{0, 1}-Solutions of Integer Linear Equation Systems

A parallel version of an algorithm for solving systems of integer linear equations with f0; 1g-variables is presented. The algorithm is based on lattice basis reduction in combination with explicit enumer-ation. 1 The algorithm A parallel version of an algorithm proposed by Kaib and Ritter 4] has been implemented with PVM to nd all f0; 1g-solutions of integer linear equation systems. For example such systems are of interest in the construction of block designs, see 1, 2, 3, 8]: It is possible to nd block designs if one nds f0; 1g-vectors x and > 0 with (1) where A is a matrix consisting of nonnegative integers. Our problem is also related to cryptography 6] and theory of numbers 7]. The algorithm { using lattice basis reduction 5] { constructs a basis for the equation kernel that consists of short integer vectors. Then the integer linear combinations of these basis vectors are enumerated and tested if they yield f0; 1g-solutions of (1). For an explicit description of the algorithm see 8]. 2 Parallelization The backtracking algorithm is now implemented using PVM. The algorithm runs along a search tree with an unpredictable number of childs at each vertex. Actually, we search all solutions of a discrete optimization problem, the optimal value of the objective function being well known in advance.