27 NP Completness

transcript

Analysis of AlgorithmsNP-Completeness

Andres Mendez-Vazquez

November 30, 2015

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Outline1 Introduction

Polynomial TimeThe Intuition P vs NP

2 Structure of the Polynomial Time ProblemsIntroductionAbstract ProblemsEncodingFormal Language FrameworkDecision Problems in The Formal FrameworkComplexity Class

3 Polynomial Time VerificationIntroductionVerification Algorithms

4 Reducibility and NP-CompletnessIntroductionNP-CompletnessAn Infamous Theorem

5 NP-Complete ProblemsCircuit SatisfiabilityHow do we prove NP-Completness?Algorithm A representationThe Correct ReductionThe Polynomial Time

Making our life easier!!!Formula Satisfiability3-CNFThe Clique ProblemFamily of NP-Complete Problems

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Polynomial Time

Algorithms Until NowAll the algorithms, we have studied this far have been polynomial-timealgorithms.

HoweverThere is a collection of algorithms that cannot being solved inpolynomial time!!!

ExampleIn the Turing’s “Halting Problem,” we cannot even say if thealgorithm is going to stop!!!

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Polynomial Time

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Polynomial Time

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The Intuition

Class PThey are algorithms that on inputs of size n have a worst case runningtime of O

(nk)for some constant k.

Class NPInformally, the Non-Polynomial (NP) time algorithms are the ones thatcannot be solved in O

(nk)for any constant k.

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The Intuition

Class PThey are algorithms that on inputs of size n have a worst case runningtime of O

(nk)for some constant k.

Class NPInformally, the Non-Polynomial (NP) time algorithms are the ones thatcannot be solved in O

(nk)for any constant k.

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There are still many thing to say about NP problems

But the one that is making everybody crazyThere is a theorem that hints to a possibility of NP = P

ThusWe have the following vision of the world of problems in computer science.

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There are still many thing to say about NP problems

But the one that is making everybody crazyThere is a theorem that hints to a possibility of NP = P

ThusWe have the following vision of the world of problems in computer science.

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The Two Views of The World

The Paradox

NP-Hard NP-Hard

NP-Complete

P=NP=NP-Complete

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However, There are differences pointing to P 6= NP

Shortest Path is in PEven with negative edge weights, we can find a shortest path for a singlesource in a directed graph G = (V ,E) in O (VE) time.

Longest Path is in NPMerely determining if a graph contains a simple path with at least a givennumber of edges is NP.

It is moreA simple change on a polynomial time problem can move it from P to NP.

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And here, a simplified classification of problems

Different Complexity ClassesComplexity Class Model of Computation Resource Constraint

P Deterministic Turing Machine Solvable using poly(n) timeNP Non-deterministic Turing Machine Verifiable in poly(n) time

PSPACE Deterministic Turing Machine Solvable using poly(n) SpaceEXPTIME Deterministic Turing Machine Solvable using 2poly(n) timeEXPSPACE Deterministic Turing Machine Space 2poly(n)

NL Non-deterministic Turing Machine Space O (log n)

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Graphically

What is contained into what

PSPACE

EXPTIME

EXSPACE

Beyond the Turing Machine

Halting Problem

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We start by formalizing the notion of polynomial time

Polynomial Time ProblemsWe generally regard these problems as tractable, but for philosophical, notmathematical, reasons.

FirstIt is possible to regard a problem with complexity O

(n100) as intractable,

really few practical problems require time complexities with such highdegree polynomial.

It is moreExperience has shown that once the first polynomial-time algorithm for aproblem has been discovered, more efficient algorithms often follow.

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Reasons

SecondFor many reasonable models of computation, a problem that can be solvedin polynomial time in one model can be solved in polynomial time inanother.

ExampleProblems that can be solved in polynomial time by a serial random-accessmachine can be solved in a Turing Machine.

ThirdThe class of polynomial-time solvable problems has nice closureproperties.Since polynomials are closed under addition, multiplication, andcomposition.

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Reasons

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Reasons

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Reasons

Why?For example, if the output of one polynomial time algorithm is fed into theinput of another, the composite algorithm is polynomial.

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Polynomial time

To understand a polynomial time we need to define:What is the meaning of an abstract problem?How to encode problems.A formal language framework.

