A Comprehensive View on P vs NP

Name : Abhay Nitin PaiReg. No. : 13MCC1032Research Facilitator : Dr. Jeganathan L.

Introduction● Problem Statement :

– A strong understanding of P vs NP is essential in order to make an attempt to solve it.

– Sometimes we tend to solve the problem just by getting the basics necessary for the problem which may lead to a dead end.

– But problems like P vs NP are not the one's which could be solved just by getting the basics.

– A comprehensive view on P vs NP is a must.

● Famous survey papers by Michael Sipser[1] and Lance Fortnow[2] already exist. Then why do we need a comprehensive view on P vs NP ?

My recepie, My Ingredients and My Kheer

Quotes and Facts

● The Turing Machine – A synonym to computer scientists for Algorithms

● Challenges– “As long as a branch of science offers an abundance of problems, so

long it is alive; a lack of problems foreshadows extinction or the cessation of independent development” - David Hilbert[1]

– Computer Science has faced many challenging problems some of which have been solved by great Mathematicians and Computer Scientists

– An example would be Entscheidungsproblem(En-shai-dungs-pob-lem) proposed by David Hilbert [1]

What are P and NP ?● P and NP are two different complexity classes defined over two different Turing

Machines. [3]

● P Class : An algorithm is a member of P class if it takes polynomial time to get to an output, deterministically.

NP

● NP Class : An algorithm is a member of NP class if it takes polynomial time to verify an output, non-deterministically.

● A turing Machine can be used synonymously for an Algorithm

● P vs NP is defined over Turing Machine.

PNP-C

[4]

Deterministic Time Algorithm A

[3]

What does P vs NP asks for ?

● Many misinterpretations may come up while understanding P vs NP● For instance assume a 3-SAT problem X

– Give all possible solution– Give one solution– What are the total number of solutions for X

– Give a solution where mi = x such that x Є {0,1}

● What P vs NP asks is that for a problem like X can we find a one or AT LEAST one solution deterministically in polynomial time ?

Methods Adapted to solve P vs NP

● Diagonalization and Relativization● Computational Circuit● Approximation● Quantum Computing● Geometric Complexity Theory● Others

Diagonalization and Relativization

● Diagonalization is a method where an NP language L is constructed so that a set of polynomial time algorithm fails to compute L on a certain input. [2]

● Cantor used diagonalization to prove real numbers are uncountable. [2]

● Similar technique was used by Alan Turing to demonstrate Halting Problem of Turing Machine.[1]

● Problem : It is not known how can a fixed NP machine can simulate an arbitary P machine

● Baker,Gill and Solovay showed no relativizable proof can settle P vs NP in either direction[2]

Computational Circuit● We can prove P ≠ NP by showing that a there exist

no small circuit that would compute a complete problem(The number of gates bounded by a fixed polynomial input)

● Saxe, Frust and Sipser showed that small circuits cannot solve parity on a small circuits that have fixed number of layers of gates [2]

● Also Razberov proved that the problem of clique does not have small monotone circuit [2][1]

● He later himself showed that the proof would fail miserably if NOT gate were to be added.[2][1]

● Computational Circuit has shown a very slow development, but for solving P vs NP it has proven to be the closest ally.

Approximation

● Approximation algorithms provide a procedure to get to a near perfect answers.

● Though they do not yeild a the perfect solution, the solution that approximation algorithm provides can be a compromise for saving time.

● Ex: Ant Colony Algorithm, Approx Vertex Cover[4], etc.

● The degree of approximation helps in comparing two different approximation algorithm. [4]

Quantum Computing

● Peter Shor showed how to factor numbers using a hypothetical quantum computer. [2]

● Lov Grover deviced a quantum algorithm that works on general NP problems but does not achives an optimal speed-up and there are evidences that the algorithm cannot be improved any further.[2]

● Though it is unlikely that a quantum computer will help to solve P vs NP, still Quantum Computing can provide a huge advantage over the Classical Computing.

● FACTS

Geometric Complexity Theory

● A different approach to measure the complexity of Algorithm

● Developers : Ketan Mulmuley and Milind Sohoni

● Geometric Complexity Theory is promised to be a right catalyst to solve P vs NP.[5]

● An explaination : How to prove P ≠ NP

● A dedicated P vs NP page has been maintained by GJ Woeginger from Eindhoven University of Technology.

● This page has provided links to authenticated digital documents to understand the basics

● Currently there are 104 descriptions for the ongoing research on P vs NP along with the links to the papers.

● The research on these links have either been accepted or still under review or in progress.

● Equal : 55 (52.38095240 %)● Not Equal : 45 (42.85714290 %)● Undecidable : 1 (0.952380952 %)● Unprovable : 2 (1.904761900 %)● Equal and not Equal : 1 (0.952380952 %)

[6]

Conclusion

● P vs NP has provided gateway to many new paradigms, problems, solutions, theories, etc.

● The impact of this problem is not only on Computer Science, but also in several other fields.

● The Research Community : a Non-Deterministic Turing Machine.

● A negation proof for a domain is more helpful than an assertion.

● We need the right catalyst !!!

References

● [1] Michael Sipser, “The History and Status of P vs NP Question",24th Annual ACM Symposium on the Theory of Computing, 1992, pp. 603-619

● [2] Lance Fortnow, “The Status of P vs NP Problem”, communications of ACM, Vol. 52 No. 9, Pages 78-86, September 2009

● [3] Richard M. Karp, “Reducibility Among Combinatorial Problems”, 1972

● [4] Introduction to Algorithms, third edition, ISBN: 9780262033848, July 2009

● [5] Ketan D. Mulmuley, “On P vs NP and Geometric Complexity Theory”, Journal of ACM(JACM), Vol. 58 Issue 2, April 2011.

● [6] The P vs NP page [http://www.win.tue.nl/~gwoegi/P-versus-NP.htm]

Acknowledgement● Dr. Jeganathan Sir

– Never follow “Operation successful patient died”

● Prof. Nish V. M.– “You need to make your own kheer, work hard so that others like it”

● Prof. Ummity Shriniwasrao– “Clear your basics”

● Family● Friends