  In this paper, we propose to study the following maximum ordinal consensus
problem: Suppose we are given a metric system (M, X), which contains k metrics
M = {\rho_1,..., \rho_k} defined on the same point set X. We aim to find a
maximum subset X' of X such that all metrics in M are "consistent" when
restricted on the subset X'. In particular, our definition of consistency will
rely only on the ordering between pairwise distances, and thus we call a
"consistent" subset an ordinal consensus of X w.r.t. M. We will introduce two
concepts of "consistency" in the ordinal sense: a strong one and a weak one.
Specifically, a subset X' is strongly consistent means that the ordering of
their pairwise distances is the same under each of the input metric \rho_i from
M. The weak consistency, on the other hand, relaxes this exact ordering
condition, and intuitively allows us to take the plurality of ordering relation
between two pairwise distances.
  We show in this paper that the maximum consensus problems over both the
strong and the weak consistency notions are NP-complete, even when there are
only 2 or 3 simple metrics, such as line metrics and ultrametrics. We also
develop constant-factor approximation algorithms for the dual version, the
minimum inconsistent subset problem of a metric system (M, P), - note that
optimizing these two dual problems are equivalent.
