ON THE RECOGNITION OF S-SYSTEMS

A sequence (A1,..., A(n)) of nonempty subsets of R(m) is an S-system if for each choice of a1 is-an-element-of A1,..., a(n) is-an-element-of A(n), the nullspace of the matrix [a1,..., a(n)] is a line that penetrates the strictly positive orthant of R(n). For the case in which each A(j) is a sign cone, S-systems are basic to the study of the sign-solvability of systems of linear equations; in that case, they can be recognized in time O(m2). The present paper provides several characterizations of general S-systems and describes a polynomial-time algorithm (based on linear programming) for the recognition of such systems when the A(j) are arbitrary finitely presented convex cones.