a comprehensive view on p vs np
TRANSCRIPT
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