On Minimum Bisection and Related Cut Problems in Trees and Tree-Like Graphs

TL;DR

This paper investigates the minimum bisection problem in trees and tree-like graphs, providing bounds on bisection width based on graph parameters and algorithms for computing such bisections efficiently.

Contribution

It establishes a new upper bound for the minimum bisection width in trees and extends the analysis to graphs with tree decompositions, including an efficient computation method.

Findings

Bound on minimum bisection width for trees: $MinBis(T) \,\leq\, 8 n \Delta(T) / diam(T)$

General upper bound for graphs with tree decompositions based on structure

Efficient algorithm for computing bisections satisfying the bounds

Abstract

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider this problem for trees and prove that the minimum bisection width of every tree $T$ on $n$ vertices satisfies $MinBis(T) \leq 8 n \Delta(T) / diam(T)$. Second, we generalize this to arbitrary graphs with a given tree decomposition $(T,X)$ and give an upper bound on the minimum bisection width that depends on the structure of $(T,X)$. Moreover, we show that a bisection satisfying our general bound can be computed in time proportional to the encoding length of the tree decomposition when the latter is provided as input.