Principled Graph Management
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Graph Generation is a recently introduced enhanced Column Generation algorithm for solving expanded Linear Programming relaxations of mixed integer linear programs without weakening the expanded relaxations which characterize these methods. To apply Graph Generation we must be able to map any given column generated during pricing to a small directed acyclic graph for which any path from source to sink describes a feasible column. This structure is easily satisfied for vehicle routing, crew scheduling and various logistics problems where pricing is a constrained shortest path problem. The construction of such graphs trades off the size/diversity of a subset of columns modeled by the graphs versus the additional computational time required to solve the problems induced by larger graphs. Graph Generation (GG) has two computational bottlenecks. The first is pricing. Pricing in GG and Column Generation (CG) is identical because of the structure of the problems solved. The second bottleneck is the restricted master problem (RMP), which is more computationally intensive in GG than in CG given the same number of columns generated. By design GG converges in fewer iterations than CG, and hence requires fewer calls to pricing. Therefore, when the computation time of GG is dominated by pricing, as opposed to solving the RMP, GG converges much faster than CG in terms of time. However GG need not converge faster than CG when the GG RMP, rather than pricing, dominates computation. In this paper we introduce Principled Graph Management (PGM), which is an algorithm to solve the GG RMP rapidly by exploiting its special structure. We demonstrate the effectiveness of PGM inside a GG solution to the classical Capacitated Vehicle Routing Problem. We demonstrate that PGM solves the GG RMP hundreds of times faster than the baseline solver and that the improvement in speed increases with problem size.
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