The vertex set of a 0/1-polytope is strongly P-enumerable

In this paper, we discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P. It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of 79-enumerability introduced by Valiant (1979). The concept of strong 79-enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0/1-polytopes. Our implementation based on the commercial LP solver CPLEX 1 is superior to general vertex enumeration algorithms. We give an example how simplifications of our algorithm lead to efficient enumeration of combinatorial objects. In this paper, we discuss the computational complexity of the following problem which is known as the vertex enumeration problem: given a rational convex polyhedron P = {x ~ Qn I Ax ~ b} defined by a system of m linear inequalities, compute the vertex set of P. The framework for the computational complexity analysis of counting problems dates back to the late seventies when it was formally introduced by Valiant [10]. In a recently electronically published classification of enumeration problems Fukuda [4] improves on these basic concepts in order to take into account the explicit generation of objects. We emphasize the distinction between counting and enumeration because we do not ask for the bare number of vertices but for listing each particular vertex.