Partitioning algorithms for weighted trees and cactus graphs

TL;DR

This paper introduces polynomial-time algorithms for (l,u)-partition problems on cactus graphs, extending known results from trees, and provides a framework for solving related NP-hard partition problems with practical efficiency.

Contribution

It demonstrates that (l,u)-partition problems, previously solvable on trees, can be extended to cactus graphs with polynomial algorithms and offers a framework for other NP-hard variants.

Findings

Polynomial-time algorithms for (l,u)-partition on cactus graphs

Extension of partitioning methods from trees to cactus graphs

Framework for solving NP-hard partition problems with pseudopolynomial runtime

Abstract

In this paper, we consider different constrained partition problems for weighted trees and cactus graphs. We focus on the (l,u)-partition problem, which is the problem of partitioning a weighted graph into connected clusters such that each cluster fulfills the lower and upper weight constraints l and u. Partitioning into a minimum, maximum or a fixed number of clusters is known to be NP-hard in general, but polynomial-time solvable on trees. We prove that these three variants of the (l,u)-partition problem can be solved for cactus graphs as well by presenting a polynomial-time algorithm. Additionally, we present an efficient method to compute the corresponding partitions. For other optimization goals or additional constraints, the partition problem becomes NP-hard - even on trees and for a lower weight bound equal to zero. We show that our method can be used as an algorithmic framework to solve other partition problems for weighted trees and cactus graphs with a pseudopolynomial runtime.