Note on Maximal Bisection above Tight Lower Bound

TL;DR

This paper investigates the parameterized complexity of finding large bisections in graphs, establishing a polynomial kernel for the problem when exceeding the tight lower bound by a parameter k.

Contribution

It introduces the Max Bisection above Tight Lower Bound problem and proves it admits a polynomial kernel with O(k^2) vertices and O(k^3) edges.

Findings

Kernelization result with O(k^2) vertices

Kernelization result with O(k^3) edges

Parameterization above tight lower bound is fixed-parameter tractable

Abstract

In a graph $G=(V,E)$, a bisection $(X,Y)$ is a partition of $V$ into sets $X$ and $Y$ such that $|X|\le |Y|\le |X|+1$. The size of $(X,Y)$ is the number of edges between $X$ and $Y$. In the Max Bisection problem we are given a graph $G=(V,E)$ and are required to find a bisection of maximum size. It is not hard to see that $\lceil |E|/2 \rceil$ is a tight lower bound on the maximum size of a bisection of $G$. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (Max-Bisec-ATLB): decide whether a graph $G=(V,E)$ has a bisection of size at least $\lceil |E|/2 \rceil+k,$ where $k$ is the parameter. We show that this parameterized problem has a kernel with $O(k^2)$ vertices and $O(k^3)$ edges, i.e., every instance of Max-Bisec-ATLB is equivalent to an instance of Max-Bisec-ATLB on a graph with at most $O(k^2)$ vertices and $O(k^3)$ edges.