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We prove a theorem on Hilbert bases analogous to Caratheodory's theorem for convex cones. The result is used to give an upper bound on the number of nonzero variables needed in optimal solutions to integer programs associated with totally dual integral systems. For integer programs arising from perfect graphs the general bounds are improved to show that if G is a perfect graph with n nodes and ''" is a vector of integral node weights, then there exists a minimum 11•-covering of the nodes that uses at most n distinct cliques. <" 1986 Academic Press, Inc I. INTRODUCTION A rational cone is a set of the form {x: Ax~O}, where A is a rational m x n matrix and 0 is the m component zero vector. By the theorems of Wey! and Minkowski (see [ 19]), C is a rational cone if and only if there exists a finite set of rational vectors {a 1 , ••• , ak} that generate C, that is, C = P, 1 a 1 +-• + },k ak:),;~ 0, i = !, ... , k }. The dimension of a rational cone C, denoted by dim C, is the cardinality of a maximal set of linearly independent vectors in C. A well-known result of Caratheodory 1s the following.

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