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Polynomial time

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Polynomial time

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Polynomial time

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What do we need to understand the polynomial time?What is an abstract problem?We define an abstract problem Q to be a binary relation on a set I ofproblem instances and a set S of problem solutions:

Q : I → S , Q(i) = s (1)

i Q(i)=s

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Abstract problem as decision problems

Something NotableThe formulation is too general to our purpose!!!

Thus, we do a restrictionThe theory of NP-completeness restricts attention to decision problems:

Those having a YES/NO solution.

ThenWe can view an abstract decision problem as a function that maps theinstance set I to the solution set 0, 1:

Q : I → 0, 1

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Q : I → 0, 1

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Q : I → 0, 1

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Q : I → 0, 1

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Example

Example of optimization problem: SHORTEST-PATHThe problem SHORTEST-PATH is the one that associates each graph Gand two vertices with the shortest path between them.

Problem, this is a optimization problemWe need a decision problem!!!

What do we do?We can cast a given optimization problem as a related decision problem byimposing a bound on the value to be optimized.

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Example

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Example

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Example PATH ProblemGiven a undirected graph G, vertices u and v, and an integer k, we needto answer the following question:

Does a path exist from u to v consisting of at most k edges?

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In a more formal way

We have the following optimization problemmin

td [t]

s.t.d[v] ≤ d[u] + w(u, v) for each edge (u, v) ∈ Ed [s] = 0

Then, we have the following decision problemPATH = 〈G, u, v, k〉 |G = (V ,E) is an undirected graph,

u, v ∈ V , k ≥ 0 is an integer and there exist a path fromu to v in G consisting of at most k edges

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In a more formal way

We have the following optimization problemmin

td [t]

s.t.d[v] ≤ d[u] + w(u, v) for each edge (u, v) ∈ Ed [s] = 0

Then, we have the following decision problemPATH = 〈G, u, v, k〉 |G = (V ,E) is an undirected graph,

u, v ∈ V , k ≥ 0 is an integer and there exist a path fromu to v in G consisting of at most k edges

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Why Encoding?

We need to represent our problems in a way that the Turing machinecan understand.That is called an encoding.

ExampleGiven a graph G = (V ,E):

We can encode each vertex 1, 2, ... as 0, 1, 10, ...Then, each edge, for example 1, 2 as 0, 1Clearly you need to encode some kind of delimiter for each element inthe description

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Why Encoding?

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Why Encoding?

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Why Encoding?

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Why Encoding?

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Why a Turing machine or a RAM machine?

Given an encoding...We need a computational device to solve the encoded problem

Some factsThis means that when the device solves a problem in reality solves theencoded version of Q.This encoded problem is called a concrete problem.This tells us how important encoding is!!!

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It is more!!!

We want time complexities of O (T (n))When it is provided with a problem instance i of length |i| = n.

ThenThe algorithm can produce the solution in O (T (n)).

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It is more!!!

We want time complexities of O (T (n))When it is provided with a problem instance i of length |i| = n.

ThenThe algorithm can produce the solution in O (T (n)).

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Using Encodings

Something NotableGiven an abstract decision problem Q mapping an instance set I to 0, 1.

ThenAn encoding e : I −→ 0, 1∗ can induce a related concrete decisionproblem, denoted as by e (Q).

IMPORTANTIf the solution to an abstract-problem instance i ∈ I is Q (i) ∈ 0, 1.Then the solution to the concrete-problem instance e (i) ∈ 0, 1∗ isalso Q (i).

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Using Encodings

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Using Encodings

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Using Encodings

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What do we want?

Something NotableWe would like to extend the definition of polynomial-time solvability fromconcrete problems to abstract problems by using encodings as the bridge.

IMPORTANT!!!We want the definition to be independent of any particular encoding.

In other wordsThe efficiency of solving a problem should not depend on how the problemis encoded.

HOWEVER, it depends quite heavily on the encoding.

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What do we want?

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What do we want?

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An example of a really BAD situation

Imagine the followingYou could have an algorithm that takes k as the sole input with analgorithm that runs in Θ(k).

Now, if the integer is provided in an unary representation (Only ones)Quite naive!!!

ThenRunning time of the algorithm is O(n) on n-length inputs, which ispolynomial.

