A Polynomial Algorithm for the Two-Variable Integer Programming Problem

ABSTRACT A polynomial time algorithm is presented for solving the following two-variable integer programming problem maximize ClXl + c2x2 subject to a, lxl + a,2x2 = O, integers, where a,j, cj, and b, are assumed to be nonnegattve integers This generahzes a result of Htrschberg and Wong, who developed a polynomial algorithm for the same problem with only one constraint (l e, where n = 1) However, the techniques used here are quite different KEY WORDS AND PHRASES integer programming, knapsack problem, polynomial algorithm, coefficient size, feasible region decomposition CR CATEGORIES 3 15, 5 25, 5 30, 5 40 Introduction We consider the following integer programming problem: (1) maximize clxa + czx2 subject to aaxl + a,2xz ~ b,, a= 1,2 .... n, and xl, x2 => 0, integers, where au, G, and b, are assumed to be nonnegative integers. We first show that the solution to (1) can be obtained easily from the solutions to at most n problems, each of which is of the form (2) maximize