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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 on a classical computer?
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?


Methodology: The study uses graph theory and matrix theory to propose a method for solving the 3k-CLIQUE problem. It creates an auxiliary graph G' with O(nk) vertices and O(n2k) edges, equivalent to the 3-CLIQUE problem on the original graph G. The study then proposes a way to determine if there exists a 3-clique in G' by checking if a certain equation holds true for each pair of vertices.
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.


Results: The main result is that any deterministic and exact algorithm that does not use an oracle must take Ω(n2k) time in the worst-case scenario to solve the 3k-CLIQUE problem. This is because there are adjacency matrices with Θ(n2k) indices such that the equation holds true, but only a constant number of indices have the equation hold false.
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.


Implications: This lower bound on the time complexity implies that P/∧≠NP, meaning that there are problems that are inherently harder to solve approximately than exactly. This is a significant result in the field of computational complexity.
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.


In conclusion, the study shows that determining if a 3k-clique exists in a graph requires at least Ω(n2k) time in the worst-case scenario for a deterministic and exact algorithm. This lower bound holds even for the fastest known algorithms, implying that some problems are fundamentally harder to solve than others.
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.


Link to Article: https://arxiv.org/abs/0310060v14
Link to Article: https://arxiv.org/abs/0310060v15
Authors:  
Authors:  
arXiv ID: 0310060v14
arXiv ID: 0310060v15


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

Revision as of 14:47, 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, which involves finding a clique of size 3k in a given undirected graph?

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.

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.

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.

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.

Link to Article: https://arxiv.org/abs/0310060v15 Authors: arXiv ID: 0310060v15

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