(Microsoft Word - INFORMS JOC)

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. The 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 (Branch & Bound, Branch & Cut …) rely on the repeated solution of the continuous version. 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] .