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For example

Now, use the more natural binary representation of the integer kNow, given a binary representation:

I Thus the input length is n = blog kc+ 1→ Θ (k) = Θ (2n)

RemarkThus, depending on the encoding, the algorithm runs in either polynomialor superpolynomial time.

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For example

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For example

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More Observations

ThusHow we encode an abstract problem matters quite a bit to how weunderstand it!!!

It is moreWe cannot talk about solving an abstract problem without specifying theencoding!!!

NeverthelessIf we rule out expensive encodings such as unary ones, the actual encodingof a problem makes little difference to whether the problem can be solvedin polynomial time.

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More Observations

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More Observations

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Some properties of the polynomial encoding

FirstWe say that a function f : 0, 1∗ → 0, 1∗ is polynomial timecomputable, if there exists a polynomial time algorithm A that, given anyinput x ∈ 0, 1∗, produces as output f (x).

SecondFor some set I of problem instances, we say that two encodings e1 and e2are polynomially related

if there exist two polynomial time computable functions f12 and f21such that for any i ∈ I , we have f12 (e1(i)) = e2(i) andf21 (e2(i)) = e1(i).

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Observation

We have thatA polynomial-time algorithm can compute the encoding e2 (i) from theencoding e1 (i), and vice versa.

Something NotableIf two encodings e1 and e2 of an abstract problem are polynomially related

We have that if the problem is polynomial-time solvable or not isindependent of which encoding we use.

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Observation

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Observation

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Observation

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An important lemma

Lemma 34.1Let Q be an abstract decision problem on an instance set I , and let e1 ande2 be polynomially related encodings on I . Then, e1(Q) ∈ P if and only ife2(Q) ∈ P.

Proof in the board

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An important lemma

Lemma 34.1Let Q be an abstract decision problem on an instance set I , and let e1 ande2 be polynomially related encodings on I . Then, e1(Q) ∈ P if and only ife2(Q) ∈ P.

Proof in the board

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Formal language framework to handle representation

Definitions1 An alphabet Σ is a finite set of symbols.2 A language L over Σ is any set of strings made up of symbols from

Σ∗.3 The empty language is ε.4 The language of all strings over Σ is denoted Σ∗.

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Special languages and operations

Union, intersection and complementL1⋂L2 = x ∈ Σ∗| x ∈ L1∧x ∈ L2

L1⋃L2 = x ∈ Σ∗| x ∈ L1∨x ∈ L2

L = x ∈ Σ∗| x 6= L

ConcatenationL = x1x2 | x1 ∈ L1 and x2 ∈ L2

Kleene closureL∗ = ε

⋃L2⋃L3⋃ ...

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L1⋃L2 = x ∈ Σ∗| x ∈ L1∨x ∈ L2

L = x ∈ Σ∗| x 6= L

⋃L2⋃L3⋃ ...

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L1⋃L2 = x ∈ Σ∗| x ∈ L1∨x ∈ L2

L = x ∈ Σ∗| x 6= L

⋃L2⋃L3⋃ ...

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L1⋃L2 = x ∈ Σ∗| x ∈ L1∨x ∈ L2

L = x ∈ Σ∗| x 6= L

⋃L2⋃L3⋃ ...

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L1⋃L2 = x ∈ Σ∗| x ∈ L1∨x ∈ L2

L = x ∈ Σ∗| x 6= L

⋃L2⋃L3⋃ ...

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Observation

We have thatFrom the point of view of language theory,

The set of instances for any decision problem Q is simply the set∑∗

with∑

= 0, 1.

Something NotableQ is entirely characterized by instances that produce a YES or ONEanswer.

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Observation

with∑

= 0, 1.

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Observation

with∑

= 0, 1.

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This allow us to define a language that is solvable by Q

We can write it down as the language

L = x ∈ Σ∗|Q(x) = 1 (2)

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Thus, we can express the duality DecisionProblem-Algorithm

ImportantThe formal-language framework allows to express concisely the relationbetween decision problems and algorithms that solve them.

Decision Problems Set of Algorithms

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Given an instance x of a problemAn algorithm A accepts a string x ∈ 0, 1∗, if given x , thealgorithm’s output is A(x) = 1.

The language accepted by an algorithm A is the set of strings

L = x ∈ 0, 1∗ |A (x) = 1 . (3)

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Given an instance x of a problemAn algorithm A accepts a string x ∈ 0, 1∗, if given x , thealgorithm’s output is A(x) = 1.

