Summary: On the problem of approximating the
number of bases of a matroid
Y. Azar \Lambda A. Z. Broder \Lambda A. M. Frieze y
March 12, 1996
In this note we consider the problem of counting the number of bases
of a matroid. The problem is of practical significance as it contains graph
reliability as a special case. This is a #PHard 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 ap
proximation in deterministic polynomial time if the matroid M is given to us
by an independence or basis oracle. Thus our result has the same flavour as
those of Elekes  and B'ar'anyi and F¨uredi  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 #PHard problems has been on randomized algorithms, in particular the
Markov chain approach initiated by Broder ; see Dyer and Frieze , F'eder
and Mihail  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  or
Lov'asz and Simonovits .