This document provides a comprehensive overview of the P vs NP problem. It discusses the problem statement, definitions of complexity classes P and NP, methods that have been used to attempt to solve P vs NP like diagonalization and relativization, computational circuits, approximation algorithms, quantum computing, and geometric complexity theory. It also mentions surveys by Sipser and Fortnow on P vs NP, as well as statistics on opinions of experts about whether P equals NP or not. The document concludes that P vs NP has significantly impacted computer science and other fields and that more work is still needed to solve this important open problem.

1. A Comprehensive View on
P vs NP
Name : Abhay Nitin Pai
Reg. No. : 13MCC1032
Research Facilitator : Dr. Jeganathan L.

2. 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 ?

3. My recepie, My Ingredients and My Kheer

4. 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]

5. 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.
P
NP-C

6. [4]

7. Deterministic Time Algorithm A
[3]

8. 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 ?

9. Methods Adapted to solve P vs NP
● Diagonalization and Relativization
● Computational Circuit
● Approximation
● Quantum Computing
● Geometric Complexity Theory
● Others

10. 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]

11. 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.

12. 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]

13. 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

14. 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

15. ● 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.

16. ● 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]

17. 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 !!!

18. 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]

19. 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