On the kernelization of ranking r-CSPs: Linear vertex-kernels for generalizations of Feedback Arc Set and Betweenness in tournaments - Perez, Anthony (Discrete Applied Mathematics, 2015)
An instance of a rankingr-constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V, an arity r⩾2, a parameter k∈N and a constraint systemc, where c is a function which maps rankings of r-sized sets S⊆V to {0,1}. The objective is to decide if there exists a ranking σ of the vertices satisfying all but at most k constraints (i.e ∑S⊆V,|S|=rc(σ(S))⩽k). We mainly focus on dense instances, that is instances where the constraint system is defined for everyr-sized subset S⊆V. Famous dense ranking r-CSPs include Feedback Arc Set and Betweenness in tournaments, two well-studied problems (Alon et al., 2009; Bessy et al., 2011; Karpinski and Schudy, 2010, 2011; Paul et al., 2011). We consider such problems from the kernelization viewpoint (Niedermeier, 2006). We prove that so-called pr-simply characterized ranking r-CSPs admit linear vertex-kernels whenever they admit constant-factor approximation algorithms. This implies that r-Dense Betweenness and r-Dense Transitive Feedback Arc Set, two natural generalizations of the previously mentioned problems (Karpinski and Schudy, 2010, 2011), admit linear vertex-kernels. Moreover, we introduce another generalization of Feedback Arc Set in Tournaments, which does not fit the aforementioned framework. Based on techniques from Coppersmith (2006) we obtain a 5-approximation, and then provide a linear vertex-kernel for this problem as well.