Theory of Computer Science E2. P, NP and Polynomial Reductions Gabriele R¨oger

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Theory of Computer Science

E2. P, NP and Polynomial Reductions

Gabriele R¨oger

University of Basel

April 29, 2019

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Theory of Computer Science

April 29, 2019 — E2. P, NP and Polynomial Reductions

E2.1 P and NP

E2.2 Polynomial Reductions

E2.3 NP-Hardness and NP-Completeness

E2.4 Summary

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E2. P, NP and Polynomial Reductions P and NP

E2.1 P and NP

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Course Overview

Theory

Background Logic Automata Theory Turing Computability

Complexity

Nondeterminism P, NP

Polynomial Reductions Cook-Levin Theorem NP-complete Problems More Computability

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Accepting a Word in Time n

Definition (Accepting a Word in Timen)

LetM be a DTM or NTM with input alphabet Σ, w ∈Σ^∗ a word andn ∈N0.

M acceptsw in timen if there is a sequence of configurations c0, . . . ,ck with k ≤n, where:

I c₀ is the start configuration for w,

I c₀`c₁ ` · · · `c_k, and

I c_k is an end configuration.

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Accepting a Language in Time f

Definition (Accepting a Language in Timef) LetM be a DTM or NTM with input alphabet Σ, L⊆Σ^∗ a language andf :N0→N0 a function.

M acceptsL in timef if:

1 for all wordsw ∈L: M accepts w in time f(|w|)

2 for all wordsw ∈/L: M does not accept w German: M akzeptiert Lin Zeitf

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P and NP

Definition (P and NP)

Pis the set of all languages Lfor which a DTM M

and apolynomialp exist such that M accepts Lin timep.

NPis the set of all languagesL for which anNTMM and apolynomialp exist such that M accepts Lin timep.

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P and NP: Remarks

I Sets of languages like P and NP that are defined in terms of computation time of TMs

(or other computation models) are called complexity classes.

I We know that P⊆NP. (Why?)

I Whether the converse is also true is an open question:

this is the famousP-NP problem.

German: Komplexit¨atsklassen, P-NP-Problem

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Example: DirHamiltonCycle ∈ NP

Example (DirHamiltonCycle∈NP)

The nondeterministic algorithm of Chapter E1 solves the problem and can be implemented on an NTM in polynomial time.

I Is DirHamiltonCycle∈P also true?

I The answer is unknown.

I So far, only exponential deterministic algorithms for the problem are known.

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Simulation of NTMs with DTMs

I Unlike DTMs, NTMs are not arealistic computation model:

they cannot be directly implemented on computers.

I But NTMs can besimulated by systematically trying all computation paths, e. g., with a breadth-first search.

More specifically:

I Let M be an NTM that accepts language Lin timef, wheref(n)≥n for alln ∈N0.

I Then we can specify a DTMM⁰ that accepts Lin time f⁰, wheref⁰(n) = 2^O(f⁽ⁿ⁾⁾.

I without proof

(cf. “Introduction to the Theory of Computation”

by Michael Sipser (3rd edition), Theorem 7.11)

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E2. P, NP and Polynomial Reductions Polynomial Reductions

E2.2 Polynomial Reductions

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Course Overview

Theory

Complexity

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Polynomial Reductions: Idea

I Reductionsare a common and powerful concept in computer science. We know them from Part D.

I The basic idea is that we solve a new problem by reducingit to a known problem.

I In complexity theory we want to use reductions

that allow us to prove statements of the following kind:

Problem A can be solved efficiently if problem B can be solved efficiently.

I For this, we need a reduction fromA toB

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Polynomial Reductions

Definition (Polynomial Reduction)

LetA⊆Σ^∗ andB ⊆Γ^∗ be decision problems.

We say thatAcan be polynomially reduced toB,

writtenA≤_pB, if there is a functionf : Σ^∗ →Γ^∗ such that:

I f can be computed inpolynomial time by aDTM

I i. e., there is a polynomialp and a DTMM such thatM computesf(w) in at mostp(|w|) steps given inputw ∈Σ^∗

I f reducesA toB

I i. e., for allw ∈Σ^∗: w ∈Aifff(w)∈B f is called apolynomial reductionfromAto B German: Apolynomiell aufB reduzierbar,

German:

polynomielle Reduktion vonAauf B

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Polynomial Reductions: Remarks

I Polynomial reductions are also calledKarp reductions (after Richard Karp, who wrote a famous paper describing many such reductions in 1972).

I In practice, of course we do not have to specify a DTM for f: it just has to be clear that f can be computed

in polynomial time by adeterministic algorithm.

