NP-Completeness: Understanding the Complexity of Decidable Languages, Slides of Theory of Automata

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NP-completeness

P and NP

languages that can be decided on a TM

with polynomial running time

P =

(problems that admit efficient algorithms)

NP = languages decidable on a nondeterministic

TM with polynomial running time

(solutions can be checked efficiently)

decidable

NP P

We believe that NP is bigger than P,

but we are not 100% sure

Equivalence of certain NP languages

• We strongly suspect that problems like

CLIQUE, SAT, etc. require time ≈2 n^ to solve

• We do not know how to prove this, but what

we can prove is that

If any one of them is tractable (by a polynomial-

time algorithm), then all of them are tractable

Equivalence of certain NP languages

• All these problems are as hard as one another
• Moreover, they are at the “frontier” of NP
  • They are at least as hard as any problem in NP NP

P

Polynomial-time reductions

• Theorem

If CLIQUE has a

polynomial-time

algorithm, so does IS

CLIQUE = {(G, k ): G is a graph with a clique of k vertices}

IS = {(G, k ): G is a graph with an independent set of k vertices}

1 2

3 4

{1, 4}, {2, 3, 4}, {1} are cliques {1, 2}, {1, 3}, {4} are independent sets

Polynomial-time reductions

• Proof: Suppose CLIQUE has an algorithm A

• We want to use it to solve IS

If CLIQUE has a polynomial-time

algorithm, so does IS

G, k

reject if not

accept if G has IS of size k

A

G ’ , k ’ reject if not

accept if G’ has clique of size k

Reducing IS to CLIQUE

G, k R G’, k’

On input ( G , k ) Construct G ’ by flipping all edges of G Set k ’ = k Output ( G ’, k ’)

independent sets in G (^) cliques in G’

If G ’ has a clique of size k ’, then G has an IS of size k

If G ’ does not have a clique of size k ’,

then G has no IS of size k ✓

Reduction recap

• We showed that

by converting an imaginary algorithm for

CLIQUE into one for IS

• To do this, we came up with a reduction that

transforms instances of IS into ones for CLIQUE

If CLIQUE has a polynomial-time algorithm,

so does IS

Meaning of poly-time reductions

• Saying L reduces to L ’ means L is easier than

L’

• In other words, if we can solve L’ , then we can also

solve L

• Theorem

• Proof

If L reduces to L’ and L’ ∈ P, then L ∈ P

x y

x ∈ L y ∈ L ’

R algorithm for L’

acc rej

algorithm accepts

Notes about reductions

• The direction of the reduction is very

important

• This makes sense, because “A is easier than B” and

“B is easier than A” mean different things

• However, it is possible that L reduces to L’

and L’ reduces to L

• This means that L and L’ are as hard as one

another

• For example, IS and CLIQUE reduce to one

Interpretation of Cook-Levin

Theorem

• Optimistic view:

• Pessimistic view:

If we manage to solve SAT, then we can also solve

CLIQUE, scheduling, and almost anything

Since we do not believe P = NP, it is unlikely that

we will ever have a fast algorithm for SAT