A survey on exact algorithms for the maximum flow and minimum‐cost flow problems

Network flow problems form an important and much‐studied family of combinatorial optimization problems, with a huge array of practical applications. Two network flow problems in particular have received a great deal of attention: the maximum flow and minimum‐cost flow problems. We review the progress that has been made on exact solution algorithms for these two problems, with an emphasis on worst‐case running times.

[1]  Christian Wulff-Nilsen,et al.  Negative-Weight Single-Source Shortest Paths in Near-linear Time , 2022, 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS).

[2]  Richard Peng,et al.  Maximum Flow and Minimum-Cost Flow in Almost-Linear Time , 2022, 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS).

[3]  Richard Peng,et al.  Faster maxflow via improved dynamic spectral vertex sparsifiers , 2021, STOC.

[4]  Adrian Vladu,et al.  Faster Sparse Minimum Cost Flow by Electrical Flow Localization , 2021, 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS).

[5]  Richard Peng,et al.  Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao , 2021, 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS).

[6]  Yin Tat Lee,et al.  Minimum cost flows, MDPs, and ℓ1-regression in nearly linear time for dense instances , 2021, STOC.

[7]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[8]  James B. Orlin,et al.  A fast maximum flow algorithm , 2020, Networks.

[9]  Aaron Sidford,et al.  Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers , 2020, SODA.

[10]  Aaron Sidford,et al.  Unit Capacity Maxflow in Almost $O(m^{4/3})$ Time , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[11]  Aaron Sidford,et al.  Faster energy maximization for faster maximum flow , 2019, STOC.

[12]  Richard Peng,et al.  A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond , 2019, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[13]  David P. Williamson,et al.  Network Flow Algorithms , 2019 .

[14]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[15]  Aleksander Madry,et al.  Computing Maximum Flow with Augmenting Electrical Flows , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[16]  Andrew V. Goldberg,et al.  Finding minimum-cost flows by double scaling , 2015, Math. Program..

[17]  Yin Tat Lee,et al.  Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(vrank) Iterations and Faster Algorithms for Maximum Flow , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[18]  Andrew V. Goldberg,et al.  Efficient maximum flow algorithms , 2014, CACM.

[19]  James B. Orlin,et al.  Max flows in O(nm) time, or better , 2013, STOC '13.

[20]  Ravindra K. Ahuja,et al.  A Fast and Simple Algorithm for the Maximum Flow Problem , 2011, Oper. Res..

[21]  Dorit S. Hochbaum,et al.  A Computational Study of the Pseudoflow and Push-Relabel Algorithms for the Maximum Flow Problem , 2009, Oper. Res..

[22]  Daniel A. Spielman,et al.  Faster approximate lossy generalized flow via interior point algorithms , 2008, STOC.

[23]  Shang-Hua Teng,et al.  Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems , 2003, STOC '04.

[24]  Mikkel Thorup,et al.  Integer priority queues with decrease key in constant time and the single source shortest paths problem , 2003, STOC '03.

[25]  Andrew V. Goldberg,et al.  On Implementing the Push—Relabel Method for the Maximum Flow Problem , 1997, Algorithmica.

[26]  Andrew V. Goldberg,et al.  Beyond the flow decomposition barrier , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[27]  Ravindra K. Ahuja,et al.  Computational investigations of maximum flow algorithms , 1997 .

[28]  Joseph Cheriyan,et al.  An Empirical Study of Min Cost Flow Algorithms for the Minimum-Cost Flow Problem , 1993, Network Flows And Matching.

[29]  David S. Johnson,et al.  Network Flows and Matching: First DIMACS Implementation Challenge , 1993 .

[30]  Steven J. Phillips,et al.  On-Line Load Balancing and Network Flow , 1993, Algorithmica.

[31]  James B. Orlin,et al.  A faster strongly polynomial minimum cost flow algorithm , 1993, STOC '88.

