A subadditive approach to the group problem of integer programming

Abstract

Solving the linear program associated with an all-integer program gives the group problem N x≡b(mod 1), x≥0 and integer, z=c x (minimize), upon relaxation of non-negativity of the basic variables, where c≥0 and N is the fractional part of the updated, non-basic columns. A method is given for solving this problem which does not require an explicit group representation and is not dependent on knowing the order of the group. From a diamond gauge function the algorithm constructs a continuous function π, which is shown to be subadditive on the unit hypercube. Such continuous functions yield valid inequalities and are used in solving the group problem.