Abstract
We present two algorithms for network flow on networks with infinite capacities and finite integer supplies and demands. The first algorithm runs inO(m√K) time on networks withm edges, whereK=O(m2/log4 m) is the value of the optimal flow, and can also be applied to the capacitated case by lettingK be the sum of thefinite capacities alone. The second algorithm runs inO(wm logK) time for arbitraryK, where w is a new parameter, thewidth of the network. These algorithms as well as other uses of the notion of width lead to results for several questions on the 2-satisfiability problem: minimizing the weight of a solution, finding the transitive closure, recognizing partial solutions, enumerating all solutions. The results have applications to stable matching, wherew corresponds to the number of people andm to the instance size (usuallym ≈ w2).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. K. Ahuja, J. B. Orlin, and R. E. Tarjan, Improved Time Bounds for the Maximum Flow Problem, Technical Report CS-TR-118-87, Department of Computer Science, Princeton University, 1987. (SIAM J. Comput., to appear.)
R. Bar-Yehuda and S. Even, A linear time approximation algorithm for the weighted vertex cover problem,J. Algorithms,2 (1981), 198–203.
K. Clarkson, A modification of the greedy algorithm for vertex cover,Inform. Process. Lett.,16 (1983), 23–25.
R. P. Dilworth, A decomposition theorem for partially ordered sets,Ann. of Math.,51 (1950), 161–166.
E. A. Dinic, Algorithm for solution of a problem of maximum flow in a network with power estimation,Soviet Math. Dokl.,11 (1970), 1277–1280.
S. Even, A. Itai, and A. Shamir, On the complexity of timetable and multicommodity flow problems,SIAM J. Comput.,5 (1976), 691–703.
T. Feder, A new fixed point approach for stable networks and stable marriages,Proc. 21st ACM Symp. on Theory of Computing (1989), pp. 513–522. (Submitted toJ. Comput. System Sci.)
T. Feder, Stable Networks and Product Graphs, Ph.D. dissertation, Technical Report STAN-CS-91-1362, Stanford University (1991).
D. Gale and L. S. Shapley, College admissions and the stability of marriage,Amer. Math. Monthly,69 (1962), 9–15.
A. V. Goldberg and R. E. Tarjan, A new approach to the maximum flow problem,Proc. 18th ACM Symp. on Theory of Computing (1986), pp. 136–146.
A. V. Goldberg and R. E. Tarjan, Finding minimum-cost circulations by successive approximation,Math. Oper. Res.,15(3) (1990), 430–466.
D. Gusfield, Three fast algorithms for four problems in stable marriage,SIAM J. Comput.,16(1) (1987), 111–128.
D. Gusfield, The structure of the stable roommate problem: efficient representation and enumeration of all stable assignments,SIAM J. Comput.,17(4) (1988), 742–769.
D. Gusfield and R. W. Irving, The parametric stable marriage problem,Inform. Process. Lett.,30 (1989), 255–259.
D. Gusfield and R. W. Irving,The Stable Marriage Problem: Structure and Algorithms, MIT Press Series in the Foundations of Computing, MIT Press, Cambridge, MA (1989).
D. Gusfield, R. Irving, P. Leather, and M. Saks, Every finite distributive lattice is a set of stable matchings for a small stable marriage,J. Combin. Theory Ser. A,44 (1987), 304–309.
D. Gusfield and L. Pitt, Equivalent approximation algorithms for node cover,Inform. Process. Lett.,22(6) (1986), 291–294.
D. Gusfield and L. Pitt, A Bounded Approximation for the Minimum Cost 2-SAT Problem, Technical Report CSE-89-4, University of California, Davis (1989).
F. Harary,Graph Theory, Addison-Wesley, Reading, MA.
D. S. Hochbaum, Approximation algorithms for the set covering and vertex cover problems,SIAM J. Comput.,11(3) (1982), 555–556.
J. E. Hopcroft and R. M. Karp, An n5/2 algorithm for maximum matching in bipartite graphs,SIAM J. Comput.,2 (1973), 225–231.
R. W. Irving and P. Leather, The complexity of counting stable marriages,SIAM J. Comput.,15(3) (1986), 655–667.
R. W. Irving, P. Leather, and D. Gusfield, An efficient algorithm for the optimal stable marriage,J. Assoc. Comput. Mach.,34(3) (1987), 532–543.
D. E. Knuth,Mariages stables et leur relations avec d'autres problèmes combinatories, Les Presses de l'Université de Montréal, Montréal, Québec (1976).
G. L. Nemhauser and R. E. Trotter, Vertex packing structural properties and algorithms,Math. Programming,8 (1975), 232–248.
C. Ng, An O(n3√logn) Algorithm for the Optimal Stable Marriage Problem, Technical Report 90-22, University of California, Irvine (1990).
J. B. Orlin, A faster strongly polynomial minimum cost flow algorithm,Proc. 20th A CM Symp. on Theory of Computing (1988), pp. 377–387.
B. Pittel, On likely solutions of a stable marriage problem, Manuscript.
G. Pólya, R. E. Tarjan, and D. R. Woods,Notes on Introductory Combinatorics, Birkhäuser, Basel (1983).
J. S. Provan and M. O. Ball, The complexity of counting cuts and of computing the probability that a graph is connected,SIAM J. Comput.,12(4) (1983), 777–788.
D. D. Sleator and R. E. Tarjan, A data structure for dynamic trees,J. Comput. System Sci.,26 (1983), 652–686.
A. Subramanian, A New Approach to Stable Matching Problems, Technical Report STAN-CS-89-1275, Stanford University (1989).
R. E. Tarjan, Depth-first search and linear graph algorithms,SIAM J. Comput.,1 (1972), 146–160.
R. E. Tarjan,Data Structures and Network Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA (1983).
Author information
Authors and Affiliations
Additional information
Communicated by Harold N. Gabow.
Rights and permissions
About this article
Cite this article
Feder, T. Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994). https://doi.org/10.1007/BF01240738
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01240738