Proving Lower Bounds to answer the P versus NP Question

Proving Lower Bounds to answer the P versus NP Question

Prerna Thakral

George Mason University

Computer Science

How did we get P versus NP?• Turing developed a model for his

computational theory, but it failed to account for• time • memory

• Divided theoretical computer science problems into two classes – P and NP.

BACKGROUND INFORMATION

What does the P class hold?• P is for Polynomial Time.

• Problems whose positive solutions can be solved in an amount of time that is polynomial to the size of the input.

What does NP class hold? • NP stands for Nondeterministic Polynomial

Time.

• Problems that can be verified in polynomial time.

Relationship between P and NP

How did P and NP come to existence?

• P became the class of those problems that were “realistically solvable.”

• NP class became important once the computer scientists realized the large number of problems contained in it that still needed to be solved.

Importance and Consequences• A proof of P equals NP:

• will lead to efficient methods for solving some important NP problems

• fundamental to many fields such as mathematics, biology, etc.

• A proof of P does not equal NP:• will show, in a formal way, that many common

problems that can be verified easily and efficiently cannot be solved efficiently.

CURRENT RESEARCH - PROVING LOWER BOUNDS

Limitations in Problem• Is seen when computer scientists

have tried to prove lower bounds on the complexity of problems in the class, NP.

• Methods such as:• Diagonalization• the use of pseudo-random generators • Circuits

are currently being used to prove lower bounds.

Terminology• Diagonalization - a basic technique

used to prove that the set A does not belong to complexity class C.

• Combinatorial Circuit - a sequence of instructions, each producing a function based on the already obtained previous functions.

Goal of the Research • Develop a new technique in determining lower

bounds by conducting an experiment between the current techniques, diagonalization, and combinatorial circuits and comparing the results to answer the P versus NP question.

EXPERIMENTMethods and Procedures

Constants in the Experiment

• Lower bounds will be computed on the Traveling Salesman Problem, an NP-complete problem.

• The traveling salesman problem will include 15 cities to be toured.

Trials One and Two• Diagonalization Technique - a set and function

A will be established.

• Circuit Technique - a circuit tree will be created from previously defined functions.

Trial Three• Set A will use the diagonalization technique and

the combinatorial circuits simultaneously to achieve higher efficiency.

EXPERIMENTAssessment

Efficiency • Efficiency is:

• measured by the time required to complete the technique and analyze the results to see if the technique produced anything meaningful.

• Time required to find a set A will be important.

• The time required to create these various circuit trees will also be noted.

Success• The experiment will be declared as successful if

the new technique which uses the two current techniques simultaneously is seen to be more efficient than the other techniques in proving lower bounds.

EXPERIMENTNext Steps

Prove P equals/does not equal NP

• I will be able to determine that the Traveling Salesman Problem is a part of the P class.

• This will allow me to determine which other NP-complete problems can be solved in polynomial time, making them a part of the P class.

Publish Results• If successful, I would like to publish my findings

in scholarly journals such as:• IEEE Journal • Communications of ACM IEEE

Journal