Tight Approximation Ratio of a General Greedy Splitting Algorithm for the Minimum k-Way Cut Problem

TL;DR

This paper analyzes a general greedy algorithm for the minimum k-way cut problem, establishing its tight approximation ratio and extending understanding beyond previously known bounds for specific cases.

Contribution

It provides a tight approximation ratio for a broad class of greedy algorithms that iteratively increase components by varying sizes, generalizing prior results.

Findings

Approximation ratio is 2 - (sum of binomial coefficients)/binomial(k,2)

Ratio is tight and improves understanding of greedy algorithms for k-way cuts

Special case ratio is 2 - h/k, with exact bounds when k-1 is divisible by h-1

Abstract

For an edge-weighted connected undirected graph, the minimum $k$-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into $k$ connected components. The problem is NP-hard when $k$ is part of the input and W[1]-hard when $k$ is taken as a parameter. A simple algorithm for approximating a minimum $k$-way cut is to iteratively increase the number of components of the graph by $h-1$, where $2 \le h \le k$, until the graph has $k$ components. The approximation ratio of this algorithm is known for $h \le 3$ but is open for $h \ge 4$. In this paper, we consider a general algorithm that iteratively increases the number of components of the graph by $h_i-1$, where $h_1 \le h_2 \le ... \le h_q$ and $\sum_{i=1}^q (h_i-1) = k-1$. We prove that the approximation ratio of this general algorithm is $2 - (\sum_{i=1}^q {h_i \choose 2})/{k \choose 2}$, which is tight. Our result implies that the approximation ratio of the simple algorithm is $2-h/k + O(h^2/k^2)$ in general and $2-h/k$ if $k-1$ is a multiple of $h-1$.