On the boolean-width of a graph: structure and applications

TL;DR

This paper investigates the boolean-width of graphs, revealing its typical small size in random graphs, its relation to treewidth in bounded degree graphs, and its connection to VC dimension, with implications for algorithm design.

Contribution

It provides new structural insights into boolean-width, including bounds for random and bounded degree graphs, and links to VC dimension to facilitate better algorithms.

Findings

Boolean-width of random graphs is O(log^2 n) almost surely.

Graphs with bounded degree have boolean-width linear in treewidth.

Boolean-cut value correlates with VC dimension, aiding approximation.

Abstract

We study the recently introduced boolean-width of graphs. Our structural results are as follows. Firstly, we show that almost surely the boolean-width of a random graph on $n$ vertices is $O(\log^2 n)$, and it is easy to find the corresponding decomposition tree. Secondly, for any constant $d$ a graph of maximum degree $d$ has boolean-width linear in treewidth. This implies that almost surely the boolean-width of a (sparse) random $d$-regular graph on $n$ vertices is linear in $n$. Thirdly, we show that the boolean-cut value is well approximated by VC dimension of corresponding set system. Since VC dimension is widely studied, we hope that this structural result will prove helpful in better understanding of boolean-width. Combining our first structural result with algorithms from Bui-Xuan et al \cite{BTV09,BTV09II} we get for random graphs quasi-polynomial $O^*(2^{O(\log ^4 n)})$ time algorithms for a large class of vertex subset and vertex partitioning problems.