  The paper develops a new technique to extract a characteristic subset from a
random source that repeatedly samples from a set of elements. Here a
characteristic subset is a set that when containing an element contains all
elements that have the same probability. With this technique at hand the paper
looks at the special case of the tournament isomorphism problem that stands in
the way towards a polynomial-time algorithm for the graph isomorphism problem.
Noting that there is a reduction from the automorphism (asymmetry) problem to
the isomorphism problem, a reduction in the other direction is nevertheless not
known and remains a thorny open problem. Applying the new technique, we develop
a randomized polynomial-time Turing-reduction from the tournament isomorphism
problem to the tournament automorphism problem. This is the first such
reduction for any kind of combinatorial object not known to have a
polynomial-time solvable isomorphism problem.
