An Improved Implicit Enumeration Approach for Integer Programming

by A. M. Geoffrion

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A synthesis of the Balasian implicit enumeration approach to integer linear programming with the approach typified by Land and Doig and by Roy, Bertier, and Nghiem. This synthesis results from the use of an imbedded linear program to compute surrogate constraints that are as "strong" as possible in a sense slightly different from that originally used by Glover. A very simple implicit enumeration algorithm fitted with optional imbedded linear programming machinery was implemented and tested extensively on an IBM 7044 computer. Use of the imbedded linear program dramatically reduced solution time in virtually every case and sufficed to render the tested algorithm superior to the other five implicit enumeration algorithms for which comparable published experience was available. The crucial issue of the sensitivity of solution time to the number of integer variables was given special attention. Sequences were run of set covering, optimal routing, and knapsack problems of varying sizes up to 90 variables. The results suggest the following working hypothesis: use of the imbedded linear program in the prescribed way reduces solution time dependence on the number of variables from exponential to low-order monomial increase. Existing evidence suggests that the present approach should permit the routine solution of practical integer problems involving hundreds of variables.

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