Integral Column Generation for Set Partitioning Problems with Side Constraints

Abstract

The integral column generation algorithm (ICG) was recently introduced to solve set partitioning problems involving a very large number of variables. This primal algorithm generates a sequence of integer solutions with decreasing costs, leading to an optimal or near-optimal solution. ICG combines the well-known column generation algorithm and a primal algorithm called the integral simplex using decomposition algorithm (ISUD). In this paper, we develop a generalized version of ICG, denoted I2CG, that can solve efficiently large-scale set partitioning problems with side constraints. This new algorithm can handle the side constraints in the reduced problem of ISUD, in its complementary problem, or in both components. Computational experiments on instances of the airline crew pairing problem (CPP) and the multidepot vehicle routing problem with time windows show that the latter strategy is the most efficient one and I2CG significantly outperforms basic variants of two popular column generation heuristics, namely, a restricted master heuristic and a diving heuristic. For the largest tested CPP instance with 1,761 constraints, I2CG can produce in less than one hour of computational time more than 500 integer solutions leading to an optimal or near-optimal solution. Summary of Contribution: In this paper, we develop a new integral column generation algorithm that can solve efficiently large-scale set partitioning problems with side constraints. The latter alter the quasi-integrality property needed for primal integral algorithms. The paper adds a methodological contribution remedying this issue. This remedy should, in our opinion, boost the use of primal exact methods, especially in the column generation context. The paper also has a computational contribution. Effectively, computational experiments on instances of the airline crew pairing problem and the multidepot vehicle routing problem with time windows are extensively discussed. We compare the proposed algorithm to basic variants of two popular column generation heuristics.