D2.1 Polynomial Reductions

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Theory of Computer Science

D2. Polynomial Reductions and NP-completeness

Gabriele R¨oger

University of Basel

May 5, 2021

Gabriele R¨oger (University of Basel) Theory of Computer Science May 5, 2021 1 / 21

Theory of Computer Science

May 5, 2021 — D2. Polynomial Reductions and NP-completeness

D2.1 Polynomial Reductions

D2.2 NP-Hardness and NP-Completeness D2.3 Summary

D2. Polynomial Reductions and NP-completeness Polynomial Reductions

D2.1 Polynomial Reductions

Polynomial Reductions: Idea

I Reductions are a common and powerful concept in computer science. We know them from Part C.

I The basic idea is that we solve a new problem byreducing it 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 from AtoB that can be computed efficiently itself

(otherwise it would be useless for efficiently solving A).

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Polynomial Reductions

Definition (Polynomial Reduction)

Let A⊆Σ^∗ andB ⊆Γ^∗ be decision problems.

We say that Acan be polynomially reduced toB,

written A≤_p B, if there is a function f : Σ^∗→Γ^∗ such that:

I f can be computed inpolynomial time by aDTM I i. e., there is a polynomialpand a DTMM such thatM

computesf(w) in at mostp(|w|) steps given inputw ∈Σ^∗ I f reducesAto B

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

German:

polynomielle Reduktion vonAauf B

Polynomial Reductions: Remarks

I Polynomial reductions are also called Karp 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.

Polynomial Reductions: Example (1)

Definition (HamiltonCycle)

HamiltonCycle is the following decision problem:

I Given: undirected graph G =hV,Ei

I Question: DoesG contain a Hamilton cycle?

Reminder:

Definition (Hamilton Cycle)

A Hamilton cycleofG is a sequence of vertices inV, π =hv₀, . . . ,v_ni, with the following properties:

I π is a path: there is an edge fromv_i to v_i+1 for all 0≤i <n I π is a cycle: v =v

Polynomial Reductions: Example (2)

Definition (TSP)

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

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

I Question: Is there a tour with total cost at most K, i. e., a permutationhs₁, . . . ,s_niof the cities with

Pn−1

i=1 cost(s_i,s_i+1) +cost(s_n,s₁)≤K? German: Problem der/des Handlungsreisenden

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Polynomial Reductions: Example (3)

Theorem (HamiltonCycle≤_p TSP) HamiltonCycle≤_p TSP.

Proof.

blackboard

Exercise: Polynomial Reduction

Definition (HamiltonianCompletion)

HamiltonianCompletionis the following decision problem:

I Given: undirected graphG =hV,Ei, number k∈N0

I Question: CanG be extended with at most k edges such that the resulting graph has a Hamilton cycle?

Show that

HamiltonCycle≤_p HamiltonianCompletion.

Reminder: P and NP

P: class of languages that are decidable in polynomial time by a deterministic Turing machine

NP: class of languages that are decidable in polynomial time by a non-deterministic Turing machine

Properties of Polynomial Reductions (1)

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

1 If A≤_p B and B ∈P, then A∈P.

2 If A≤_p B and B ∈NP, then A∈NP.

3 If A≤_p B and A∈/ P, then B ∈/P.

4 If A≤_p B and A∈/ NP, then B∈/ NP.

5 If A≤_p B and B ≤_p C , then A≤_p C .

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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 DTMM_B that accepts B in time p, where p is a polynomial.

I There is a DTMM_f that computes a reduction fromA toB in time q, whereq is a polynomial.

. . .

Properties of Polynomial Reductions (3)

Proof (continued).

Consider the machineM that first behaves likeM_f, and then (afterM_f stops) behaves likeM_B on the output ofM_f. M acceptsA:

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

I Becausef is a reduction, w ∈A ifff(w)∈B.

. . .

Properties of Polynomial Reductions (4)

Proof (continued).

Computation time ofM on inputw:

I firstM_f runs on inputw: ≤q(|w|) steps I thenM_B runs on inputf(w): ≤p(|f(w)|) steps I |f(w)| ≤ |w|+q(|w|) because in q(|w|) steps,

M_f can write at most q(|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.

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≤_p B with reduction f andB≤_p C with reductiong. Theng ◦f is a reduction of AtoC.

The computation time of the two computations in sequence

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D2. Polynomial Reductions and NP-completeness NP-Hardness and NP-Completeness

D2.2 NP-Hardness and NP-Completeness

NP-Hardness and NP-Completeness

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

B is called NP-hardifA≤_p B for allproblemsA∈NP.

B is called NP-completeifB ∈NP andB is NP-hard.

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

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 If A∈P foranyNP-complete problem, then P = NP. (Why?) I That means that either there are efficient algorithms

forallNP-complete problems or fornone of them.

I Do NP-complete problems actually exist?

D2. Polynomial Reductions and NP-completeness Summary

D2.3 Summary

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D2. Polynomial Reductions and NP-completeness Summary

Summary

I polynomial reductions: A≤_p B if

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

I A≤_pB implies thatAis “at most as difficult”asB I polynomial reductions aretransitive

I NP-hardproblems B: A≤_p B forallA∈NP

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