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P Vs NP Solution – Advances in Computational Complexity, Status and Future Scope International Journal of Computer Applications (0975 – 8887) Volume 177 – No. 9, October 2019 P vs NP Solution – Advances in Computational Complexity, Status and Future Scope Amit Sharma Sunil Kr. Singh AMIE CSE Research Scholar CSE Department, CCET The Institution of Engineers (INDIA), India Degree Wing, Chandigarh, India ABSTRACT 2. THE P vs NP PROBLEM The significance & stature of the P vs NP problem is so Study of algorithms is a fundamental task in computer imperative that even the failed attempts at proof have science. Researchers & programmers aim to develop efficient furnished unprecedented breakthroughs and valuable insights. algorithms for data processing. The efficiency of an algorithm While the scientists and researchers do not expect the problem M is measured in terms of complexity function f(n) which to be solved in foreseeable future, the P vs NP question has outputs the required running time or storage space in term of been the harbinger of advancement of the theory of size n of the input data, relative to a number of key operations computation and complexity theory in particular. Multitude of it has to manipulate. By bounding the time or space research papers have been published on number of topics requirement and model of computation deterministic or non- which have begged numerous accolades and awards. This deterministic Turing machine, algorithms have been classified paper presents and highlights a non-technical review of series into various complexity classes. of complex mathematical research and enlists the notable awards & advances from each subsequent effort. The paper The general classes of mathematical problems which can be also presents the limitations of existing and proposed solved by a deterministic Turing machine in a polynomial techniques and highlights the direction of active future time are classified as P problems. For instance, the time research towards P vs NP solution. complexity of bubble sort algorithm for an unsorted array of n numbers is f(n2) and thus its complexity lie in P class. So, the Keywords class P is the class of decision problems that are easy for computers to solve. P vs NP, Cryptography, Complexity theory, P vs NP attempts and limitations, Geometric complexity theory, quantum The general classes of a mathematical problem whose computing, One-way function, incompleteness theorem. solution is efficiently verifiable by a deterministic Turing machine in polynomial time but not easily solved are 1. INTRODUCTION classified as NP problems. NP stands for non-deterministic P versus NP is perhaps the most fundamental and most polynomial time i.e. they can be solved by non-deterministic essential contemporary problem in mathematics and computer Turing machine in polynomial time. For instance, a brute science, whose relevance and significance with time has force algorithm for cracking passwords containing n symbols grown beyond bounds. In the first half of the twentieth in the worst possible case will have to go through all the century, pioneering and prominent research papers on permutations of n symbols. The execution time of such decidability, computability & complexity of algorithmic problem is not in polynomial time but in exponential time. So, problems led to the development of computational complexity even with fastest imaginable machines, for the modest value theory. In 1936, Alan Turing in his historic paper “On of n (say 30) it would take billions of years to find it. computable numbers, with an application to the However, for a given password, it is very easy to verify if it [1] Entscheidungsproblem” presented a formal mathematical correct or incorrect. model of a computation machine Turing Machine which could simulate any algorithm. Thus, he is regarded as the father of NP-Hard is the class of problems such that every problem in modern theoretical computer science. NP can be reduced to any problem in NP-hard, i.e. complexity of NP-hard problems is greater than or equal to the hardest In 1961, Stephen Cook in his landmark paper “The complexity problems in NP. Some problems in NP-Hard are in NP but [2] of theorem-proving procedure” introduced the P vs NP there are problems in NP-hard which are outside of NP, some problem. Since then, researchers, complexity theorists, of which may not even be decidable. NP-complete is the class mathematicians, programmers, and amateurs have been of problems which lie in both NP and NP-Hard. grappling with the legendary problem and still the possibility of the resolution eludes the brightest of minds. The P versus NP problem is to ascertain - This question is so fundamental that Clay Mathematics Either P = NP i.e. problems being easy to solve are Institute, California has named it among 