# Space-Efficient Vertex Separators for Treewidth

## TL;DR

This paper introduces a space-efficient algorithm for computing vertex separators and tree decompositions in graphs with bounded treewidth, enabling solutions to several problems using minimal memory.

## Contribution

It presents the first space-efficient algorithms for tree decomposition and related problems in graphs with certain treewidth bounds, using only O(n) bits.

## Key findings

- Algorithm computes vertex separators with O(n) bits.
- Provides an O(1)-approximation for tree decomposition.
- Solves Vertex Cover, Independent Set, and other problems with limited memory.

## Abstract

For $n$-vertex graphs with treewidth $k = O(n^{1/2-\epsilon})$ and an arbitrary $\epsilon>0$, we present a word-RAM algorithm to compute vertex separators using only $O(n)$ bits of working memory. As an application of our algorithm, we give an $O(1)$-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in $c^k n (\log \log n) \log^* n$ time using $O(n)$ bits for some constant $c > 0$. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover, Independent Set, Dominating Set, MaxCut and $q$-Coloring by using $O(n)$ bits as long as the treewidth of the graph is smaller than $c' \log n$ for some problem dependent constant $0 < c' < 1$.