A Faster Algorithm for Independent Cut
Vsevolod Chernyshev, Johannes Rauch, Dieter Rautenbach, Liliia Redina
TL;DR
This work tackles the Independent Cut problem by introducing solid subgraphs and quasi-coverings as a structural framework. The authors reduce IC to a partition problem on a quasi-covering set and a corresponding 2-SAT instance, enabling an exact algorithm with runtime $\\mathcal{O}^*(2^{(\\frac{1}{2}-α_Δ)n})$ for graphs of maximum degree $\\Delta$, where $α_Δ=\\frac{1}{2+4\\lfloor \\frac{Δ}{2} \\rfloor}$. Consequently, IC on general graphs achieves $\\mathcal{O}^*(\\sqrt{2}^n)$ time, and IC is fixed-parameter tractable for graphs with minimum degree $δ \\ge βn$ for some $β>\\frac{1}{2}$, with runtime $2^{\\frac{1}{(2β-1)^2}n}\\cdot n^{O(1)}$. The key technical contributions include a windmill-based construction that bounds the size of the quasi-covering, and a polynomial-time reduction to 2-SAT that certifies the existence of an independent cut when a satisfying assignment exists. These results yield improved exact algorithms and provide ETH-based lower bounds, sharpening the landscape for IC in dense and high-degree graphs.
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 $\mathcal{O}^*\left( 2^{(\frac{1}{2}-α_Δ)n} \right)$ on graphs of order $n$ and maximum degree at most $Δ$, where $α_Δ=\frac{1}{2+4\lfloor \fracΔ{2} \rfloor}$. Furthermore, we show that the problem is fixed-parameter tractable on graphs of order $n$ and minimum degree at least $βn$ for some $β> \frac{1}{2}$, where $β$ is the parameter.