Linear vertex-kernels for several dense ranking r-CSPs

TL;DR

This paper demonstrates that certain dense ranking constraint satisfaction problems admit linear vertex-kernels when they have constant-factor approximation algorithms, including new generalizations of Feedback Arc Set in Tournaments.

Contribution

It establishes linear vertex-kernels for a class of ranking r-CSPs and introduces a new generalization with a 5-approximation and linear kernel.

Findings

r-Betweenness in Tournaments admits a linear kernel.

r-Transitive Feedback Arc Set in Tournaments admits a linear kernel.

A new generalization of Feedback Arc Set in Tournaments with a 5-approximation and linear kernel.

Abstract

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.