NP ≡ Polynomial NTM I

NP ≡ Polynomial NTM I

Polynomial verifier ⇔ polynomial NTM Idea:

“⇒” NTM by guessing certificate

“⇐” using NTM’s accepting branch as certificate Proof:

“⇒”: now we have a verifier V in time n^k

NP ≡ Polynomial NTM II

Recall the definition below

A = {w | V accepts hw , ci for some strings c}

We have

|c| ≤ n^k

because to handle hw , ci in n^k, |c| should be bounded by n^k

Use an NTM to

1 nondeterministically select c

2 run V on hw , ci

NP ≡ Polynomial NTM III

That is, run c in parallel and each is polynomial We have that for any w ∈ A, the NTM accepts it in polynomial time

“⇐”: now w is accepted by a polynomial NTM Let c be any accepting branch

Note that for polynomial NTM, each branch is polynomial

Then we have a verifier V that handles input hw , ci in polynomial time

Note: the definition of V requires only “some c.”

So finding one is sufficient

SUBSET-SUM I

Given x₁, . . . , x_k and t, is sum of a subset = t?

Formally

{hs, ti | s = {x₁, . . . , x_k} and ∃ {y₁, . . . , y_l} ⊂ {x₁, . . . , x_k} such that X

y_i = t}

Example

h{4, 11, 16, 21, 27}, 25i OK as 4 + 21 = 25

SUBSET-SUM II

Note: allow repetition here

h{4, 11, 11, 16, 21, 27}, 25i We prove that this problem is NP Idea: the subset is the certificate.

Consider any input

hhs, ti, ci We

1 check if P c_i = t

SUBSET-SUM III

2 check if all c_i ∈ s

If both pass, accept; otherwise, reject Here

length of c < length of s

The verification can be done in polynomial time

P vs. NP I

Roughly

P: problems decided quickly NP: problems verified quickly Question: is P = NP?

This is one of the greatest unsolved problems Most believe P 6= NP

NP-completeness I

It has been shown that some problems in NP are related

For certain NP problems:

If ∃ a polynomial algorithm for one NP ⇒ P = NP These problems are called NP-complete problems They are useful to study the issue of P versus NP To prove P 6= NP: only need to focus on

NP-complete problems

To prove P=NP: need only polynomial algorithms for an NP-complete problem