Losing Treewidth by Separating Subsets

TL;DR

This paper develops approximation algorithms for deleting vertices or edges to reduce a graph's complexity, specifically targeting classes with bounded treewidth, improving previous results and introducing simpler, uniform approaches.

Contribution

It introduces a general framework linking graph partitioning problems to graph modification problems, leading to improved approximation algorithms for treewidth reduction.

Findings

Improved approximation ratios for k-Treewidth Vertex Deletion and Planar-F Vertex Deletion.

First uniform approximation algorithms under natural parameterization.

APX-hardness results for edge deletion problems.

Abstract

We study the problem of deleting the smallest set $S$ of vertices (resp. edges) from a given graph $G$ such that the induced subgraph (resp. subgraph) $G \setminus S$ belongs to some class $\mathcal{H}$. We consider the case where graphs in $\mathcal{H}$ have treewidth bounded by $t$, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called $k$-Subset Vertex Separator and $k$-Subset Edge Separator, respectively. For the vertex deletion setting, our framework combined with the current best result for $k$-Subset Vertex Separator, yields a significant improvement in the approximation ratios for basic problems such as $k$-Treewidth Vertex Deletion and Planar-$F$ Vertex Deletion. Our algorithms are simpler than previous works and give the first uniform approximation algorithms under the natural parameterization. For the edge deletion setting, we give improved approximation algorithms for $k$-Subset Edge Separator combining ideas from LP relaxations and important separators. We present their applications in bounded-degree graphs, and also give an APX-hardness result for the edge deletion problems.