Algorithm for maximizing a linear function on the set of integral points of a convex polyhedron

Abstract

In this paper we present a new algorithm for solving the following problem of integral linear programming: find the maximum of the linear function fo (x) = clx: qc2x~ -5 "-5 c,~x,~ ( i ) on the finite set of integral points of the convex polyhedron V, defined by the finite system of linear inequalities f~ (x) = ai:xl + ai~x2 + . . . + a~xn <~ hi, (2) ~= 1,m. It is assumed here that aij, b i, cj (i = i, m, j = I, n) belong to the ring of integers Z. In Sec. i the algorithm of [i] for finding the set of integral points of the polyhedron V is generalized; in Sec. 2 an algorithm for solving the problem of integral linear programming for the case when the set of its admissible solutions is bounded is described.