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: What is the minimum time complexity required for a deterministic and exact algorithm to solve the 3k-CLIQUE problem on a classical computer? | ||
Methodology: The study uses | Methodology: The study uses an auxiliary graph technique to reduce the 3k-CLIQUE problem to the 3-CLIQUE problem, which is then solved using existing algorithms. The time complexity of these algorithms is then used to derive the lower bound for the 3k-CLIQUE problem. | ||
Results: The | Results: The research finds 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. | ||
Implications: This | Implications: This result implies that the problem of finding a clique of size 3k in a graph is computationally hard, as it requires at least Ω(n2k) time in the worst-case scenario. This lower bound is confirmed by the fact that the fastest known deterministic and exact algorithm that solves 3k-CLIQUE has a running time of Θ(nωk), where ω ≥ 2. | ||
Link to Article: https://arxiv.org/abs/ | Significance: This research contributes to the understanding of the computational complexity of graph problems and provides a lower bound for the 3k-CLIQUE problem, which is useful for comparing the efficiency of different algorithms. | ||
Link to Article: https://arxiv.org/abs/0310060v9 | |||
Authors: | Authors: | ||
arXiv ID: | arXiv ID: 0310060v9 | ||
[[Category:Computer Science]] | [[Category:Computer Science]] | ||
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[[Category:3K]] | [[Category:3K]] | ||
[[Category:Problem]] | [[Category:Problem]] | ||
[[Category: | [[Category:Time]] | ||
[[Category:Complexity]] | [[Category:Complexity]] | ||
Latest revision as of 14:48, 24 December 2023
Title: The Computational Complexity of 3k-CLIQUE
Research Question: What is the minimum time complexity required for a deterministic and exact algorithm to solve the 3k-CLIQUE problem on a classical computer?
Methodology: The study uses an auxiliary graph technique to reduce the 3k-CLIQUE problem to the 3-CLIQUE problem, which is then solved using existing algorithms. The time complexity of these algorithms is then used to derive the lower bound for the 3k-CLIQUE problem.
Results: The research finds 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.
Implications: This result implies that the problem of finding a clique of size 3k in a graph is computationally hard, as it requires at least Ω(n2k) time in the worst-case scenario. This lower bound is confirmed by the fact that the fastest known deterministic and exact algorithm that solves 3k-CLIQUE has a running time of Θ(nωk), where ω ≥ 2.
Significance: This research contributes to the understanding of the computational complexity of graph problems and provides a lower bound for the 3k-CLIQUE problem, which is useful for comparing the efficiency of different algorithms.
Link to Article: https://arxiv.org/abs/0310060v9 Authors: arXiv ID: 0310060v9