A Simplified Mathematical Study of Local Computation Algorithms for Efficient Hypergraphs Coloring

This paper presents a comprehensive mathematical exploration of local computation algorithms specifically tailored for efficient hypergraph coloring. We develop an axiomatic mathematical foundation for hypergraphs and formally define proper vertex coloring constraints. Our novel local approach iteratively improves the defect of monochromatic hyperedges through strategic local updates, leveraging the Lovász Local Lemma and Moser-Tardos framework for theoretical guarantees. Theoretical analysis demonstrates algorithm convergence to a proper coloring with an expected runtime of O(m⋅Δ), where m represents the number of hyperedges, provided that appropriate conditions are met for both the number of colors k and the hyperedge size Δ. We validate our approach through extensive comparative experimentation against leading algorithms Chatterjee et al., Harris & Srinivasan, and Davis & Kim across various hypergraph configurations. Results show our algorithm achieves superior convergence rates with an average of 1.12-1.5 iterations compared to 2.0-2.35 for competing methods. While our implementation exhibits higher computational overhead on medium-scale hypergraphs, it demonstrates excellent scalability properties with increasing graph size and consistently maintains perfect 100% coloring success rates. These results confirm the theoretical foundations while demonstrating that our approach is both practical and efficient for applications requiring correct hypergraph coloring with minimal iterations.