  We study ordinal embedding relaxations in the realm of parameterized
complexity. We prove the existence of a quadratic kernel for the {\sc
Betweenness} problem parameterized above its tight lower bound, which is stated
as follows. For a set $V$ of variables and set $\mathcal C$ of constraints
"$v_i$ \mbox{is between} $v_j$ \mbox{and} $v_k$", decide whether there is a
bijection from $V$ to the set $\{1,\ldots,|V|\}$ satisfying at least $|\mathcal
C|/3 + \kappa$ of the constraints in $\mathcal C$. Our result solves an open
problem attributed to Benny Chor in Niedermeier's monograph "Invitation to
Fixed-Parameter Algorithms." The betweenness problem is of interest in
molecular biology. An approach developed in this paper can be used to determine
parameterized complexity of a number of other optimization problems on
permutations parameterized above or below tight bounds.
