BU CS 332 Theory of Computation Lecture 22: Reading: NP - - PowerPoint PPT Presentation

BU CS 332 – Theory of Computation

Lecture 22:

• NP‐Completeness Example
• Space Complexity
• Savitch’s Theorem

Reading: Sipser Ch 8.1‐8.2

Mark Bun April 22, 2020

NP‐completeness

Definition: A language is NP‐complete if 1) and 2) Every language is poly‐time reducible to , i.e.,

• (“

is NP‐hard”) Theorem: If and

• for some NP‐complete

language , then is also NP‐complete

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(3‐CNF Satisfiability)

Definition(s):

• A literal either a variable of its negation

,

• A clause is a disjunction (OR) of literals

Ex.

• A 3‐CNF is a conjunction (AND) of clauses where each

clause contains exactly 3 literals Ex.

• …
• Cook‐Levin Theorem:

is NP‐complete

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Some general reduction strategies

• Reduction by simple equivalence
• Ex. 𝐽𝑂𝐸𝐹𝑄𝐹𝑂𝐸𝐹𝑂𝑈 𝑇𝐹𝑈 𝑊𝐹𝑆𝑈𝐹𝑌 𝐷𝑃𝑊𝐹𝑆

and 𝑊𝐹𝑆𝑈𝐹𝑌 𝐷𝑃𝑊𝐹𝑆 𝐽𝑂𝐸𝐹𝑄𝐹𝑂𝐸𝐹𝑂𝑈 𝑇𝐹𝑈

• Reduction from special case to general case
• Ex. 𝑊𝐹𝑆𝑈𝐹𝑌 𝐷𝑃𝑊𝐹𝑆 𝑇𝐹𝑈 𝐷𝑃𝑊𝐹𝑆
• Gadget reductions
• Ex. 3𝑇𝐵𝑈 𝐽𝑂𝐸𝐹𝑄𝐹𝑂𝐸𝐹𝑂𝑈 𝑇𝐹𝑈

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Independent Set

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An independent set in an undirected graph 𝐻 is a set of vertices that includes at most one endpoint of every edge.

𝐽𝑂𝐸𝐹𝑄𝐹𝑂𝐸𝐹𝑂𝑈 𝑇𝐹𝑈

• 𝐻, 𝑙 𝐻 is an undirected graph containing an independent set with 𝑙 vertices
• Is there an independent set of size  6?
• Yes.
• Is there an independent set of size  7?
• No.

independent set

Independent Set is NP‐complete

1) 2) Reduce

• Proof. “On input 𝜒 , where 𝜒 is a 3CNF formula,
• 1. Construct graph 𝐻 from 𝜒
• 𝐻 contains 3 vertices for each clause, one for each literal.
• Connect 3 literals in a clause in a triangle.
• Connect literal to each of its negations.

2. Output 𝐻, 𝑙 , where 𝑙 is the number of clauses in 𝜒.”

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Example of the reduction

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𝜒 𝑦 ∨ 𝑦 ∨ 𝑦 ∧ 𝑦 ∨ 𝑦 ∨ 𝑦 ∧ 𝑦 ∨ 𝑦 ∨ 𝑦

Proof of correctness for reduction

Let 𝑙 = # clauses and 𝑚 = # literals in 𝜒 Claim: 𝜒 is satisfiable iff 𝐻 has an ind. set of size 𝑙 ⟹ Given a satisfying assignment, select one literal from each

• triangle. This is an ind. set of size 𝑙

⟸ Let 𝑇 be an ind. set of size 𝑙

• 𝑇 must contain exactly one vertex in each triangle
• Set these literals to true, and set all other variables in an arbitrary

way

• Truth assignment is consistent and all clauses satisfied

Runtime: 𝑃𝑙 𝑚 which is polynomial in input size

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Space Complexity

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Complexity measures we’ve studied so far

• Deterministic time TIME
• Nondeterministic time NTIME
• Classes P, NP

Many other resources of interest: Space (memory), randomness, parallel runtime / #processors, quantum entanglement, interaction, communication, …

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Space analysis

Space complexity of a TM (algorithm) = maximum number

• f tape cell it uses on a worst‐case input

Formally: Let . A TM runs in space if on every input

∗,

halts on using at most cells For nondeterministic machines: Let . An NTM runs in space if on every input

∗,

halts on using at most cells on every computational branch

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Space complexity classes

Let A language if there exists a basic single‐ tape (deterministic) TM that 1) Decides , and 2) Runs in space A language if there exists a single‐ tape nondeterministic TM that 1) Decides , and 2) Runs in space

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Example: Space complexity of SAT

Theorem: Proof: The following deterministic TM decides using linear space On input where is a Boolean formula:

• 1. For each truth assignment to the variables
• f :

2. Evaluate

• n
• 3. If any evaluation

, accept. Else, reject.

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Example: NFA analysis

Theorem: Let

• ∗

Then

• .

Proof: The following NTM decides

in linear space

On input where is an NTM: 1. Place a marker on the start state of . 2. Repeat times where is the # of states of : 3. Nondeterministically select . 4. Adjust the markers to simulate all ways for to read 5. Accept if at any point none of the markers are on an accept

• state. Else, reject.

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Example

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0,1 ε 2 3 1 1

Space vs. Time

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Space vs. Time

How about the opposite direction? Can low‐space algorithms be simulated by low‐time algorithms?

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Reminder: Configurations

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A configuration is a string where and

∗

• Tape contents =

(followed by blanks )

• Current state =
• Tape head on first symbol of

Example:

• Start configuration:

Accepting configuration: = Rejecting configuration: =

Reminder: Configurations

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Consider a TM with

• states
• tape alphabet
• space

How many configurations are possible when this TM is run on an input

• Observation: If a TM enters the same configuration twice

when run on input it loops forever Corollary: A TM running in space also runs in time

Savitch’s Theorem

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Savitch’s Theorem: Deterministic vs. Nondeterministic Space

Theorem: Let be a function with . Then

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