Given a graph G cellularly embedded on a surface Σ of genus g, a
cut graph is a subgraph of G such that cutting Σ along G yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
ε>0, we show how to compute a (1+ε) approximation of
the shortest cut graph in time f(ε,g)n3.
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest