A number theoretic reformulation and decomposition method for integer programming

Integer programming problems, and especially knapsack and finite abelian problems, can be exactly replaced by equivalent problems of ''smaller'' size. This reformulation theoretically provides a new method of solution for such problems, but the main advantages lie in reducing coefficient magnitudes and in removing selected constraints, while a disadvantage is the large increase in the number of variables. By modifying the dynamic programming approach so as effectively to avoid generating a large number of these new variables, an algorithm to overcome this difficulty is developed. Applied to the solution of large non-prime group problems this provides an algorithm that appears on average to compare favourably with the deterministic algorithms of Hu and Gomory.