## Algorithms & Optimization

The P=NP? problem was emerged in 1971. This is one of the longstanding unresolved problems in computer science. Its solution is directly related to the determination of the time complexity of NP-complete problems. Not all problems can be solved quickly.

**So, it is very difficult to figure out exactly which ones could be solved and which ones couldn’t.**

Steve Cook and Dick Karp decided that any efficient algorithm is that which runs in polynomial time:

**P – the class of polynomial-time problems****The time complexity of these problems is***O(n^*for some constant^{c})*c*. Cook, Karp, and others defined another class of so-called NP-complete problems all of which exhibit precisely the same phenomena:

Stephen Cook. The complexity of theorem-proving procedures. In Conference Record of Third Annual ACM Symposium on Theory of Computing, pages 151–158, 1971.[Referat_on_P=NP]

**no NP-complete problem can be solved by any known polynomial algorithm****if there is a polynomial algorithm for any NP-complete problem, then there are polynomial algorithms for all NP-complete problems****NPC-the class of NP-complete problems****Time Complexity of NP-complete problems has not been decided yet**

This is one of the most difficult unresolved problems in Computer Science.

It’s accepted that the problem Does P=NP? has been open since it was posed in 1971.## It is the one of “The Millennium Prize Problems” established by Clay Mathematics Institute of Cambridge, Massachusetts (CMI).

**The nature of NPC Problems**

More than 1,000 diverse algorithmic problems in NPC exhibit precisely the same phenomena:

*no NP-complete problem can be solved by any known polynomial algorithm;**if there is a polynomial algorithm for any NP-complete problem, then there are polynomial algorithms for all NP-complete problems.*

*NPC class problem contains around 1000 different algorithmic problems*- Hamiltonian path problem
- Traveling salesman problem
- Scheduling and Matching problem
- Coloring Maps and Graphs
- 3SAT etc.

*The problem exists in the following areas**Combinatorics**Operation Researches**Economics**Graph & Game Theory**Statistics**Logic**High- Energy Physics and X-Ray crystallography**Protein Design etc.*

Any simple Timetable Problem is NP-complete. After about seventeen years of research, Karlen G. Gharibyan finally invented the Peaceful Coexistence Algorithm. This is a polynomial time algorithm that solves the General Timetable Problem (GTP) for any instance.

## Karlen G. Gharibyan proved that P=NP. Nobody rejected it.

This means, that each problem in NP may be solved in polynomial time, including

3SATproblem.

Here is the demo how works Peaceful Coexistence Algorithm®