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# Real-world Applications of a constructive P=NP proof

• Difficulty Level : Basic
• Last Updated : 07 Jul, 2021

Prerequisite : NP-Completeness

Real-world Applications of constructive  P=NP  proof :
The polynomial class of problems, also known as P, are solvable in polynomial time. However, the other class of problems are not solvable in polynomial time but the solution can be verified rather quickly. These are known as non-polynomially solvable deterministic problems.

The P versus NP problem is a major unsolved problem in the domain of computer science. The current assumption is that P!=NP, since the reverse would mean that there is no fundamental gap in recognising a problem and obtaining its corresponding solution. However, many scientists believe that this subject is at a stage where the space algorithm needs more exploration and a definite solution for the P = NP will be obtained. It would imply that the solution to a problem can both be found as well as verified in polynomial time.

A constructive proof of the P = NP problem would imply that the solutions are identified by a specified reasonable bound, a bounding polynomial and a detailed description of the algorithms and their functionality will be available.

The NP-complete problems encompass a wide range of applications and therefore, the real-world applications of the P = NP proof can be both positive as well as negative. If , then we would be able to solve a large number of decisions , searching, counting , sampling as well as optimisation problems with a much greater efficiency. Development of constructive proof, would mean that almost all the NP problems could be solved deterministically in polynomial time. It will provide applications and answers to a large number of concrete problems, such as designing a better bridge, or finding a better drug, and also providing redressal to a large number of scientific theories or “laws of nature”. It would change the dynamics of the engineering problems.

We could have some of the greater real-world applications of the same :

• Prime Factorization Algorithms –
Since a large number of cryptographic algorithms, for instance, the RSA, would be easy to develop and compute. The RSA encryptions would be easy to understand even with a small key size.
• Vehicle Routing (Travelling Salesman Problem) –
The transportation sector makes up 10% of the EU’s GDP. Being able to find the shortest path to any possible pair of points between cities, would make it very easy to deliver and travel and would save a huge amount of money.
• Facility Location –
It would be possible to find an optimal place for the relocation of the factories, where the supplies could be easily transported to the stores. This would increase efficiency while reducing factory expenses.
• Circuit designing –
Large boolean circuits are solved based on approximations. Hardware dependency can be largely reduced if the circuit solving can be efficiently computed. Minimization would be easily done.
• Scheduling Algorithms –
The conflicts that arise with day to day problems, like deciding on exam dates, organization of some important events, would easily be possible to minimize them.
• Plane ticket booking –
Distance modelling is an important factor to be considered while booking plane or train tickets. If P = NP, then it would be possible to find the optimal sequence of points to follow while visiting the cities. It would be easy to have the cheapest sequence of flights that visit some number of cities ending at the desired city. For instance, it may be more feasible to go from A -> B -> C, than reach A -> C directly.
• Compilers –
Most compilers use graph coloring for register allocation in their architecture. The register allocation will be optimised to include a large number of registers for storage. The existing solutions, for instance, the chordal graph, provide approximate solutions, where an exact solution would mean faster processing to a great extent.
• Graph optimisations –
An independent set, 3SAT , graph coloring would lead to its application in a wide range of arenas.
• Computer-aided designs and artificial intelligence algorithms’ design and implementation would become much easier.
• Protein structure prediction problems would be easily solvable in polynomial time, which would lead to advances in technology and the sciences.
• Cryptography –
Development of a constructive proof for a P = NP problem, for instance, a 3 SAT problem, would break a large number of existing cryptographic systems :
• Public key cryptography
• Symmetric ciphers
• Cryptographic hashing, including the bitcoin system
• Operation research –
For instance, the development of a polynomially bounded algorithm for this problem would improve the logistics to a great extent.
• Logical and very short proofs of mathematical theorems can be easily found using the Occam razor principle. It would further lead to the ease of the process of learning by finding the smallest program consistent with the data.
• Weather predictions and forecasts will be made with greater accuracy.
• Artificial intelligence –
Translation will be easily done since we would have the correct output from a finite set of inputs. Perfect speech recognition can be done easily.
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