Betweenness Parameterized Above Tight Lower Bound

TL;DR

This paper proves the existence of a quadratic kernel for the Betweenness problem parameterized above its tight lower bound, advancing fixed-parameter tractability results and addressing an open problem in the field.

Contribution

It establishes a quadratic kernel for the Betweenness problem above its tight lower bound, solving an open problem and providing a framework for similar problems.

Findings

Quadratic kernel for Betweenness problem above lower bound

Solves an open problem in fixed-parameter algorithms

Framework applicable to other permutation problems

Abstract

We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the {\sc Betweenness} problem parameterized above its tight lower bound, which is stated as follows. For a set $V$ of variables and set $\mathcal C$ of constraints "$v_i$ \mbox{is between} $v_j$ \mbox{and} $v_k$", decide whether there is a bijection from $V$ to the set $\{1,\ldots,|V|\}$ satisfying at least $|\mathcal C|/3 + \kappa$ of the constraints in $\mathcal C$. 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.