The general Steiner problem in Boolean space and application

Ting, S.-T. and S.-Y. Zhao, The general Steiner problem in Boolean space and application, Discrete Mathematics 90 (1991) 75-84. The Steiner problem is to find the minimum point P of sum of distances C;=‘=,p=@(P)whereP,i=l,..., n, are given points in Euclidean plane. It is solved for n = 3, [l]. When n > 3 even approximation is unsatisfactory [3]. In Boolean space the distance of Boolean points A, B is defined as the symmetric difference: D(A, B) = A Cl3 B = AB' + A'B. Let x1, . . . , x, be given Boolean points, ci the Boolean weight of xi, the general Steiner problem is to find the point X such that @(X) = Z:Z1 c,D(x,, X) b e minimum. This paper shows the minimum points fill up the sub-space (a’, /3) where CY = CyZ1 cixl, /? = CYZ1 cixi. This is a Boolean optimization problem. The paper gives an application to genuine (not pseudo) Boolean optimization to the real world by regarding Boolean elements as sets, X as target set in consideration, xi as sets of basic factors effecting X, G(X) gives the total contradiction of X to the x/s. (@(X)1 (the number of individuals in the set Q(X)) g ives a numerical measure of the contradiction. The smaller I@(X)1 indicates the better. The treatment of the general Steiner problem is reduced to a question of solving the Boolean equation. It can be applied to a wider realm such as genuine (not pseudo) Boolean programming. 1. The general Steiner problem in Boolean space The Boolean space is defined by a set of geometrical axioms, and was proved equivalent to Boolean algebra. The Boolean points are also Boolean elements so * This paper was initiated by the late Ting about thirty years ago and proceeded with by Zhao. 0012-365X/91/$03.50