Betweenness parameterized above tight lower bound

Abstract

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 “ v i is between v j and v k ”, decide whether there is a bijection from V to the set { 1 , … , | V | } satisfying at least | C | / 3 + κ 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.