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Polynomial Reductions: Example (1)

Definition (HamiltonCycle)

HamiltonCycleis the following decision problem:

I Given: undirected graph G =hV,Ei

I Question: DoesG contain a Hamilton cycle?

Reminder:

Definition (Hamilton Cycle)

AHamilton cycleof G is a sequence of vertices inV, π=hv₀, . . . ,v_ni, with the following properties:

I π is a path: there is an edge fromvi to vi+1 for all 0≤i <n

I π is a cycle: v₀ =v_n

I π is simple: v_i 6=v_j for all i 6=j with i,j <n

I π is Hamiltonian: all nodes ofV are included in π

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Polynomial Reductions: Example (2)

Definition (TSP)

TSP(traveling salesperson problem) is the following decision problem:

I Given: finite set S 6=∅ of cities, symmetric cost function cost:S×S →N0, cost boundK ∈N0

I Question: Is there a tour with total cost at mostK, i. e., a permutationhs₁, . . . ,s_ni of the cities with

Pn−1

i=1 cost(s_i,s_i+1) +cost(s_n,s₁)≤K?

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Polynomial Reductions: Example (3)

Theorem (HamiltonCycle≤_pTSP) HamiltonCycle≤_pTSP.

Proof.

blackboard

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Properties of Polynomial Reductions (1)

Theorem (Properties of Polynomial Reductions) Let A, B and C decision problems.

1 If A≤_pB and B ∈P, then A∈P.

2 If A≤_pB and B ∈NP, then A ∈NP.

3 If A≤_pB and A∈/P, then B ∈/P.

4 If A≤_pB and A∈/NP, then B ∈/ NP.

5 If A≤_pB and B ≤_pC , then A≤_pC .

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Properties of Polynomial Reductions (2)

Proof.

for 1.:

We must show that there is a DTM acceptingA in polynomial time.

We know:

I There is a DTM M_B that acceptsB in time p, wherep is a polynomial.

I There is a DTM Mf that computes a reduction from AtoB in time q, whereq is a polynomial.

. . .

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Properties of Polynomial Reductions (3)

Proof (continued).

Consider the machineM that first behaves like M_f, and then (afterMf stops) behaves likeMB on the output of Mf. M acceptsA:

I M behaves on input w asMB does on input f(w), so it acceptsw if and only iff(w)∈B.

I Because f is a reduction,w ∈A iff f(w)∈B.

. . .

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Properties of Polynomial Reductions (4)

Proof (continued).

Computation time ofM on input w:

I firstM_f runs on input w: ≤q(|w|) steps

I then MB runs on inputf(w): ≤p(|f(w)|) steps

I |f(w)| ≤ |w|+q(|w|) because inq(|w|) steps,

M_f can write at mostq(|w|) additional symbols onto the tape total computation time ≤q(|w|) +p(|f(w)|)

≤q(|w|) +p(|w|+q(|w|)) this is polynomial in |w| A∈P.

. . .

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Properties of Polynomial Reductions (5)

Proof (continued).

for 2.:

analogous to 1., only thatM_B andM are NTMs of 3.+4.:

equivalent formulations of 1.+2. (contraposition) of 5.:

LetA≤_pB with reduction f andB ≤_pC with reduction g. Theng ◦f is a reduction ofA toC.

The computation time of the two computations in sequence

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E2. P, NP and Polynomial Reductions NP-Hardness and NP-Completeness

E2.3 NP-Hardness and

NP-Completeness

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Course Overview

Theory

Complexity

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NP-Hardness and NP-Completeness

Definition (NP-Hard, NP-Complete) LetB be a decision problem.

B is called NP-hard ifA≤_pB for allproblemsA∈NP.

B is called NP-completeif B∈NP and B is NP-hard.

German: NP-hart (selten: NP-schwer), NP-vollst¨andig

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NP-Complete Problems: Meaning

I NP-hard problems are “at least as difficult”

as all problems in NP.

I NP-complete problems are “the most difficult” problems in NP: allproblems in NP can be reduced to them.

I IfA∈P for anyNP-complete problem, then P = NP. (Why?)

I That means that either there are efficient algorithms for allNP-complete problems or for none of them.

I Do NP-complete problems actually exist?

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E2. P, NP and Polynomial Reductions Summary

E2.4 Summary

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E2. P, NP and Polynomial Reductions Summary

Summary

I P: languages accepted by DTMsin polynomial time

I NP: languages accepted by NTMs in polynomial time

I polynomial reductions: A≤_pB if

there is a total functionf computable in polynomial time, such that for all words w: w ∈Aiff f(w)∈B

I A≤_pB implies that Ais “at most as difficult”as B

I polynomial reductions are transitive

I NP-hard problemsB: A≤_pB for allA∈NP

I NP-completeproblems B: B ∈NP and B is NP-hard