Breaking the $n^k$ Barrier for Minimum $k$-cut on Simple Graphs

TL;DR

This paper presents a new algorithm for the minimum k-cut problem on simple, unweighted graphs that runs faster than the previous n^k barrier, indicating a fundamental difference from weighted graphs.

Contribution

The authors develop an algorithm with runtime O(n^{(1- ext{epsilon})k}) for simple graphs, breaking the longstanding n^k barrier and highlighting a separation between weighted and unweighted cases.

Findings

Achieved faster-than-n^k runtime for simple graphs.

Established a separation between weighted and unweighted minimum k-cut problems.

Demonstrated that the n^k barrier is not tight for simple graphs.

Abstract

In the minimum $k$-cut problem, we want to find the minimum number of edges whose deletion breaks the input graph into at least $k$ connected components. The classic algorithm of Karger and Stein runs in $\tilde O(n^{2k-2})$ time, and recent, exciting developments have improved the running time to $O(n^k)$. For general, weighted graphs, this is tight assuming popular hardness conjectures. In this work, we show that perhaps surprisingly, $O(n^k)$ is not the right answer for simple, unweighted graphs. We design an algorithm that runs in time $O(n^{(1-\epsilon)k})$ where $\epsilon>0$ is an absolute constant, breaking the natural $n^k$ barrier. This establishes a separation of the two problems in the unweighted and weighted cases.