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A Ranking r-Constraint Satisfaction Problem (ranking r-CSP) consists of aground set of vertices V, an arity r >= 2, a parameter k and a constraintsystem 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 verticessatisfying all but at most k constraints. Famous ranking r-CSP include theFeedback Arc Set in Tournaments and Betweenness in Tournaments problems. Weconsider these problems from the kernelization viewpoint. We prove thatso-called l_r-simply characterized ranking r-CSPs admit linear vertex-kernelswhenever they admit constant-factor approximation algorithms. This implies thatr-Betweenness in Tournaments and r-Transitive Feedback Arc Set In Tournaments,two natural generalizations of the previously mentioned problems, admit linearvertex-kernels. Moreover, we introduce another generalization of Feedback ArcSet in Tournaments, which does not fit the aforementioned framework. We obtaina 5-approximation and a linear vertex-kernel for this problem.

Linear vertex-kernels for several dense ranking r-CSPs