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**1 - 2**of**2**### Prtnted m U.S.A. INTEGER PROGRAMMING WITH A FIXED NUMBER OF VARIABLES*

"... It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers. The integer linear programming problem is formulated äs follows. Let n and m be positive integers, A an m X «-matrix with integral ..."

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It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers. The integer linear programming problem is formulated äs follows. Let n and m be positive integers, A an m X «-matrix with integral coefficients, and b e T&quot;. The question is to decide whether there exists a vector χ e / &quot; satisfying the System of m inequalities Ax < b. No algorithm for the solution of this problem is known which has a running time that is bounded by a polynomial function of the length of the data. This length may, for our purposes, be defined to be n · m · log(iz + 2), where a denotes the maximum of the absolute values of the coefficients of A and b. Indeed, no such polynomial algorithm is likely to exist, since the problem in question is NP-complete [3], [12]. In this paper we consider the integer linear programming problem with a fixed value of n. In the case n = l it is trivial to design a polynomial algorithm for the solution of the problem. For n = 2, Hirschberg and Wong [5] and Kannan [6] have given polynomial algorithms in special cases. A complete treatment of the case n = 2 was