship at Cornell University. Joseph Naor's work was supported by Contract ONR N00014-88-K-0166 while he was a post-doctoral fellow in the Department of Computer Science at Stanford University.
Rights and permissions
About this article
Cite this article
Khuller, S., Naor, J.(. Flow in planar graphs with vertex capacities. Algorithmica 11, 200–225 (1994). https://doi.org/10.1007/BF01240733
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01240733