Treewidth computation and extremal combinatorics

TL;DR

This paper establishes a combinatorial bound on certain vertex subsets in graphs and applies it to develop faster algorithms for computing and analyzing treewidth, minimal separators, and potential maximal cliques.

Contribution

It introduces a new extremal combinatorics result and leverages it to create more efficient algorithms for treewidth-related problems.

Findings

Improved algorithms for computing treewidth with exponential time bounds.

Faster enumeration of minimal separators and potential maximal cliques.

Significant reduction in computational complexity compared to previous methods.

Abstract

For a given graph G and integers b,f >= 0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most n\binom{b+f}{b} such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide algorithms that for a given n-vertex graph G - compute the treewidth of G in time O(1.7549^n) by making use of exponential space and in time O(2.6151^n) and polynomial space; - decide in time O(({\frac{2n+k+1}{3})^{k+1}\cdot kn^6}) if the treewidth of G is at most k; - list all minimal separators of G in time O(1.6181^n) and all potential maximal cliques of G in time O(1.7549^n). This significantly improves previous algorithms for these problems.