Betweenness parameterized above tight lower bound

We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the betweenness problem parameterized above its tight lower bound, which is stated as follows. For a set v of variables and set c of constraints “vi is between vj and vk”, decide whether there is a bijection from v to the set {1,…,|v|} satisfying at least |c|/3+? of the constraints in c. Our result solves an open problem attributed to benny chor in niedermeier's monograph “invitation to fixed-parameter algorithms”. The betweenness problem is of interest in molecular biology. An approach developed in this paper can be used to determine parameterized complexity of a number of other optimization problems on permutations parameterized above or below tight bounds.