Optimal Flows in Dynamic Networks and Algorithms for their Finding

The minimum cost flow problem and the maximum flow problem on networks with time-dependent characteristics and nonlinear cost functions on arcs are considered. The algorithms for determining optimal solutions of the single-commodity and multicommodity network flow problems based on the time-expanded network method are elaborated. Some applications of the optimal flow problems are provided.

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

[2]  Dmitrii Lozovanu,et al.  Algorithms and the calculation of Nash equilibria for multi-objective control of time-discrete systems and polynomial-time algorithms for dynamic c , 2007, Eur. J. Oper. Res..

[3]  D. R. Fulkerson,et al.  Constructing Maximal Dynamic Flows from Static Flows , 1958 .

[4]  R. Tarjan.  A simple version of Karzanov's blocking flow algorithm , 1984 .

[5]  Malachy Carey,et al.  An approach to modelling time-varying flows on congested networks , 2000 .

[6]  Lisa Fleischer,et al.  Approximating Fractional Multicommodity Flow Independent of the Number of Commodities , 2000, SIAM J. Discret. Math..

[7]  S. Vajda,et al.  Integer Programming and Network Flows , 1970 .

[8]  D. Lozovanu,et al.  Network models of discrete optimal control and dynamic games with p players , 2001 .

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

[10]  Warren B. Powell,et al.  Stochastic and dynamic networks and routing , 1995 .

[11]  D. Lozovanu,et al.  Optimal multicommodity flows in dynamic networks and algorithms for their finding , 2005 .

[12]  Harold N. Gabow,et al.  Scaling algorithms for network problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  Martin Skutella,et al.  The Quickest Multicommodity Flow Problem , 2002, IPCO.

[14]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[15]  Chak-Kuen Wong,et al.  Time-varying minimum cost flow problems , 2001, Eur. J. Oper. Res..

[16]  Deguang Cui,et al.  Dynamic network flow model for short-term air traffic flow management , 2004, IEEE Trans. Syst. Man Cybern. Part A.

[17]  Gerhard J. Woeginger,et al.  Minimum Cost Dynamic Flows: The Series-Parallel Case , 1995, IPCO.

[18]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[19]  Eitan Altman,et al.  A survey on networking games in telecommunications , 2006, Comput. Oper. Res..

[20]  Alan J. Hoffman,et al.  SOME RECENT APPLICATIONS OF THE THEORY OF LINEAR INEQUALITIES TO EXTREMAL COMBINATORIAL ANALYSIS , 2003 .

[21]  Gerhard J. Woeginger,et al.  One, Two, Three, Many, or: Complexity Aspects of Dynamic Network Flows with Dedicated Arcs , 1996, WG.

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

[23]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[24]  Arjang A. Assad,et al.  Multicommodity network flows - A survey , 1978, Networks.

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

[26]  Andrew V. Goldberg,et al.  An efficient implementation of a scaling minimum-cost flow algorithm , 1993, IPCO.

[27]  Ariel Orda,et al.  Architecting noncooperative networks , 1995, Eighteenth Convention of Electrical and Electronics Engineers in Israel.

[28]  Dmitrii Lozovanu,et al.  The minimum cost multicommodity flow problem in dynamic networks and an algorithm for its solving , 2005, Comput. Sci. J. Moldova.

[29]  Lisa Fleischer,et al.  Faster Algorithms for the Quickest Transshipment Problem , 2001, SIAM J. Optim..

[30]  Dmitrii Lozovanu,et al.  Minimum Cost Multicommodity Flows in Dynamic Networks and Algorithms for their Finding , 2007 .

[31]  Andrew V. Goldberg,et al.  Finding minimum-cost circulations by canceling negative cycles , 1989, JACM.

[32]  Dmitrii Lozovanu,et al.  Optimal Dynamic Multicommodity Flows in Networks , 2006, Electron. Notes Discret. Math..

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

[34]  M. Fonoberova.  On determining the minimum cost flows in dynamic networks , 2006 .

[35]  J. Castro,et al.  An implementation of linear and nonlinear multicommodity network flows , 1996 .

[36]  Richard D. McBride,et al.  Progress Made in Solving the Multicommodity Flow Problem , 1998, SIAM J. Optim..

[37]  Ariel Orda,et al.  Virtual path bandwidth allocation in multiuser networks , 1997, TNET.

[38]  S. Kumar,et al.  An Incremental Algorithm for the Maximum Flow Problem , 2003, J. Math. Model. Algorithms.

[39]  Dmitrii Lozovanu,et al.  The maximum flow in dynamic networks , 2004, Comput. Sci. J. Moldova.

[40]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .

[41]  Gerhard J. Woeginger,et al.  Minimum-cost dynamic flows: The series-parallel case , 2004, Networks.

[42]  Ching-Yao Huang,et al.  Non-cooperative uplink power control in cellular radio systems , 1998, Wirel. Networks.

[43]  Jay E. Aronson,et al.  A survey of dynamic network flows , 1989 .

[44]  George L. Nemhauser,et al.  A Dynamic Network Flow Problem with Uncertain Arc Capacities: Formulation and Problem Structure , 2000, Oper. Res..

[45]  Lisa Fleischer,et al.  Universally maximum flow with piecewise‐constant capacities , 2001, Networks.

[46]  Irena Lasiecka,et al.  Analysis and Optimization of Differential Systems , 2002, IFIP — The International Federation for Information Processing.

[47]  David L. Jensen,et al.  On the computational behavior of a polynomial-time network flow algorithm , 1992, Math. Program..

[48]  Dmitrii Lozovanu,et al.  Optimal flow in dynamic networks with nonlinear cost functions on edges , 2002, Analysis and Optimization of Differential Systems.

[49]  E. Kay,et al.  Graph Theory. An Algorithmic Approach , 1975 .

[50]  Jordi Castro,et al.  Solving Difficult Multicommodity Problems with a Specialized Interior-Point Algorithm , 2003, Ann. Oper. Res..

[51]  S. Pallottino,et al.  The maximum flow problem: A max-preflow approach , 1991 .

[52]  O. SIAMJ.,et al.  A SPECIALIZED INTERIOR-POINT ALGORITHM FOR MULTICOMMODITY NETWORK FLOWS∗ , 1996 .

[53]  Éva Tardos,et al.  “The quickest transshipment problem” , 1995, SODA '95.

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

[55]  Gerhard J. Woeginger,et al.  One, two, three, many, or: complexity aspects of dynamic network flows with dedicated arcs , 1998, Oper. Res. Lett..

[56]  Dmitrii Lozovanu,et al.  Game-Theoretic Approach for Solving Multiobjective Flow Problems on Networks , 2005, Comput. Sci. J. Moldova.

[57]  Frank Harary,et al.  Graph Theory , 2016 .