[32]  Robert E. Tarjan,et al.  A faster deterministic maximum flow algorithm , 1992, SODA '92.

[33]  Andrew V. Goldberg,et al.  Finding Minimum-Cost Circulations by Successive Approximation , 1990, Math. Oper. Res..

[34]  Noga Alon,et al.  Generating Pseudo-Random Permutations and Maximum Flow Algorithms , 1990, Inf. Process. Lett..

[35]  Kurt Mehlhorn,et al.  Can A Maximum Flow be Computed on o(nm) Time? , 1990, ICALP.

[36]  Torben Hagerup,et al.  A randomized maximum-flow algorithm , 1989, 30th Annual Symposium on Foundations of Computer Science.

[37]  Robert E. Tarjan,et al.  Faster Scaling Algorithms for Network Problems , 1989, SIAM J. Comput..

[38]  A. Goldberg,et al.  A new approach to the maximum-flow problem , 1988, JACM.

[39]  Andrew V. Goldberg,et al.  Solving minimum-cost flow problems by successive approximation , 1987, STOC.

[40]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[41]  Éva Tardos,et al.  An O(n2(m + n log n) log n) min-cost flow algorithm , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[42]  Éva Tardos,et al.  A strongly polynomial minimum cost circulation algorithm , 1985, Comb..

[43]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[44]  Robert E. Tarjan,et al.  A data structure for dynamic trees , 1981, STOC '81.

[45]  Amnon Naamad,et al.  An O(EVlog²V) Algorithm for the Maximal Flow Problem , 1980, J. Comput. Syst. Sci..

[46]  Zvi Galil,et al.  An O(V5/3E2/3) algorithm for the maximal flow problem , 1980, Acta Informatica.

[47]  Zvi Galil,et al.  A new algorithm for the maximal flow problem , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[48]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[49]  M. Klein.  A Primal Method for Minimal Cost Flows with Applications to the Assignment and Transportation Problems , 1966 .

[50]  Robert G. Busacker,et al.  A PROCEDURE FOR DETERMINING A FAMILY OF MINIMUM-COST NETWORK FLOW PATTERNS , 1960 .

[51]  D. R. Fulkerson,et al.  An Out-of-Kilter Method for Minimal-Cost Flow Problems , 1960 .

[52]  A. Orden.  The Transhipment Problem , 1956 .

[53]  Inge Li Gørtz,et al.  COMP251: Network flows , 2014 .

[54]  W. Marsden.  I and J , 2012 .

[55]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

[56]  J. Orlin.  Working Paper Alfred P. Sloan School of Management Genuinely Polynominal Simplex and Non-simplex Algorithms for the Minimum Cost Flow Problem Genuinely Polynominal Simplex and Non-simplex Algorithms for the Minimum Cost Flow Problem , 2008 .

[57]  A. Schrijver.  On the History of Combinatorial Optimization (Till 1960) , 2005 .

[58]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[59]  R. Burkard,et al.  Assignment Problems , 1998, IFIP Congress: Fundamentals - Foundations of Computer Science.

[60]  Jesper Larsson Träff,et al.  An Implementation of the Binary Blocking Flow Algorithm , 1998, WAE.

[61]  D. Sleator.  An 0 (nm log n) algorithm for maximum network flow , 1980 .

[62]  A. V. Karzanov,et al.  Determining the maximal flow in a network by the method of preflows , 1974 .

[63]  N. Tomizawa,et al.  On some techniques useful for solution of transportation network problems , 1971, Networks.

[64]  E. A. Dinic.  Algorithm for solution of a problem of maximal flow in a network with power estimation , 1970 .

[65]  G. Minty.  Monotone networks , 1960, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[66]  M. Iri.  A NEW METHOD OF SOLVING TRANSPORTATION· NETWORK PROBLEMS , 1960 .

[67]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956, Canadian Journal of Mathematics.

[68]  Péter Kovács,et al.  Minimum-cost flow algorithms: an experimental evaluation , 2015, Optim. Methods Softw..