A primal (all-integer) integer programming algorithm

Abstract

The algorithm is most closely related to three existing procedures: the simplex method of G. B. Dantzig for linear programming problems, the Gumory all-integer integer programming algurithm, and the direct algorithm for integer prugramming uf Ben-Israel and Charnes. The algorithm is similar to the Gomory all·integer algorithm in these respects: (i) it is an all-integcr algorithm; (ii) it uses t he same c ut generation proccdure; (iii) it uses the cut row as the pivot row; and (iv) the pivot coefficient always has unit value. While the dual method provides the vehicle for moving from tableau to tableau in the Gomory all-integer algorithm. the simplex method has the analag()us role in the primal algorithm. Thus in a general sense this algorithm is a primal analog t() the (dual) Comory all-integer algo rithm. The direct algorithm of Be n-Israel and Charnes also has the above similarities to the Gomory all-integer al~orithm, but has one significant difference: an iteration or cycle of the direct algorithm must frequently include the solution of an "auxiliary problem" (which is itself an int eger prugramming problem) or a determination that no solution to the "auxiliary problem" exists. In contrasl. the cycles of the primal algorithm include only the adjoining of a ComUl'y cut and the executiun of the change of basis procedure of the simplex method. The procedure uf the algorithm and the proof of finit eness are founded on a classification of cycles of the algorithm and on two theore ms. Two types of prorf'dural restrictions are imposed as a basis for proving fin iteness: (a) selection of the incoming variable is subjected to regulation (beyond that required by t he simplex met hod). and t he rules applied are a function uf the type of cycle being executed; (b) selection of the row Llsed as the source of the data for the