Polynomial-time Approximation Scheme for Minimum k-cut in Planar and Minor-free Graphs

TL;DR

This paper presents the first polynomial-time approximation scheme for the minimum k-cut problem in planar and minor-free graphs, significantly improving previous approximation factors and advancing the understanding of graph cut problems.

Contribution

The paper introduces the first polynomial-time approximation scheme for the k-cut problem in planar and minor-free graphs, surpassing the longstanding 2-approximation barrier.

Findings

Established a polynomial-time approximation scheme for k-cut in planar graphs.

Extended the scheme to minor-free graphs.

Achieved approximation factors better than 2 for these graph classes.

Abstract

The $k$-cut problem asks, given a connected graph $G$ and a positive integer $k$, to find a minimum-weight set of edges whose removal splits $G$ into $k$ connected components. We give the first polynomial-time algorithm with approximation factor $2-\epsilon$ (with constant $\epsilon > 0$) for the $k$-cut problem in planar and minor-free graphs. Applying more complex techniques, we further improve our method and give a polynomial-time approximation scheme for the $k$-cut problem in both planar and minor-free graphs. Despite persistent effort, to the best of our knowledge, this is the first improvement for the $k$-cut problem over standard approximation factor of $2$ in any major class of graphs.