The Computational Complexity of 3k-CLIQUE: Difference between revisions
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Title: The Computational Complexity of 3k-CLIQUE | Title: The Computational Complexity of 3k-CLIQUE | ||
Research Question: | Research Question: How fast can a deterministic and exact algorithm solve the 3k-CLIQUE problem on a classical computer? | ||
Methodology: The study uses a graph | Methodology: The study uses a graph theory approach to solve the 3k-CLIQUE problem. It proposes a method where an auxiliary graph G′ is created, which has a 3-clique if and only if the 3k-CLIQUE problem is true for the original graph G. The Hadamard product of the adjacency matrix of G′ is then used to determine if there is a 3-clique in G′. | ||
Results: The main result is that the fastest deterministic and exact algorithm that solves the 3k-CLIQUE problem must run in Ω( n2k) time in the worst-case scenario on a classical computer, where n is the number of vertices in the graph. This | Results: The main result is that the fastest deterministic and exact algorithm that solves the 3k-CLIQUE problem must run in Ω( n2k) time in the worst-case scenario on a classical computer, where n is the number of vertices in the graph. This implies that P/negationslash ≠ NP, which is a significant contribution to the complexity theory. | ||
Implications: This research has implications for the field of computational complexity | Implications: This research has practical implications for the field of computer science, particularly in the areas of graph theory and computational complexity. It sets a lower bound on the time complexity for solving the 3k-CLIQUE problem, which has applications in various areas such as combinatorics, artificial intelligence, and network analysis. The results also contribute to our understanding of the relationship between NP and P/negationslash classes. | ||
Link to Article: https://arxiv.org/abs/ | Link to Article: https://arxiv.org/abs/0310060v12 | ||
Authors: | Authors: | ||
arXiv ID: | arXiv ID: 0310060v12 | ||
[[Category:Computer Science]] | [[Category:Computer Science]] | ||
[[Category:Clique]] | |||
[[Category:3K]] | [[Category:3K]] | ||
[[Category: | [[Category:Problem]] | ||
[[Category:Graph]] | [[Category:Graph]] | ||
[[Category:Complexity]] | [[Category:Complexity]] | ||
Revision as of 14:46, 24 December 2023
Title: The Computational Complexity of 3k-CLIQUE
Research Question: How fast can a deterministic and exact algorithm solve the 3k-CLIQUE problem on a classical computer?
Methodology: The study uses a graph theory approach to solve the 3k-CLIQUE problem. It proposes a method where an auxiliary graph G′ is created, which has a 3-clique if and only if the 3k-CLIQUE problem is true for the original graph G. The Hadamard product of the adjacency matrix of G′ is then used to determine if there is a 3-clique in G′.
Results: The main result is that the fastest deterministic and exact algorithm that solves the 3k-CLIQUE problem must run in Ω( n2k) time in the worst-case scenario on a classical computer, where n is the number of vertices in the graph. This implies that P/negationslash ≠ NP, which is a significant contribution to the complexity theory.
Implications: This research has practical implications for the field of computer science, particularly in the areas of graph theory and computational complexity. It sets a lower bound on the time complexity for solving the 3k-CLIQUE problem, which has applications in various areas such as combinatorics, artificial intelligence, and network analysis. The results also contribute to our understanding of the relationship between NP and P/negationslash classes.
Link to Article: https://arxiv.org/abs/0310060v12 Authors: arXiv ID: 0310060v12