BU CS 332 Theory of Computation Lecture 20: Reading: More on NP - - PowerPoint PPT Presentation

BU CS 332 – Theory of Computation

Lecture 20:

• More on NP
• P vs. NP

Reading: Sipser Ch 7.3‐7.5

Mark Bun April 13, 2020

Goals of complexity theory

Ultimate goal: Classify problems according to their feasibility and inherent computational difficulty Decision problems which can be solved efficiently Can we exhibit general classes of problems which are either in

• r provably not in ?

Some problems provably require exponential time! (Chapter 9) : A fundamental and practically important class of problems which have defied classification, but nevertheless exhibits important structure ( completeness)

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Nondeterministic Time and NP

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Nondeterministic time

Let A NTM runs in time if on every input

,

halts on within at most steps on every computational branch

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Deterministic vs. nondeterministic time

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Deterministic Nondeterministic

accept or reject reject accept

𝒖𝒐

reject accept reject

Deterministic vs. nondeterministic time

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𝑥 𝑥 𝑥 𝑥 Finite control 𝑥 ⊔ # 𝑥 𝑥 1 3 3 7 Input 𝑥 to 𝑂 (read‐only) Simulation tape (run 𝑂 on 𝑥 using nondeterministic choices from tape 3) Address in computation tree

Theorem: Let be a function. Every NTM running in time has an equivalent single‐tape TM running in time Proof: Simulate NTM by 3‐tape TM

Deterministic vs. nondeterministic time

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Theorem: Let be a function. Every NTM running in time has an equivalent single‐tape TM running in time Proof: Simulate NTM by 3‐tape TM

• # leaves:
• # nodes:

Running time: To simulate one root‐to‐node path: Total time:

input simulation address

Deterministic vs. nondeterministic time

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Theorem: Let be a function. Every NTM running in time has an equivalent single‐tape TM running in time Proof: Simulate NTM by 3‐tape TM in time We know that a 3‐tape TM can be simulated by a single‐ tape TM with quadratic overhead, hence we get running time

= · =

Difference in time complexity

Extended Church‐Turing Thesis: At most polynomial difference in running time between all (reasonable) deterministic models At most exponential difference in running time between deterministic and nondeterministic models

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Nondeterministic time

Let A NTM runs in time if on every input

,

halts on within at most steps on every computational branch is a class (i.e., set) of languages: A language if there exists an NTM that 1) Decides , and 2) Runs in time

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NTIME explicitly

A language if there exists an NTM such that, on every input

∗

• 1. Every computational branch of

halts in either the accept or reject state within steps 2. iff there exists an accepting computational branch of

• n input

3. iff every computational branch of rejects on input (or dies with no applicable transitions)

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Complexity class

Definition: is the class of languages decidable in polynomial time on a nondeterministic TM

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Hamiltonian Path

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The following nondeterministic algorithm accepts in time

. , where

, adjacency matrix encoding On input : (Vertices of are numbers )

• 1. Nondeterministically guess a sequence
• f numbers
• 2. Check that
• is a permutation: Every

number appears exactly once

• 3. Check that

, , and there is an edge from every to

• 4. Accept if all checks pass, otherwise, reject.

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An alternative characterization of

“Languages with polynomial‐time verifiers” A verifier for a language is a deterministic algorithm such that iff there exists a string such that accepts Running time of a verifier is only measured in terms of is a polynomial‐time verifier if it runs in time polynomial in

• n every input

(Without loss of generality, is polynomial in , i.e.,

for some constant

)

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has a polynomial‐time verifier

Certificate : Verifier : On input : (Vertices of are numbers )

• 1. Check that
• is a permutation: Every

number appears exactly once

• 2. Check that

, , and there is an edge from every to

• 3. Accept if all checks pass, otherwise, reject.

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NP is the class of languages with polynomial‐ time verifiers

Theorem: A language iff there is a polynomial‐ time verifier for Proof:

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