The language accepted by an algorithm A is the set of strings

L = x ∈ 0, 1∗ |A (x) = 1 . (3)

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Nevertheless

We have a problemEven if language L is accepted by an algorithm A.The algorithm will not necessarily reject a string x /∈ L provided as input toit.

I Example: The algorithm could loop forever.

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Nevertheless

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Nevertheless

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Nevertheless

We need to be more stringentA language L is decided by an algorithm A if every binary string in L is acceptedby A and every binary string not in L is rejected by A.

Algorithm A

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Finally

Thus, a language L is decided by A, ifGiven a string x ∈ 0, 1∗, only one of two things can happen:

An algorithm A accepts, if given x ∈ L the algorithm outputsA(x) = 1.An algorithm A rejects, if if given x /∈ L the algorithm outputsA(x) = 0.

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Finally

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Finally

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Now, it is possible to define acceptance in polynomial time

We have thatA language L is accepted in polynomial time by an algorithm A, if itis accepted by A in polynomial time:

I There exists a constant k such that for any n-length string x ∈ L ⇒algorithm A accepts x in time O

ThusA language L is decided in polynomial time by an algorithm A, ifthere exists a constant k such that for any n-length string x ∈ 0, 1∗:

I The algorithm correctly decides whether x ∈ L in time O(nk).

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Example of polynomial accepted problems

Example of polynomial accepted problem

PATH = 〈G, u, v, k〉 |G = (V ,E) is an undirected graph,u, v ∈ V , k ≥ 0 is an integer and there exist a path fromu to v in G consisting of at most k edges

What does the polynomial times accepting algorithm do?One polynomial-time accepting algorithm does the following

I It verifies that G encodes an undirected graph.I It calculate the shortest path between vertices and compares the

number of edges of that path with k.I If it finds such a path outputs ONE and halt.I If it does not, it runs forever!!! ⇐= PROBLEM!!!

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What we will like to have...

A decision algorithmBecause we want to avoid the infinite loop, we do the following...

StepsIt verifies that G encodes an undirected graph.It calculate the shortest path between vertices and compares thenumber of edges of that path with k.If it finds such a path outputs ONE and halt.If it does not find such a path output ZERO and halt.

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HoweverThere are problemsAs the Turing’s Halting Problem where:

There exists an accepting algorithmBut no decision algorithm exist

What!?It turns out there are perfectly decent computational problems for whichno algorithms exist at all!

For example, an arithmetical version of what will talk later, the SATproblemGiven a polynomial equation in many variables, perhaps:

x3yz + 2y4z2 − 7xy5z = 6

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x3yz + 2y4z2 − 7xy5z = 6

Actually

Something NotableThere is no algorithm that solves this problem.No algorithm at all, polynomial, exponential, doubly exponential, or worse!

I Such problems are called unsolvable.

This was discovered byThe first unsolvable problem was discovered in 1936 by Alan M. Turing, then astudent of mathematics at Cambridge, England.

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Actually

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Actually

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ActuallySomething Notable

There is no algorithm that solves this problem.No algorithm at all, polynomial, exponential, doubly exponential, or worse!

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What did he do?

Basic Idea1 Suppose that given a program p and an input x.

1 There is an algorithm, called TERMINATE, that takes p and x and tellus if p will ever terminate in x.

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What did he do?

Basic Idea1 Suppose that given a program p and an input x.

1 There is an algorithm, called TERMINATE, that takes p and x and tellus if p will ever terminate in x.

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ThenWe have the following programfunction PARADOX(z : file)

1 if TERMINATES(z, z) goto 1

Notice what paradox doesIt terminates if and only if program z does not terminate when given itsown code as input.

What if run PARADOX(PARADOX)Funny PARDOX!!!

1 Case I : The PARADOX terminates -> Then TERMINATE saysfalse!!!

2 Case II : The PARADOX never terminates -> Then TERMINATEsays true!!!

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Complexity Classes

ThenWe can informally define a complexity class as a set of languages.

NowThe membership to this class is determined by a complexity measure,such as running time, of an algorithm that determines whether a givenstring x belongs to language L.

HoweverThe actual definition of a complexity class is somewhat more technical.

