Craig Alan Feinstein's Evidence That P ≠ NP: Difference between revisions

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Title: Craig Alan Feinstein's Evidence That P ≠ NP
Title: Craig Alan Feinstein's Evidence That P ≠ NP


Abstract: Craig Alan Feinstein, a researcher in computer science, proposed a method to solve the Subset-Sum problem, which is a well-known NP-complete problem. His method, called algorithm A, can solve the problem in O(2n^2) time, assuming constant-time arithmetic and linear-time sorting. Feinstein argued that algorithm A is the best method for solving the problem for large inputs, as it efficiently solves two subproblems that are related to each other. This evidence suggests that P ≠ NP, meaning that problems that can be solved by deterministic polynomial-time algorithms are not equivalent to problems that can be solved by nondeterministic polynomial-time algorithms.
Abstract: Craig Alan Feinstein, a researcher in computer science, presented evidence that the classes of decision problems that can be solved by deterministic polynomial-time algorithms (P) and nondeterministic polynomial-time algorithms (NP) are not equal. He focused on the SUBSET-SUM problem and proposed an algorithm (algorithm A) that could solve this problem in O(2n^2) time, assuming constant-time arithmetic and linear-time sorting. Feinstein argued that algorithm A had the best running-time for large problems and could be improved further by sorting sets S- ∪ (S- + an+1) and S+ instead of sorting sets S- and S+, reducing the running-time to 8 units of time. This improved strategy was shown to be more efficient than other methods, providing evidence that P ≠ NP.


Main Research Question: Can the class of decision problems that can be solved by deterministic polynomial-time algorithms (P) be equivalent to the class of decision problems that can be solved by nondeterministic polynomial-time algorithms (NP)?
Main Research Question: Is P = NP?


Methodology: Feinstein focused on the Subset-Sum problem, which is a common NP problem. He proposed algorithm A, which sorts the input vectors in ascending order and then compares elements from each list until a match is found or one list runs out of elements. Feinstein argued that this method is the best for solving the problem for large inputs because it efficiently solves two related subproblems.
Methodology: Feinstein focused on the SUBSET-SUM problem, a well-known NP problem. He proposed an algorithm (algorithm A) that could solve this problem in O(2n^2) time. He then argued that algorithm A had the best running-time for large problems and could be improved further by sorting sets S- ∪ (S- + an+1) and S+ instead of sorting sets S- and S+, reducing the running-time to 8 units of time.


Results: Feinstein's algorithm A can solve the Subset-Sum problem in O(2n^2) time, assuming constant-time arithmetic and linear-time sorting. This is an improvement over other methods, as it efficiently solves two related subproblems. This evidence suggests that P ≠ NP, as algorithm A is the best method for solving the problem for large inputs.
Results: Feinstein presented an algorithm (algorithm A) that could solve the SUBSET-SUM problem in O(2n^2) time. He argued that this algorithm had the best running-time for large problems and could be improved further by sorting sets S- ∪ (S- + an+1) and S+ instead of sorting sets S- and S+, reducing the running-time to 8 units of time.


Implications: If Feinstein's method is correct, it would mean that P ≠ NP, which is a long-standing open question in the field of computer science. This would have significant implications for the field, as it would mean that there are problems that can be solved more efficiently by nondeterministic algorithms than by deterministic algorithms. This could potentially lead to new algorithms and techniques for solving complex problems.
Implications: Feinstein's evidence suggests that P ≠ NP. His improved algorithm for the SUBSET-SUM problem shows that it is possible to solve NP problems more efficiently than previously thought, providing new insights into the P vs. NP problem.


Link to Article: https://arxiv.org/abs/0310060v5
Link to Article: https://arxiv.org/abs/0310060v6
Authors:  
Authors:  
arXiv ID: 0310060v5
arXiv ID: 0310060v6


[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:S]]
[[Category:Time]]
[[Category:Time]]
[[Category:Algorithm]]
[[Category:Np]]
[[Category:Problem]]
[[Category:Problem]]
[[Category:Can]]
[[Category:Feinstein]]
[[Category:Np]]

Revision as of 14:48, 24 December 2023

Title: Craig Alan Feinstein's Evidence That P ≠ NP

Abstract: Craig Alan Feinstein, a researcher in computer science, presented evidence that the classes of decision problems that can be solved by deterministic polynomial-time algorithms (P) and nondeterministic polynomial-time algorithms (NP) are not equal. He focused on the SUBSET-SUM problem and proposed an algorithm (algorithm A) that could solve this problem in O(2n^2) time, assuming constant-time arithmetic and linear-time sorting. Feinstein argued that algorithm A had the best running-time for large problems and could be improved further by sorting sets S- ∪ (S- + an+1) and S+ instead of sorting sets S- and S+, reducing the running-time to 8 units of time. This improved strategy was shown to be more efficient than other methods, providing evidence that P ≠ NP.

Main Research Question: Is P = NP?

Methodology: Feinstein focused on the SUBSET-SUM problem, a well-known NP problem. He proposed an algorithm (algorithm A) that could solve this problem in O(2n^2) time. He then argued that algorithm A had the best running-time for large problems and could be improved further by sorting sets S- ∪ (S- + an+1) and S+ instead of sorting sets S- and S+, reducing the running-time to 8 units of time.

Results: Feinstein presented an algorithm (algorithm A) that could solve the SUBSET-SUM problem in O(2n^2) time. He argued that this algorithm had the best running-time for large problems and could be improved further by sorting sets S- ∪ (S- + an+1) and S+ instead of sorting sets S- and S+, reducing the running-time to 8 units of time.

Implications: Feinstein's evidence suggests that P ≠ NP. His improved algorithm for the SUBSET-SUM problem shows that it is possible to solve NP problems more efficiently than previously thought, providing new insights into the P vs. NP problem.

Link to Article: https://arxiv.org/abs/0310060v6 Authors: arXiv ID: 0310060v6

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