Lecture 6 - Polynomial Time Reductions

Question 1

What is a polynomial time reduction?

Answer 1

a mapping from one decision problem to another such that:

for every instance f(problem 1) of problem 2 can be constructed in poly-time

(instance of problem 2 can be constructed in polytime)

f(problem 1) is a yes instance of problem 2 if and only if problem 1 is a yes instance

Question 2

What is the way to write PTRs?

Answer 2

PROBLEM 1 ∝ PROBLEM 2

Question 3

What is the transitivity property of ptrs?

Answer 3

reduciton from 1 to 2 and 2 to 3 means that 1 can be reduced to 3

Question 4

Does the thing we reduce to have the same answer as the original instance?

Answer 4

yes (either yes or no)

Question 5

What is the relevance property of PTRs?

Answer 5

p1 ∝ p2

• this means p1 is no harder than p2
  if p2 is in P, then p1 must be in P
• if we can solve p2 then we can solve p1 without much extra effort
• p2 could be hrader to solve (we only map to easy to solve instances of p2)

Question 6

What can we reduce?

Answer 6

Any np-complete problem can be reduced to another.