有原则的图形管理
Principled Graph Management
摘要：
图生成算法是最近提出的一种改进的列生成算法，用于求解混合整数线性规划的扩展线性规划松弛问题，而不削弱扩展松弛问题。要应用Graph Generation，我们必须能够将定价过程中生成的任何给定列映射到一个小的有向无圈图，对于该图，从源到宿的任何路径都描述了一个可行列。这种结构很容易满足车辆路径，船员调度和各种物流问题的定价是一个受约束的最短路问题。这种图的构造权衡了由图建模的列的子集的大小/多样性与解决由较大图引起的问题所需的额外计算时间。图生成（GG）有两个计算瓶颈。首先是定价。由于所解决问题的结构，GG和列生成（CG）中的定价是相同的。第二个瓶颈是受限主问题（RMP），在生成相同数量的列的情况下，GG中的计算量比CG中的计算量更大。通过设计，GG比CG在更少的迭代中收敛，因此需要更少的定价调用。因此，当GG的计算时间由定价决定时，与求解RMP相反，GG在时间上比CG收敛得快得多。然而，当GG RMP而不是定价主导计算时，GG不需要比CG更快地收敛。本文介绍了一种利用图的特殊结构快速求解GG RMP的算法--原理图管理算法（PGM）。我们证明了PGM内的GG解决方案的经典容量限制的车辆路径问题的有效性。我们证明了PGM解决GG RMP的速度比基线求解器快数百倍，并且速度的提高随着问题的大小而增加。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
