Optimal Bounds for the $k$-cut Problem

TL;DR

This paper establishes tight bounds and efficient algorithms for the $k$-cut problem, showing the probability of finding fixed $k$-cuts and bounding their number, advancing understanding of graph partitioning complexity.

Contribution

It provides tight bounds on the number of minimum $k$-cuts and an algorithm to compute the $k$-cut value with near-optimal runtime, using novel extremal and probabilistic analysis.

Findings

Probability of finding any fixed $k$-cut is $oldsymbol{ ext{Omega}_k(n^{- ext{alpha}k})}$.

Number of minimum $k$-cuts is $oldsymbol{O_k(n^k)}$, tight up to lower-order factors.

Algorithm computes $oldsymbol{ ext{lambda}_k}$ in roughly $oldsymbol{n^k ext{polylog}(n)}$ time.

Abstract

In the $k$-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other hand, lower bounds from conjectures about the $k$-clique problem imply that $\Omega(n^{(1-o(1))k})$ time is likely needed. Recent results of Gupta, Lee \& Li have given new algorithms for general $k$-cut in $n^{1.98k + O(1)}$ time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed $k$-cut of weight $\alpha \lambda_k$ with probability $\Omega_k(n^{-\alpha k})$, where $\lambda_k$ denotes the minimum $k$-cut weight. This also gives an extremal bound of $O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute $\lambda_k$ with roughly $n^k \mathrm{polylog}(n)$ runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight $k$-clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than $2 \lambda_k/k$, using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process.