The problem of whether and how one can compute the twin-width of a graph -- along with an accompanying contraction sequence -- lies at the forefront of the area of algorithmic model theory. While significant effort has been aimed at obtaining a fixed-parameter approximation for the problem when parameterized by twin-width, here we approach the question from a different perspective and consider whether one can obtain (near-)optimal contraction sequences under a larger parameterization, notably the feedback edge number $k$. As our main contributions, under this parameterization we obtain (1) a linear bikernel for the problem of either computing a $2$-contraction sequence or determining that none exists and (2) an approximate fixed-parameter algorithm which computes an $\ell$-contraction sequence (for an arbitrary specified $\ell$) or determines that the twin-width of the input graph is at least $\ell$. These algorithmic results rely on newly obtained insights into the structure of optimal contraction sequences, and as a byproduct of these we also slightly tighten the bound on the twin-width of graphs with small feedback edge number.

翻译：图是否可计算孪生宽度及其伴随收缩序列的问题，是算法模型论领域的前沿课题。尽管已有大量研究致力于在孪生宽度参数化下获得该问题的固定参数近似解，此处我们从不同视角切入，探究在更大参数化框架（尤其是反馈边数 $k$）下能否获得（近）最优收缩序列。作为主要贡献，在该参数化下我们实现了：（1）针对计算 $2$-收缩序列或判定其不存在问题的一个线性双核；（2）一个近似固定参数算法，可计算指定 $\ell$ 的 $\ell$-收缩序列，或判定输入图的孪生宽度至少为 $\ell$。这些算法结果依赖于对最优收缩序列结构的新发现，并由此小幅收紧了小反馈边数图的孪生宽度上界。