  A Ranking r-Constraint Satisfaction Problem (ranking r-CSP) consists of a
ground set of vertices V, an arity r >= 2, a parameter k and a constraint
system c, where c is a function which maps rankings of r-sized subsets of V to
{0,1}. The objective is to decide if there exists a ranking of the vertices
satisfying all but at most k constraints. Famous ranking r-CSP include the
Feedback Arc Set in Tournaments and Betweenness in Tournaments problems. We
consider these problems from the kernelization viewpoint. We prove that
so-called l_r-simply characterized ranking r-CSPs admit linear vertex-kernels
whenever they admit constant-factor approximation algorithms. This implies that
r-Betweenness in Tournaments and r-Transitive Feedback Arc Set In Tournaments,
two natural generalizations of the previously mentioned problems, admit linear
vertex-kernels. Moreover, we introduce another generalization of Feedback Arc
Set in Tournaments, which does not fit the aforementioned framework. We obtain
a 5-approximation and a linear vertex-kernel for this problem.
