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# About Summarization Of Recent Work About P Versus NP Problem (Essay Sample)

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Summarization of recent work about P versus NP problem

The P versus NP problem has been a major topic of discussion among computer experts. With the quickly developing area of circuit complexity, many researchers believed that the P versus NP problem would be settled soon; whether every algorithmic problem with efficiently verifiable solutions has efficiently solvable solutions. Unfortunately, circuit complexity and other techniques have stalled thereby pushing away the hope of ever separating the P and NP problem.

Nevertheless, the computer science field has dramatically developed in the recent four decades ago since Steve Cook’s NP-completeness paper presentation titled “The Complexity of Theorem-proving Procedures” in early May 1971.

The relation between complexity classes P and NP is studied in computational complexity theory. The most common resources are time (how many steps it takes to solve the problem) and space (how much memory it takes to solve the problem). In this theory, the P consists of all those decision problems that can be solved in a deterministic sequential machine in an amount of time that is polynomial in size of the input; the NP consists of those decision problems whose solution can be verified in a polynomial time given the right information, P ⊆ NP. This paper sought to examine some of the recent contributions to the problem.

Opinions on the recent work

In the year 2009, Zeilberger, in his attempt to contribute to the NP problem, established proof that that P=NP. He came up with an algorithm which was capable of providing a solution to the Subset Sum problem. His approach involved converting the problem into solving an underlying integral. He argued that adopting rigorous interval analysis, instead of non-rigorous floating point computations, it was possible to establish an approximate solution to the integral, while also establishing error bounds, thereby finding answers to the problem in polynomial time. The rigorous estimate, involved answering more than ten thousand Linear Programming problems, each with more than one hundred thousand variables (Zeilberger 10). The credibility of his findings was supported by independent checks carried out by four other different computers, running on different platforms and different programming languages.

Blinder (2009) in his publication on possible new approach to solving open problems in computer science opined that P is not equal to NP. To support his position, Blinder argued that there existed language variation between the two Classes of questions. He provided proof that indeed there existed a language that was held in NP but was not contained in co-NP which by implication could be translated to P not being equal to NP. His findings however cannot be used to generalize the conclusion since they were not based on any empirical studies: He rejected the hypothesis that P was Equal to NP by basing on theoretical arguments (Blinder 16). In fact, Blinder himself later pointed out that he had established serious procedural flaws in his work.

The most recent research on the P vs NP problem was done by LaPlante in March 2015, in his research, he attempted to develop a polynomial time algorithm for offering solutions to the clique problem such as establishing whether there existed a complete sub-graph for a given undirected graph vertices and edges. He came up with an algorithm which could efficiently evaluate the problem K-clique, highest number of cliques as well as the entire enumeration of the maximal cliques for individual graphs (Laplante 6). His approach was sharply criticized by (Cardenas, Holts and Meyers 19) in their paper on refutation of the Clique based P=NP as proposed by Laplante. They argue that the algorithms proposed by (Laplante 4) were flawed and could not be relied upon to yield desirable outcomes. Unfortunately, the researche...

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Summarization of recent work about P versus NP problem

The P versus NP problem has been a major topic of discussion among computer experts. With the quickly developing area of circuit complexity, many researchers believed that the P versus NP problem would be settled soon; whether every algorithmic problem with efficiently verifiable solutions has efficiently solvable solutions. Unfortunately, circuit complexity and other techniques have stalled thereby pushing away the hope of ever separating the P and NP problem.

Nevertheless, the computer science field has dramatically developed in the recent four decades ago since Steve Cook’s NP-completeness paper presentation titled “The Complexity of Theorem-proving Procedures” in early May 1971.

The relation between complexity classes P and NP is studied in computational complexity theory. The most common resources are time (how many steps it takes to solve the problem) and space (how much memory it takes to solve the problem). In this theory, the P consists of all those decision problems that can be solved in a deterministic sequential machine in an amount of time that is polynomial in size of the input; the NP consists of those decision problems whose solution can be verified in a polynomial time given the right information, P ⊆ NP. This paper sought to examine some of the recent contributions to the problem.

Opinions on the recent work

In the year 2009, Zeilberger, in his attempt to contribute to the NP problem, established proof that that P=NP. He came up with an algorithm which was capable of providing a solution to the Subset Sum problem. His approach involved converting the problem into solving an underlying integral. He argued that adopting rigorous interval analysis, instead of non-rigorous floating point computations, it was possible to establish an approximate solution to the integral, while also establishing error bounds, thereby finding answers to the problem in polynomial time. The rigorous estimate, involved answering more than ten thousand Linear Programming problems, each with more than one hundred thousand variables (Zeilberger 10). The credibility of his findings was supported by independent checks carried out by four other different computers, running on different platforms and different programming languages.

Blinder (2009) in his publication on possible new approach to solving open problems in computer science opined that P is not equal to NP. To support his position, Blinder argued that there existed language variation between the two Classes of questions. He provided proof that indeed there existed a language that was held in NP but was not contained in co-NP which by implication could be translated to P not being equal to NP. His findings however cannot be used to generalize the conclusion since they were not based on any empirical studies: He rejected the hypothesis that P was Equal to NP by basing on theoretical arguments (Blinder 16). In fact, Blinder himself later pointed out that he had established serious procedural flaws in his work.

The most recent research on the P vs NP problem was done by LaPlante in March 2015, in his research, he attempted to develop a polynomial time algorithm for offering solutions to the clique problem such as establishing whether there existed a complete sub-graph for a given undirected graph vertices and edges. He came up with an algorithm which could efficiently evaluate the problem K-clique, highest number of cliques as well as the entire enumeration of the maximal cliques for individual graphs (Laplante 6). His approach was sharply criticized by (Cardenas, Holts and Meyers 19) in their paper on refutation of the Clique based P=NP as proposed by Laplante. They argue that the algorithms proposed by (Laplante 4) were flawed and could not be relied upon to yield desirable outcomes. Unfortunately, the researche...

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