Linear Vertex-kernels for Several Dense Ranking r -Constraint Satisfaction Problems

Abstract

A ranking r -constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V, an arity $r \geqslant 2$, a parameter k ∈ ℕ and a constraint system c, where c is a function which maps rankings (i.e. orderings) of r-sized sets S ⊆ V to {0,1} [16]. The objective is to decide if there exists a ranking σ of the vertices satisfying all but at most k constraints (i.e. $\sum_{S \subseteq V, |S| = r} c(\sigma(S)) \leqslant k$). Famous ranking r-CSPs include Feedback Arc Set in Tournaments and Dense Betweenness [4,15]. In this paper, 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 -Dense Betweenness and r -Dense Transitive Feedback Arc Set [15], two natural generalizations of the previously mentioned problems, admit linear vertex-kernels. Both cases were left opened by Karpinksi and Schudy [16]. We also consider another generalization of Feedback Arc Set in Tournaments for constraints of arity $r \geqslant 3$, that does not fit the aforementioned framework. Based on techniques from [11], we obtain a 5-approximation and then provide a linear vertex-kernel. As a main consequence of our result, we obtain the first constant-factor approximation algorithm for a particular case of the so-called Dense Rooted Triplet Inconsistency problem [9].