7-millennium prize the same as problems having solutions that are easy problems[3] and offered $1 million to anyone who provides a to check. To put it simply, the P = NP problem is verified proof. With each attempt the complexity of the the search for a way to solve problems that require problem seems to increase, thereby reiterating the general full exhaustion without actually having to try each consensus on the limitation of existing techniques, thus of colossal combinations. reinstating the requirement of a novel technique. Despite the Or P i.e. to prove that there are some failures to find a proof, the incessant quest for the Holy Grail problems that are easily verified but not easily of mathematics and computer science, P vs NP has solved. In other words, to prove that NP complete bequeathed the beacon amidst the haze of uncertainty over problem will never have efficient solutions. computational complexity perspectives. 25 International Journal of Computer Applications (0975 – 8887) Volume 177 – No. 9, October 2019 After decades of study and research, a legitimate proof still clustering and TSP, heuristics and average case complexity of eludes the likes of genius researchers as it does that of the TSP, graph partitioning, narrowing the problem space, amateur enthusiasts. For around 3000 important NP-complete simulated annealing, random algorithms, improved problems no one has managed to prove P = NP, so practical exponential time algorithms, parallelization, dynamic experience overwhelmingly suggests that P≠ NP. But in programming etc. is being used to tackle NP-complete absence of a sound mathematical proof, the legitimacy of the problems. assumption remains questionable. So, P vs NP is still open and up for the challenge, providing an opportunity to the 4. ADVANCES IN COMPLEXITY sharpest minds to gain an access to instant fame and riches. THEORY The P vs NP problems stems from the limitation of 3. SIGNIFICANCE OF P vs NP computation in response to abstract mathematics The relentless fervor, with which mathematicians and Entscheidungsproblem. researchers have endeavored the profound pursuit of unlocking the enigmatic intricacies of P vs NP reflects its 4.1 Entscheidungsproblem: German mathematician David significance. The rationale it draws so much attention is the Hilbert, at the 1928 International Congress of stunning implications of the answer. The resolution of the Mathematics, asked three fundamental questions about the problem either way would foster the understanding of the completeness, consistency, and decidability of mathematics. computation problems and quest for efficient algorithms The third question is famously known by its German name enormously, and have vast practical consequences. The class Entscheidungsproblem – “Is every statement in mathematics of NP-complete problems is extremely rich and diverse. There decidable?” i.e. is there an algorithm which can be applied to are numerous NP-complete problems in combinatorics, every statement that will tell us in finite time whether or not meteorology, economics, biology, mathematics, string the statement is true or false. Until 1930, Hilbert himself processing, mathematical programming, optimization, believed and advocated the answer all three problems is “Yes” artificial intelligence, cryptography, operation research, graph and that there is no such thing as an unsolvable problem. theory, game theory, industrial processes etc. whose efficient 4.2 Gödel's incompleteness theorem: In 1931 Kurt Gödel solution will have far reaching consequences. published a paper On Formally Undecidable Propositions of If P = NP, its implications are so profound that the humanity Principia Mathematica and Related Systems I.[4] In this will have a giant leap. Every aspect of knowledge in art & paper, Gödel’s first incompleteness theorem shows that in any science, from learning to the application will be much more consistent effectively generated formal system, there are efficient. Resource optimization in logistics, human-like statements which can neither be proved nor disapproved and neural network in artificial intelligence, automation of is thus incomplete. Gödel’s second incompleteness theorem mathematical proofs, correct coding of DNA sequence in states any formal system can prove its own consistency if and biology, scientific theory for a given data in physics, market only if it is inconsistent. behavior in economics, design of engineering system with 4.3 Tarski’s undefinability theorem: In 1936, following given constraints, predicting earthquakes in meteorology, Gödel's undecidability and incompleteness proofs, Alfred anything & everything will benefit from fast solution of NP Tarski published Tarski's undefinability theorem[5] which problems. But, it will have its serious repercussions too, says
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