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Complexity Classes

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Complexity Classes

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We can use this framework to say the followingP = L ⊆ 0, 1∗ | There exists an algorithm A that decides L inpolynomial time

I In fact, P is also the class of languages that can be accepted inpolynomial time

Theorem 34.2P=L| L is accepted by a polynomial-time algorithm

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Exercises

From Cormen’s book solve34.1-134.1-234.1-334.1-434.1-534.1-6

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What is verification?

Intuitive definitionGiven an instance of a decision problem:

For example 〈G, u, v, k〉 of PATH.

ThenWe are given :

A path p from A to F .Then, check if the length of p is at most k (i.e. Belongs toPATH), then it is called a “certificate.”

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ThenWe are given :

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ThenWe are given :

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ThenWe are given :

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It is more

In factWe would like to be able to verify in polynomial time the certificate ofcertain types of problems.For example:

I Polynomial time problems.I Non-Polynomial time problems.

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It is more

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It is more

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It is more

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Example of verifiable problemsHamiltonian cycleA Hamiltonian cycle of an undirected graph G = (V , E) is asimple cycle that contains each vertex in V .

Certificate

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As a formal language

Does a graph G have a Hamiltonian cycle?

HAM − CYCLES = 〈G〉 |G is a Hamiltonian graph (4)

How do we solve this decision problem?Can we even solve it?

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Decision algorithm for Hamiltonian

Given an instance < G > and encode itIf we use the “reasonable” encoding of a graph as its adjacencymatrix.

1 2 3 4 512345

0 0 0 1 00 0 0 1 10 0 0 0 11 1 0 0 10 1 1 1 0

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We can then say the followingIf the number of vertices is m = Ω (

We have then√

n︷︸︸︷1 2 3 4 5

0 0 0 1 00 0 0 1 10 0 0 0 11 1 0 0 10 1 1 1 0

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We can then say the followingIf the number of vertices is m = Ω (

We have then√

n︷︸︸︷1 2 3 4 5

0 0 0 1 00 0 0 1 10 0 0 0 11 1 0 0 10 1 1 1 0

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The encoding size is√

n ×√

n = n = |〈G〉|

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Then, I decide to go NAIVE!!!

The algorithm does the followingIt lists the all permutations of the vertices of G and then checks eachpermutation to see if it is a Hamiltonian path.

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Complexity

Performance analysis on the previous algorithmWe have then m = Ω(

Then, for our naive algorithm produce m! permutations.Then Ω(m!) = Ω(

√n!) = Ω(2

√n) EXPONENTIAL TIME!!!

Something NotableStill, with no-naive algorithm the Hamiltonian is not solvable in polynomialtime!

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Complexity

√n!) = Ω(2

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Complexity

√n!) = Ω(2

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Complexity

√n!) = Ω(2

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Verification Algorithms

A more formal definition in terms of formal languagesA verification algorithm is a two-argument algorithm A, where:

1 There is an input string x (Instance of the problem).2 There is a binary string y, certificate (The possible solution).

Then, a two-argument algorithm A verifiesA verifies x by using a certificate y.Then, it verifies x by taking y and outputting ONE i.e. A (x, y) = 1.

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Finally we have

The language verified by a verification algorithm isL = x ∈ 0, 1∗|∃y ∈ 0, 1∗such that A(x, y) = 1

Important remark!For any string x /∈ L there must be no certificate proving x ∈ L(consistency is a must).

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Finally we have

The language verified by a verification algorithm isL = x ∈ 0, 1∗|∃y ∈ 0, 1∗such that A(x, y) = 1

Important remark!For any string x /∈ L there must be no certificate proving x ∈ L(consistency is a must).

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The NP class

DefinitionThe complexity class NP is the class of the languages that can beverified by a polynomial time algorithm.

L = x ∈ 0, 1∗|∃y ∈ 0, 1∗ with |y| = O(|x|c) such that A(x, y) = 1

NoteWe say that A verifies language L in polynomial time.Clearly, the size of the certificate must be polynomial in size!!!

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The NP class

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The NP class

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Observation of the class NP and co-NP

ExampleHAM-CYCLE is NP, thus NP class is not empty.Observation: L ∈ P → L ∈ NP or P ⊆ NP.

Now, Do we have P = NP?There is evidence that P 6= NP basically because

I the existence of languages that are NP-Complete.

And actually, it is worseWe still cannot answer if L ∈ NP → L ∈ NP (closure undercomplement).

