We present a Cost Decomposition approach for the linear Multicommodity Min-Cost Flow problem, where the mutual capacity constraints are dualized and the resulting Lagrangean Dual is solved with a dual-ascent algorithm belonging to the class of Bundle methods. Although decomposition approaches to block-structured Linear Programs have been reported not to be competitive with general-purpose software, our extensive computational comparison shows that, when carefully implemented, a decomposition algorithm can outperform several other approaches, especially on problems where the number of commodities is "large" with respect to the size of the graph. Our specialized Bundle algorithm is characterized by a new heuristic for the trust region parameter handling, and embeds a specialized Quadratic Program solver that allows the efficient implementation of strategies for reducing the number of active Lagrangean variables. We also exploit the structural properties of the single-commodity Min-Cost Flow subproblems to reduce the overall computational cost. The proposed approach can be easily extended to handle variants of the problem.
Keywords: Multicommodity Flows, Bundle methods, Decomposition Methods, Lagrangean Dual
IntroductionThe Multicommodity Min-Cost Flow problem (MMCF), i.e. the problem of shipping flows of different nature (commodities) at minimal cost on a network, where different commodities compete for the resources represented by the arc capacities, has been widely addressed in the literature since it models a wide variety of transportation and scheduling problems [28,58,68,3,2,6,11] . MMCF is a structured Linear Program (LP), but the instances arising from practical applications are often huge and the usual solution techniques are not efficient enough: this is especially true if the solution is required to be integral, since then the problem is NP-hard and most of the solution techniques From an algorithmic viewpoint, MMCF has motivated many important ideas that have later found broader application: examples are the column generation approach [22] and the Dantzig-Wolfe decomposition algorithm [21] . This "pushing" effect is still continuing, as demonstrated by a number of interesting recent developments [59,31,32,33] .MMCF problems also arise in finding approximate solutions to several hard graph problems [10,43] . Recently, some ε-approximation approaches have been developed for the problem [36,60,64] , making MMCF one of the few LPs for which approximations algorithms of practical interest are known [52,34] .
