A Faster Algorithm for Independent Cut

TL;DR

This paper presents a significantly faster algorithm for the independent cut problem, improving runtime bounds and establishing fixed-parameter tractability for certain graph classes.

Contribution

It introduces an improved algorithm with a better exponential runtime and proves fixed-parameter tractability on graphs with high minimum degree.

Findings

Runtime improved to *(1.4143^n)

Established fixed-parameter tractability for graphs with minimum degree > 0.5n

Derived runtime bounds based on maximum degree __________________________________________________

Abstract

The previously fastest algorithm for deciding the existence of an independent cut had a runtime of $\mathcal{O}^*(1.4423^n)$, where $n$ is the order of the input graph. We improve this to $\mathcal{O}^*(1.4143^n)$. In fact, we prove a runtime of $\mathcal{O}^*\left( 2^{(\frac{1}{2}-\alpha_\Delta)n} \right)$ on graphs of order $n$ and maximum degree at most $\Delta$, where $\alpha_\Delta=\frac{1}{2+4\lfloor \frac{\Delta}{2} \rfloor}$. Furthermore, we show that the problem is fixed-parameter tractable on graphs of order $n$ and minimum degree at least $\beta n$ for some $\beta > \frac{1}{2}$, where $\beta$ is the parameter.