Abstract
There are three types of incremental methods, primal, dual, and primal-dual. In Chap. 6, we touched all of them for linear programming (LP). This chapter is contributed specially to primal-dual methods for further exploring techniques about primal-dual with a special interest in the minimum cost flow. Actually, the minimum cost flow is a fundamental optimization problem on networks. The shortest path problem and the assignment problem can be formulated as its special cases. We begin with the study of the assignment problem.
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References
D.P. Bertsekas: A simple and fast label correcting algorithm for shortest paths, Networks, 23(8): 703–709 (1993).
R.G. Busacker, P.G. Gowen: A procedure for determining a family of minimum cost network flow patterns, Operations Research Office Technical Report 15, John Hopkins University, Baltimore; 1961.
Mmanu Chaturvedi, Ross M. McConnell: A note on finding minimum mean cycle. Inf. Process. Lett., 127: 21–22 (2017).
J. Edmonds, R. Karp: Theoretical improvements in algorithmic efficiency for network flow problems, Journal of the ACM, 19(2): 248–264 (1972).
M.A. Engquist: A successive shortest path algorithm for the assignment problem, Research Report, Center for Cybernetic Studies (CCS) 375, University of Texas, Austin; 1980.
Thomas R. Ervolina, S. Thomas McCormick: Two strongly polynomial cut cancelling algorithms for minimum cost network flow, Discrete Applied Mathematics, 4: 133–165 (1993).
D.R. Fulkerson: An out-of-kilter method for minimal cost flow problems, Journal of the Society for Industrial and Applied Mathematics, 9(1):18–27 (1961).
Andrew V. Goldberg, Robert E. Tarjan: Finding minimum-cost circulations by canceling negative cycles, Journal of the ACM, 36 (4): 873–886 (1989).
Andrew V. Goldberg, Robert E. Tarjan: Finding minimum-cost circulations by successive approximation. Math. Oper. Res., 15(3): 430–466 (1990).
F. Guerriero, R. Musmanno: Label correcting methods to solve multicriteria shortest path problems, Journal of Optimization Theory and Applications, 111(3): 589–613 (2001).
Refael Hassin: The minimum cost flow problem: A unifying approach to existing algorithms and a new tree search algorithm, Mathematical Programming, 25: 228–239 (1983).
R. Jonker, A. Volgenant: A shortest augmenting path algorithm for dense and sparse linear assignment problems, Computing, 38(4): 325–340 (1987).
A. Karczmarz, J. Lacki: Simple label-correcting algorithms for partially dynamic approximate shortest paths in directed graphs, Proceedings, Symposium on Simplicity in Algorithms, Society for Industrial and Applied Mathematics, pp. 106–120, 2020.
R. M. Karp: A characterization of the minimum cycle mean in a digraph, Discrete Mathematics, 23(3): 309–311 (1978).
Morton Klein: A primal method for minimal cost flows with applications to the assignment and transportation problems, Management Science, 14 (3): 205–220 (1967).
H.W. Kuhn: The Hungarian method for the assignment problem, Naval Research Logistics Quarterly, 2: 83–97 (1955).
H.W. Kuhn: Variants of the Hungarian method for assignment problems, Naval Research Logistics Quarterly, 3: 253–258 (1956).
J. Munkres: Algorithms for the assignment and transportation problems, Journal of the Society for Industrial and Applied Mathematics, 5(1): 32–38 (1957).
James B. Orlin: A polynomial time primal network simplex algorithm for minimum cost flows, Mathematical Programming, 78(2): 109–129 (1997).
A.J. Skriver, K.A. Andersen: A label correcting approach for solving bicriterion shortest-path problems, Computers & Operations Research, 27(6): 507–524 (2000).
E. Tardos: A strongly polynomial minimum cost circulation algorithm, Combinatorica, 5(3): 247–255 (1985).
N. Tomizawa: On some techniques useful for solution of transportation network problems, Networks, 1(2): 173–194 (1971).
F.B. Zhan, C.E. Noon: A comparison between label-setting and label-correcting algorithms for computing one-to-one shortest paths, Journal of Geographic information and decision analysis, 4(2): 1–11 (2000).
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Du, DZ., Pardalos, P., Hu, X., Wu, W. (2022). Primal-Dual Methods and Minimum Cost Flow. In: Introduction to Combinatorial Optimization. Springer Optimization and Its Applications, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-031-10596-8_7
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