“The quickest transshipment problem”

A dynamic network consists of a graph with capacities and transit times on its edges. The quickest transshipment problem is defined by a dynamic network with several sources and sinks; each source has a specified supply and each sink has a specified demand. The problem is to send exactly the right amount of flow out of each source and into each sink in the minimum overall time. Variations of the quickest transshipment problem have been studied extensively; the special case of the problem with a single sink is commonly used to model building evacuation. Similar dynamic network flow problems have numerous other applications; in some of these, the capacities are small integers and it is important to find integral flows. There are no polynomial-time algorithms known for most of these problems. In this paper we give the first polynomial-time algorithm for the quickest transshipment problem. Our algorithm provides an integral optimum flow. Previously, the quickest transshipment problem could only be solved efficiently in the special case of a single source and single sink.

[1]  Gen-Huey Chen,et al.  On the Quickest Path Problem , 1991, ICCI.

[2]  Éva Tardos,et al.  Polynomial time algorithms for some evacuation problems , 1994, SODA '94.

[3]  James B. Orlin,et al.  Optimal Rounding of Fractional Dynamic Flows when Transit Times are Zero , 1997 .

[4]  Horst W. Hamacher,et al.  Modeling of building evacuation problems by network flows with side constraints , 1988 .

[5]  B. Hoppe,et al.  A polynomial-time algorithm for the evacuation problem , 1994 .

[6]  Bruce Hajek,et al.  Optimal dynamic routing in communication networks with continuous traffic , 1982, CDC 1982.

[7]  Bruce Hoppe,et al.  Efficient Dynamic Network Flow Algorithms , 1995 .

[8]  Geoffrey N. Berlin,et al.  The use of directed routes for assessing escape potential , 1978 .

[9]  H. D. Ratliff,et al.  Note—Some Equivalent Objectives for Dynamic Network Flow Problems , 1982 .

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

[11]  Xiaotie Deng,et al.  Deterministic load balancing in computer networks , 1990, Proceedings of the Second IEEE Symposium on Parallel and Distributed Processing 1990.

[12]  J. Ben Rosen,et al.  Algorithms for the quickest path problem and the enumeration of quickest paths , 1991, Comput. Oper. Res..

[13]  Spyros Tragoudas,et al.  On the Computation of Fast Data Transmissions in Networks with Capacities and Delays , 1995, WADS.

[14]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[15]  Nimrod Megiddo,et al.  Combinatorial optimization with rational objective functions , 1978, Math. Oper. Res..

[16]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[17]  Bruce E. Hajek,et al.  Optimal dynamic routing in communication networks with continuous traffic , 1982, 1982 21st IEEE Conference on Decision and Control.

[18]  Richard L. Francis,et al.  Network models for building evacuation , 1982 .

[19]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

[20]  Edward Minieka,et al.  Maximal, Lexicographic, and Dynamic Network Flows , 1973, Oper. Res..

[21]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[22]  Maurice Queyranne,et al.  A combinatorial algorithm for minimizing symmetric submodular functions , 1995, SODA '95.

[23]  Clifford Stein,et al.  Scheduling in a Ring with Unit Capacity Links , 1994 .

[24]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[25]  Horst W. Hamacher,et al.  On the use of lexicographic min cost flows in evacuation modeling. , 1987 .

[26]  Nimrod Megiddo.  Combinatorial Optimization with Rational Objective Functions , 1979, Math. Oper. Res..

[27]  Nimrod Megiddo,et al.  Optimal flows in networks with multiple sources and sinks , 1974, Math. Program..

[28]  Rainer E. Burkard,et al.  The quickest flow problem , 1993, ZOR Methods Model. Oper. Res..

[29]  Cynthia A. Phillips,et al.  Task Scheduling in Networks , 1997, SIAM J. Discret. Math..

[30]  Y. H. Chin,et al.  The quickest path problem , 1990, Comput. Oper. Res..

[31]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

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

[33]  Richard L. Francis,et al.  Network models for building evacuation , 1982 .