On the Problem of Approximating the Number of Bases of a Matroid

In this note we consider the problem of counting the number of bases of a matroid. The problem is of practical signiicance as it contains graph reliability as a special case. This is a #P-Hard problem and the main focus in recent research has been on trying to approximate the number of bases. The main result of this paper is that it is impossible to get a good approximation in deterministic polynomial time if the matroid M is given to us by an independence or basis oracle. Thus our result has the same avour as those of Elekes 5] and BB arr anyi and F uredi 1] on the problem of computing the volume of a convex body given by a membership oracle. It should be noted that the main thrust of recent work on approximation for #P-Hard problems has been on randomized algorithms, in particular the Markov chain approach initiated by Broder 2]; see Dyer and Frieze 3], F eder and Mihail 6] for examples of this approach to counting matroid bases. It is to be hoped that randomisation can triumph in this case as it does for computing the volume of a convex body { Dyer, Frieze and Kannan 4] or Lovv asz and Simonovits 8]. We need very little from the theory of matroids, but see Welsh 12] or Oxley 9] for any of the basic deenitions we use. Our computational model is that of Robinson and Welsh 11]. We assume that we have an oracle which