Pré-Publication, Document De Travail Année : 2025

New Insights and Developments on the Unique Games Conjecture

Frank Vega

Résumé

This paper presents a novel approximation algorithm for the vertex cover problem, a well-known NP-hard optimization challenge in graph theory. Our algorithm achieves an approximation ratio of at most 1.75, improving upon the standard 2-approximation methods. By leveraging minimum edge covers, bipartite matching techniques, and Kőnig's theorem, we develop a polynomial-time solution that guarantees a cover size of at most 1.75 times the optimal. The algorithm operates by first computing a minimum edge cover, then decomposing the resulting subgraph into bipartite components. For each component, we apply the Hopcroft-Karp algorithm to find a maximum matching, followed by Kőnig's theorem to derive a minimum vertex cover. This approach exploits the structural properties of bipartite graphs to achieve improved approximation quality. We provide a rigorous proof of the algorithm's approximation ratio and analyze its time complexity, which is $O(n^{3})$ in the worst case for a graph with $n$ vertices. This study contributes to the ongoing research in approximation algorithms for NP-hard problems and provides compelling evidence that challenges the Unique Games Conjecture. However, our algorithm's increased runtime and its approximation ratio nearing 2 render it impractical for real-world applications, as demonstrated by our experimental results. Thus, our solution holds value primarily as a theoretical contribution which leaves open the possibility for future advancements and novel breakthroughs in the fields of Graph Theory and Combinatorial Optimization.

Fichier principal
Vignette du fichier
Unique Games Conjecture.pdf (353.21 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-04972401 , version 1 (01-03-2025)

Licence

Identifiants

  • HAL Id : hal-04972401 , version 1

Citer

Frank Vega. New Insights and Developments on the Unique Games Conjecture. 2025. ⟨hal-04972401⟩
36 Consultations
270 Téléchargements

Partager

  • More