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arXiv · 0907.5427

Betweenness Parameterized Above Tight Lower Bound

Gregory Gutin·Eun Jung Kim·Matthias Mnich·Anders Yeo
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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.

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Gregory Gutin, Eun Jung Kim, Matthias Mnich, Anders Yeo. 2009-07-30. Betweenness Parameterized Above Tight Lower Bound. https://arxiv.org/abs/0907.5427

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