1. Proving Lower Bounds to
answer the P versus NP
Question
Prerna Thakral
George Mason University
Computer Science

2. BACKGROUND
INFORMATION

3. How did we get P versus NP?
• Turing developed a model for his
computational theory, but it failed to
account
• time
• memory
• From this – scientists divided
theoretical computer science
problems into two classes – P and
NP.

4. What does the P class hold?
• P is for Polynomial Time.
• Consists of all those problems whose positive
solutions can be solved in an amount of time
that is polynomial to the size of the input.

5. What does NP class hold?
• NP stands for Nondeterministic Polynomial
Time.
• This class consists of all those problems that
can be verified in polynomial time.

6. Relationship between P and NP

7. Importance and Consequences
• A proof of P equals NP could have striking
practical consequences.
• Will lead to efficient methods for solving some
important NP problems, which are fundamental to
many fields such as mathematics, biology, etc.
• A proof of P does not equal NP will have just as
great consequences.
• Will show, in a formal way, that many common
problems that can be verified easily and efficiently
cannot be solved efficiently.

8. CURRENT RESEARCH -
PROVING LOWER
BOUNDS

9. Limitations in Problem
• A limitation 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 and
circuits are currently being used
to prove lower bounds.

10. Terminology
• Diagonalization is a basic technique
used to prove that the set A does not
belong to complexity class C.
• A combinatorial circuit is a sequence
of instructions, each producing a
function based on the already
obtained previous functions.

11. 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 develop a new technique to answer the
question whether P equals NP.

12. EXPERIMENT
Methods and Procedures

13. Constants in the Experiment
• Lower bounds will be
computed on the
Travelling Salesman
Problem, an NP-
complete problem.
• The travelling salesman
problem will include 15
cities to be toured by
finding a path with the
shortest distance,
visiting each city only
once.

14. Trials One and Two
• Diagonalization technique - a set and function
A will be established and used to show that it
does not belong to the complexity class EXP,
which will conclude that set A is a part of the
complexity class NP.
• A circuit tree will be created from previously
defined functions. Other circuit trees will also
be created by limiting the depth of the tree and
restricting the original set and function A.

15. Trial Three
• Set A will use the diagonalization technique
and the combinatorial circuits simultaneously to
achieve higher efficiency than efficiency that
would have reached by using the two
techniques individually.

16. EXPERIMENT
Assessment

17. Efficiency
• Efficiency will be 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, such that it does not
belong to the complexity class, EXP will be
important.
• The time required to create these various circuit
trees will also be noted, depending on whether the
depth of the tree was limited or if the original
function itself was restricted.

18. 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.

19. EXPERIMENT
Next Steps

20. Prove P equals/does not equal NP
• By knowing how to restrict my classes, P and
NP, I will be able to determine that the
Travelling 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.