A Faster Randomized Algorithm for Vertex Cover: An Automated Approach

2025-10-10

AI Summary

This paper addresses the Vertex Cover problem by proposing the first automated framework for generating branching rules and enhancing the Measure & Conquer technique to enable systematic optimization of randomized branching algorithms. Methodologically, it integrates local structural analysis, adaptive randomized branching strategies, and structural properties of bounded-degree graphs (Δ ≤ 6), significantly improving both branching efficiency and flexibility. The main contributions are: (1) the first automated derivation mechanism for branching rules; (2) the current best randomized time bounds for cubic graphs—O*(1.07625ⁿ) in terms of input size n and O*(1.13132ᵏ) in terms of solution size k; and (3) new state-of-the-art randomized time complexities for Vertex Cover on both bounded-degree graphs (Δ ≤ 6) and general graphs, advancing the frontier of algorithmic performance in this domain.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

This work introduces two techniques for the design and analysis of branching algorithms, illustrated through the case study of the Vertex Cover problem. First, we present a method for automatically generating branching rules through a systematic case analysis of local structures. Second, we develop a new technique for analyzing randomized branching algorithms using the Measure & Conquer method, offering greater flexibility in formulating branching rules. By combining these innovations with additional techniques, we obtain the fastest known randomized algorithms in different parameters for the Vertex Cover problem on graphs with bounded degree (up to 6) and on general graphs. For example, our algorithm solves Vertex Cover on subcubic graphs in $O^*(1.07625^n)$ time and $O^*(1.13132^k)$ time, respectively. For graphs with maximum degree 4, we achieve running times of $O^*(1.13735^n)$ and $O^*(1.21103^k)$, while for general graphs we achieve $O^*(1.25281^k)$.

Problem

• Automating branching rule generation for Vertex Cover
• Analyzing randomized branching with Measure & Conquer
• Achieving fastest randomized algorithms for bounded-degree graphs

Innovation

• Automated branching rule generation via local structure analysis
• Randomized branching analysis using Measure & Conquer method
• Combined techniques achieve fastest randomized Vertex Cover algorithms