This document discusses the P vs NP problem in computer science. It defines the complexity classes P and NP, where P contains problems solvable in polynomial time and NP contains problems verifiable in polynomial time. The document notes that many important problems like the traveling salesman problem are in NP but not known to be in P. It also discusses deterministic vs non-deterministic algorithms, 3CNF satisfiability as an NP-complete problem, and Cook's theorem which states satisfiability being in P is equivalent to P=NP.

1. By Kirti Gautam

2. CREDITS
• Abdul Bari, YouTube
https://www.youtube.com/watch?v=e2cF8a5aAhE
• Wikipedia
https://en.wikipedia.org/wiki/P_versus_NP_problem
• JavaTpoint
https://www.javatpoint.com/daa-3-cnf-satisfiability

3. Polynomial Time
• Linear Search- n
• Binary Search- log n
• Matrix multiplication- n3
• Merge Sort- n log n
• Insertion Sort – n2
Exponential Time
• 0/1 Knapsack-2n
• Traveling SP -2n
• Sum of subsets – 2n
• Graph Coloring-2n
• Hamiltonian Cycle-2n

4. DETERMINISTIC AND NON DETERMINISTIC
• Deterministic algorithm, for a given particular input, the computer will always
produce the same output going through the same states.
• In Non-Deterministic algorithm, The compiler may produce different output in
different runs. In fact non-deterministic algorithms can’t solve the problem in
polynomial time and can’t determine what is the next step.

5. NON - DETERMINISTIC
• Algorithm Search (A,n,key) {
j=choice();
if(key=A[j]){
write (j);
Success(); }
write();
Failure(); }

6. P AND NP
• P is a class of problems which can be
solved in a polynomial time.
• NP is a class of problems which is no
solved in polynomial time, but can be
solved with exponential time. These are
the problems for which the correctness
of the problem is difficult to check
rather than solving it.
NP
P

7. CNF SATISFIABILITY
• In 3CNF SAT, we have at least 3 clauses, and in clauses, you will have almost 3
literals or constants.
• For example –
(XvYvZ) ᶺ (XvYvZ) ᶺ (XvYvZ)

8. NP HARD AND NP COMPLETE
• NP Hard problems are those problems which
can be reduced by Satisfiability problem.
• SF is directly proportional to L (any hard
problem).
• NP complete problems are NP hard problems
whose non-deterministic solution is known.

9. COOK’S THEOREM
• It says that satisfiability is in P if and only if P=NP.