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Another way to see this

The co − NP classThe class called co-NP is the set of languages L such that L ∈ NP

Something NotableWe can restate the question of whether NP is closed under complement aswhether NP = co −NP

In addition because P is closed under complementWe have P ⊆ NP ∩ co −NP, however no one knows whetherP = NP ∩ co −NP.

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The four possibilities between the complexity classes

We have that

P=NP=co-NPNP=co-NP

Pco-NP NP co-NP NP

What we believe is the real relationship

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Exercises

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Before entering reducibility

Why P 6= NP?Existence of NP-Complete problems.Problem!!! There is the following property:

I If any NP-Complete problem can be solved in polynomial time,then every problem in NP has a polynomial time solution.

Not only thatThe NP-Complete problems are the hardest in the NP class, and thisis related the concept of polynomial time reducibility.

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Reducibility

Rough definitionA problem M can be reduced to M ′ if any instance of M can be easilyrephrased in terms of M ′.

Formal definitionA language L is polynomial time reducible to a language L′ writtenL ≤p L′ if there exist a polynomial time computable functionf : 0, 1∗ → 0, 1∗ such that for all x ∈ 0, 1∗

x ∈ L ⇐⇒ f (x) ∈ L′

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Reducibility

Rough definitionA problem M can be reduced to M ′ if any instance of M can be easilyrephrased in terms of M ′.

Formal definitionA language L is polynomial time reducible to a language L′ writtenL ≤p L′ if there exist a polynomial time computable functionf : 0, 1∗ → 0, 1∗ such that for all x ∈ 0, 1∗

x ∈ L ⇐⇒ f (x) ∈ L′

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Graphically

The Mapping

From the figureHere, f is called a reduction function, and the polynomial timealgorithm F that computes f is called reduction algorithm.

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Graphically

The Mapping

From the figureHere, f is called a reduction function, and the polynomial timealgorithm F that computes f is called reduction algorithm.

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Properties of f

Polynomial time reductionsPolynomial time reductions give us a powerful tool for proving that variouslanguages belong to P.

How?Lemma: If L1,L2 ⊆ 0, 1∗ are languages such that L1 ≤p L2, then

L2 ∈ P implies that L1 ∈ P.Proof

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Properties of f

Polynomial time reductionsPolynomial time reductions give us a powerful tool for proving that variouslanguages belong to P.

How?Lemma: If L1,L2 ⊆ 0, 1∗ are languages such that L1 ≤p L2, then

L2 ∈ P implies that L1 ∈ P.Proof

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NP-Completeness

DefinitionA language L ⊆ 0, 1 ∗ is a NP-Complete problem (NPC) if:

1 L ∈ NP2 L′ ≤p L for every L′ ∈ NP

NoteIf a language L satisfies property 2, but not necessarily property 1, we saythat L is NP-Hard (NPH).

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NP-Completeness

DefinitionA language L ⊆ 0, 1 ∗ is a NP-Complete problem (NPC) if:

1 L ∈ NP2 L′ ≤p L for every L′ ∈ NP

NoteIf a language L satisfies property 2, but not necessarily property 1, we saythat L is NP-Hard (NPH).

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By The Way

NP can also be defined asThe set of decision problems that can be solved in polynomial time on anon-deterministic Turing machine.

Actually, it looks like a multi-threaded backtracking

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By The WayNP can also be defined asThe set of decision problems that can be solved in polynomial time on anon-deterministic Turing machine.

Actually, it looks like a multi-threaded backtracking

Branching to all possibilitiesHeight

For some

Branching to all possibilities Branching to all possibilities Branching to all possibilities

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Now, the infamous theorem

TheoremIf any NP-Complete problem is polynomial time solvable, thenP = NP. Equivalently, if any problem in NP is not polynomial timesolvable, then no NP-Complete problem is polynomial time solvable.

ProofSuppose that L ∈ P and also that L ∈ NPC . For any L′ ∈ NP, we haveL′ ≤p L by property 2 of the definition of NPC. Thus, by the previousLemma, we have that L’ ∈ P.

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Now, the infamous theorem

TheoremIf any NP-Complete problem is polynomial time solvable, thenP = NP. Equivalently, if any problem in NP is not polynomial timesolvable, then no NP-Complete problem is polynomial time solvable.

