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Title: The Computational Complexity of 3k-CLIQUE
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, which involves finding a clique of size 3k in a given undirected graph?
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 a mathematical approach to prove a lower bound on the time complexity of solving the 3k-CLIQUE problem. It introduces an auxiliary graph G′ and shows that determining whether the Hadamard product of two matrices representing G′ equals zero is equivalent to solving the 3k-CLIQUE problem on the original graph G. This implies that any deterministic and exact algorithm must take at least Ω( n2k) time in the worst-case scenario, where n is the number of vertices in the graph.
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 main result of the study is the lower bound of Ω( n2k) on the time complexity of deterministic and exact algorithms for solving the 3k-CLIQUE problem. This bound is confirmed by the fact that the fastest known deterministic and exact algorithm, published in 1985, has a running time of Θ( nωk), where ω≥2.
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: The lower bound on the time complexity has implications for the complexity of the 3k-CLIQUE problem and the algorithms used to solve it. It suggests that there may be limitations in the efficiency of deterministic and exact algorithms for solving this problem, and it may motivate research into alternative approaches, such as probabilistic algorithms or approximation algorithms.
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.


In conclusion, the study provides a lower bound on the time complexity of deterministic and exact algorithms for solving the 3k-CLIQUE problem, which is equivalent to finding a clique of size 3k in a given undirected graph. The results imply that there may be limitations in the efficiency of current algorithms, which could motivate further research into alternative approaches.
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/0310060v15
Link to Article: https://arxiv.org/abs/0310060v9
Authors:  
Authors:  
arXiv ID: 0310060v15
arXiv ID: 0310060v9


[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Clique]]
[[Category:3K]]
[[Category:3K]]
[[Category:Clique]]
[[Category:Problem]]
[[Category:Time]]
[[Category:Complexity]]
[[Category:Complexity]]
[[Category:Time]]
[[Category:Problem]]

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

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