Finding Minimum Spanning Forests in a Graph

TL;DR

This paper introduces a new NP-complete graph partitioning problem motivated by computational topology and proposes two approximation algorithms, spectral clustering and dynamic programming, demonstrating their effectiveness on test graphs.

Contribution

The paper defines a novel graph partitioning problem, proves its NP-completeness, and presents two approximation algorithms with empirical performance analysis.

Findings

The problem is NP-complete.

Spectral clustering and dynamic programming algorithms produce near-optimal solutions.

Algorithms perform well on test graphs.

Abstract

We introduce a graph partitioning problem motivated by computational topology and propose two algorithms that produce approximate solutions. Specifically, given a weighted, undirected graph $G$ and a positive integer $k$, we desire to find $k$ disjoint trees within $G$ such that each vertex of $G$ is contained in one of the trees and the weight of the largest tree is as small as possible. We are unable to find this problem in the graph partitioning literature, but we show that the problem is NP-complete. We then propose two approximation algorithms, one that uses a spectral clustering approach and another that employs a dynamic programming strategy, which produce near-optimal partitions on a family of test graphs. We describe these algorithms and analyze their empirical performance.