Send-and-Split Method for Minimum-Concave-Cost Network Flows

Abstract

Many problems from inventory, production and capacity planning, and from network design, exhibit scale economies and can be formulated in terms of finding minimum-additive-concave-cost nonnegative network flows. We reduce the problem to an equivalent one in which the arc flow costs are nonnegative and give a dynamic-programming method, called the send-and-split method, to solve it. The main work of the method entails repeatedly solving set-splitting and minimum-cost-chain problems. In uncapacitated networks with n nodes, a arcs, and d + 1 demand nodes, i.e., nodes with nonzero exogenous demand, the algorithm requires up to n2−13d + s2d operations (additions and comparisons) where s = n log2n + 3a is the number of operations required to solve a minimum-cost-chain problem with nonnegative arc costs on the augmented graph formed by appending a node and an arc thereto from each node in the graph. If also the network is k-planar, i.e., the graph is planar with all demand nodes lying on the boundary of k faces, the method requires at most n2−kd3k + sd2k operations. The algorithm can be applied to capacitated networks because they can be reduced to equivalent uncapacitated ones. These results unify, significantly generalize (e.g., to cyclic problems), and sometimes improve upon (e.g., for tandem facilities) known polynomial-time dynamic-programming algorithms for Wagner and Whitin's (Wagner, H. M., Whitin, T. M. 1958. Dynamic version of the economic lot size model. Management Sci. 5 89–96.) dynamic economic-order-quantity problem, Zangwill's (Zangwill, W. I. 1969. A backlogging model and a multi-echelon model of a dynamic economic lot size production system—A network approach. Management Sci. 15 506–527.) generalization to tandem facilities, and Veinott's (Veinott, Jr., A. F. 1969. Minimum concave-cost solution of Leontief substitution models of multi-facility inventory systems. Oper. Res. 17 262–291.) increasing-capacity warehousing problem. The networks for the finite-period versions of these problems are each 1-planar. The method improves upon Zangwill's (Zangwill, W. I. 1968. Minimum concave cost flows in certain networks. Management Sci. 14 429–450.) related O(and) running-time dynamic-programming method for finding minimum-additive-concave-cost non-negative flows in circuitless single-source networks. We also implement the method to solve in polynomial time the (d + 1)-demand-node and k-planar versions of the minimum-cost forest and Steiner problems in graphs. The running time for the (d + 1)-demand-node Steiner problem in graphs is comparable to that of Dreyfus and Wagner's (Dreyfus, S. E., Wagner, R. A. 1971. The Steiner problem in graphs. Networks 1 195–207.) method.

Dynamic lot size problems with one-way product substitution

“…The models that we consider in this paper can also be formulated as special instances of the minimum concave-cost network flow problem (Erickson et al 1987, Guisewite and Pardalos 1993, Lamar 1993, Veinott 1969, and Zangwill 1968. Efficient algorithms have been developed to solve this network flow problem on some special networks, for example, strong-series-parallel networks (Ward 1999) and networks with a fixed number of sources and nonlinear arc costs (Tuy et al 1995).…”

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

“…However, for the leaves of the tree decomposition that are added in step (6.2) of the Thinning procedure, the cost of a subset of portal edges is calculated as, e.g., the cost of the minimum Steiner tree interconnecting these portals in the corresponding brick. Because the bricks are planar, this cost can be calculated by the algorithm of [12] for Steiner tree or [4] for 2-edge connectivity. Because all the portal edges of this brick are present in this bag (recall that we do not delete parallel portal edges after contractions), all possible solutions restricted to the corresponding brick will be considered.…”

A Polynomial-time Approximation Scheme for Planar Multiway Cut

“…This is done via a dynamic programming inside the brick. The details are omitted here since a similar DP has been described in [8,17].…”