An improved local search based algorithm for $k^-$-star partition

2025-08-12

AI Summary

This paper studies the $k^-$-star partition problem: given a simple undirected graph $G = (V, E)$, partition $V$ into vertex-disjoint stars, each containing at most $k$ vertices, while minimizing the number of stars. Being NP-hard, this problem is addressed by an enhanced local search algorithm that identifies *critical vertices* and employs three structured local update operations. Crucially, we introduce amortized analysis—first applied to this problem—for proving approximation guarantees. Theoretically, the algorithm achieves a $left(frac{k}{2} - frac{k-2}{8k-14} ight)$-approximation ratio in $O(|V|^3)$ time, constituting the current best-known result and significantly improving upon all prior approximations. Our core innovation lies in the synergistic integration of the critical-vertex mechanism with amortized analysis, enabling fine-grained structural control over the solution and yielding a rigorous, improved performance bound.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the $k^-$-star partition problem that aims to find a minimum collection of vertex-disjoint stars, each having at most $k$ vertices to cover all vertices in a simple undirected graph $G = (V, E)$. Our main contribution is an improved $O(|V|^3)$-time $(frac k2 - frac {k-2}{8k-14})$-approximation algorithm. Our algorithm starts with a $k^-$-star partition with the least $1$-stars and a key idea is to distinguish critical vertices, each of which is either in a $2$-star or is the center of a $3$-star in the current solution. Our algorithm iteratively updates the solution by three local search operations so that the vertices in each star in the final solution produced cannot be adjacent to too many critical vertices. We present an amortization scheme to prove the approximation ratio in which the critical vertices are allowed to receive more tokens from the optimal solution.

Problem

• Finding minimum vertex-disjoint stars in a graph
• Improving approximation algorithm for star partition
• Optimizing local search operations for critical vertices

Innovation

• Improved O(|V|^3)-time approximation algorithm
• Distinguishes critical vertices for optimization
• Uses three local search operations iteratively