Adaptive Partitioning in Column Generation for Nested Paths

Abstract

We study a class of nested path problems, in which every path-based variable can be decomposed into a sequence of subpaths. Subpaths must satisfy local resources, while paths must satisfy additional global resources. This paper develops a new exact pricing algorithm in column generation for these problems that avoids the enumeration of non-dominated subpaths. The algorithm relies on adaptive partitioning of subpaths into buckets characterizing the consumption of global path resources. The algorithm represents each bucket by its subpath of minimum reduced cost, and iterates between pessimistic and optimistic pricing steps to combine subpaths into paths while maintaining upper and lower bounds on the minimum reduced cost. An adaptive refinement procedure closes the gap in a finite number of iterations. We demonstrate the effectiveness of the algorithm on two applications. For the balanced multi-period capacitated vehicle routing problem, we obtain speed-ups of up to a factor of 13 over a state-of-the-art column generation benchmark, and the resulting branch-price-and-cut algorithm solves three times as many instances to optimality as a subpath-based baseline. For the robust railway crew scheduling problem, we obtain speed-ups of up to a factor of three and produce primal solutions within 1% of optimality.