On the Min-Max Star Partitioning Number

TL;DR

This paper introduces a new star partitioning problem in graphs, providing polynomial algorithms for optimal solutions, analyzing special cases, and demonstrating NP-completeness in weighted variants.

Contribution

It formulates a novel star partitioning problem, develops efficient algorithms, and extends solutions to hypergraphs and multigraphs, with complexity analysis and hardness results.

Findings

Polynomial-time algorithm for optimal star partitioning.

Explicit solutions for specific graph classes.

NP-completeness of weighted decision variant.

Abstract

In this paper, we introduce a novel star partitioning problem for simple connected graphs $G=(V,E)$. The goal is to find a partition of the edges into stars that minimizes the maximum number of stars a node is contained in while simultaneously satisfying node-specific capacities. We design and analyze an efficient polynomial time algorithm with a runtime of $\mathcal{O}(|E|^2)$ that determines an optimal partition. Moreover, we explicitly provide a closed form of an optimal value for some graph classes. We generalize our algorithm to find even an optimal star partition of linear hypergraphs, multigraphs, and graphs with self-loop. We use flow techniques to design an algorithm for the star partitioning problem with an improved runtime of $\mathcal{O}(\log(\Delta) \cdot |E| \cdot \min\{|V|^{\frac{2}{3}},|E|^{\frac{1}{2}}\})$, where $\Delta$ is maximum node degree in $G$. In contrast to the unweighted setting, we show that a node-weighted decision variant of this problem is \texttt{strongly NP-complete} even without capacity constraints. Furthermore, we provide an extensive comparison to the problem of minimizing the minimum indegree satisfying node capacity constraints.