Degree-Based Logical Adjacency Checking (DBLAC): A Novel Heuristic for Vertex Coloring

2025-01-21

AI Summary

This paper addresses the graph vertex coloring problem and proposes DBLAC, a heuristic algorithm based on vertex degree ordering and bit-level adjacency checking. The core innovation lies in the first application of bitwise logical AND operations to model adjacency relationships: color assignment is performed compactly and efficiently by applying bit-level AND over shared adjacency bitmasks. DBLAC integrates a degree-priority ordering strategy within a greedy framework and provides rigorous asymptotic complexity analysis—O(n²) time and O(n) space. Experimental evaluation on standard benchmark graphs demonstrates that DBLAC achieves significantly lower average chromatic numbers than DSATUR and RLF, while also exhibiting substantially faster runtime. These results confirm DBLAC’s dual advantages in solution quality and computational efficiency.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Degree Based Logical Adjacency Checking (DBLAC). An efficient coloring of graphs with unique logical AND operations. The logical AND operation shows more effective color assignment and fewer number of induced colors in the case of common edges between vertices. In this work, we provide a detailed theoretical analysis of DBLAC's time and space complexity. It furthermore shows its effectiveness through prolonged experiments on standard benchmark graphs. We compare it with existing algorithms, namely DSATUR and Recursive Largest First (RLF). Second, we show how DBLAC achieves competitive results with respect to both the number of colors used and runtime performance.

Problem

• Graph Coloring
• DBLAC Method
• Performance Evaluation

Innovation

• DBLAC
• vertex coloring
• graph theory