A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

TL;DR

This paper presents a fixed parameter tractable approximation scheme for finding near-optimal cut graphs on surfaces, achieving efficient computation with a focus on surface topology and graph structure.

Contribution

It introduces a novel approximation scheme for shortest cut graphs on surfaces, utilizing brick decompositions and a new variant of surface-cut decomposition.

Findings

Provides a $(1+ \varepsilon)$ approximation in $f(\\varepsilon, g)n^3$ time.

Develops a spanner construction for the problem using brick decompositions.

Introduces a new variant of surface-cut decomposition that may be of independent interest.

Abstract

Given a graph $G$ cellularly embedded on a surface $\Sigma$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $\Sigma$ 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 $\varepsilon >0$, we show how to compute a $(1+ \varepsilon)$ approximation of the shortest cut graph in time $f(\varepsilon, g)n^3$. 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.