  The approximate computation of twin-width has attracted significant attention
already since the moment the parameter was introduced. A recently proposed
approach (STACS 2024) towards obtaining a better understanding of this question
is to consider the approximability of twin-width via fixed-parameter algorithms
whose running time depends not on twin-width itself, but rather on parameters
which impose stronger restrictions on the input graph. The first step that
article made in this direction is to establish the fixed-parameter
approximability of twin-width (with an additive error of 1) when the runtime
parameter is the feedback edge number.
  Here, we make several new steps in this research direction and obtain:
  - An asymptotically tight bound between twin-width and the feedback edge
number;
  - A significantly improved fixed-parameter approximation algorithm for
twin-width under the same runtime parameter (i.e., the feedback edge number)
which circumvents many of the technicalities of the original result and
simultaneously avoids its formerly non-elementary runtime dependency;
  - An entirely new fixed-parameter approximation algorithm for twin-width when
the runtime parameter is the vertex integrity of the graph.
