Ordinal embedding relaxations parametrized 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 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 + k of the constraints in C. Our result solves an open problem attributed to Benny Chor in Niedermeier’s monograph "Invitation to Fixed-Parameter Algorithms." 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.