Improved integral simplex using decomposition for the set partitioning problem
Integral simplex using decomposition (ISUD) is a method that efficiently solves set partitioning problems. It is an iterative method that starts from a known integer solution and moves through a sequence of integer solutions, decreasing the cost at each iteration. At each iteration, the method decomposes the original problem into a reduced problem (RP) and a complementary problem (CP). Given an integer solution to RP (that is also solution to the original problem), CP finds a descent direction having the minimum ratio between its cost and the number of its positive variables. We loop on until an optimal or near-optimal solution to the original problem is reached. In this paper, we introduce a modified model for CP. The new model finds a descent direction that minimizes the ratio between the cost of the direction and an overestimation of the number of variables taking one in the next solution. The new CP presents higher chances of finding improved integer solutions without branching. We present results for the same large instances (with up to 570,000 columns) as the ones previously used to test ISUD. For all the instances, optimality is always reached with a speedup factor of at least five.
KeywordsPrimal algorithms Set partitioning problem Integral simplex
Mathematics Subject Classification90C99 90C27 90C10
- Rosat S, El Hallaoui I, Soumis F, Lodi A (2014) Integral simplex using decomposition with primal cuts. In: Experimental algorithms. Springer International Publishing, pp 22–33Google Scholar
- Rosat S, El Halaoui I, Soumis F, Chakour D (2017) Influence of the normalization constraint on the integral simplex using decomposition. Discrete Appl Math 217(1):53–70Google Scholar
- Saxena A (2003) Set-partitioning via integral simplex method. Doctoral dissertation, Ph. D. thesis, Carnegie Mellon University, PittsburghGoogle Scholar
- Zaghrouti A, El Hallaoui I, Soumis F (2013) Improving ILP solutions by zooming around an improving direction. Les Cahiers du GERAD G-2013-107Google Scholar