A computational comparison of gomory and knapsack cuts

Abstract

A common method of solving integer programs is to solve the problem first as a linear program (LP) then add constraints that cut off noninteger solutions from the set of LP feasible solutions. As soon as an optimal LP solution is all integer, then it is an optimal solution to the integer program. The method of Gomory can generate a variety of different cuts but there is a dearth of reports on systematic testing of the effectiveness of different cuts. We report extensive computational comparisons between a number of different cuts, including a successful one not previously publicised. It has been known for some time that Gomory cuts can be unsuccessful because of slow convergence with the accompanying difficulties of computer round-off error. Recently a method has been proposed for generating, for 0–1 integer problems, cuts that are usually tighter than Gomory cuts and thus give faster convergence. This method of knapsack cuts is tested in comparison with Gomory cuts for moderate size problems and is found to be superior for 0–1 problems having dense constraint matrices but only slightly better than Gomory cuts for problems with sparse matrices. On the other hand, knapsack cuts applied to general integer problems reformulated as 0–1 are found to be less successful than Gomory cuts applied to the original integer problem.