ProofSuppose that L ∈ P and also that L ∈ NPC . For any L′ ∈ NP, we haveL′ ≤p L by property 2 of the definition of NPC. Thus, by the previousLemma, we have that L’ ∈ P.

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However

Most Theoretical Computer Scientist have the following view

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Our first NPC - Circuit Satisfiability

We have basic boolean combinatorial elements.

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Basic definition

DefinitionA boolean combinatorial circuit consist of one or more booleancombinatorial elements interconnected with wires.

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Circuit satisfiability problem

ProblemGiven a boolean combinatorial circuit composed of AND, OR, and NOTgates, Is it satisfiable? Output is ONE!!!

FormallyCIRCUIT−SAT = 〈C 〉 |C is a satisfiable boolean combinatorial circuit

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ProblemGiven a boolean combinatorial circuit composed of AND, OR, and NOTgates, Is it satisfiable? Output is ONE!!!

FormallyCIRCUIT−SAT = 〈C 〉 |C is a satisfiable boolean combinatorial circuit

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Example: An assignment that outputs ONE

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First, It is NP-Problem

LemmaThe circuit-satisfiability belong to the class NP.

ProofWe need to give a polynomial-time algorithm A such that

1 One of the inputs to A is a boolean combinatorial circuit C .2 The other input is a certificate corresponding to an assignment of

boolean values to the wires in C .

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Second, It is NP-Hard

The general idea for A is:For each circuit gate check that the output value is correctlycomputed and corresponds to the values provided by the certificate.

ThenThen if the output of the entire circuit is one, the algorithm Aoutputs 1, otherwise 0.Because the certificate is polynomial in size with respect to the circuitC =⇒ A runs in polynomial time.

I Actually, with a good implementation, linear time is enough.

FinallyA cannot be fooled by any certificate to believe that a unsatisfiablecircuit is accepted. Then CIRCUIT-SAT ∈ NP.

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Proving CIRCUIT-SAT is NP-Hard

LemmaThe circuit sat problem is NP-hard.

Proof:Given a language L ∈ NP, we want a polynomial-time algorithm F thatcan compute a reduction map f such that:

It maps every binary string x to a circuit C = f (x) such that x ∈ L ifand only if C ∈CIRCUIT-SAT.

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FirstGiven a L ∈ NP, there exists an algorithm A that verifies L inpolynomial time.Now T (n) = O

(nk)denotes the worst case time of A, and the

length of the certificate is O(nk).

ThusThe algorithm F to be constructed will use the two input algorithm A tocompute the reduction function f .

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Basic ideas: A Computer Program

A ProgramIt can be seen as a sequence of instructions!!!

Each instructionIt encodes an operation to be performed, addresses of operand in memory,and a final address to store the result.

Each program has a counter, PCThis counter keeps tracking of the instruction to be executed.It increments automatically upon fetching each instruction.It can be changed by an instruction, so it can be used to implementloops and branches.

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Configuration

Something NotableAt any point during the execution of a program, the computer’s memoryholds the entire state of the computation.

ThusWe call any particular state of computer memory a configuration.

IMPORTANTWe can view the execution of an instruction as mapping one configurationto another.

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Configuration

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Configuration

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We have thatThe computer hardware that accomplishes this mapping can beimplemented as a boolean combinational circuit.

ThenWe denote this boolean circuit as M .

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We have thatThe computer hardware that accomplishes this mapping can beimplemented as a boolean combinational circuit.

ThenWe denote this boolean circuit as M .

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What we wantLet L be any language in NPThere must exist an algorithm A that verifies L in polynomial time

ThusThe algorithm F that we shall construct uses the two-input algorithm A tocompute the reduction function f .

NowLet T (n) = O

(nk) denote the worst-case running time of algorithm A on a

n-length input for some k ≥ 1.Remember:

I The running time of A is actually a polynomial in the total input size,which includes both an input string and a certificate.

I The certificate is polynomial in the length n of the input.F Thus the running time is polynomial in n.

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NowLet T (n) = O

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NowLet T (n) = O

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NowLet T (n) = O

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NowLet T (n) = O

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NowLet T (n) = O

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Basic ideas: Algorithm A representation

We can represent the computation of A as a sequence ofconfigurations.

Start with configuration c0, then finish with configuration cT(n).

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ThenThe idea

The Combinatorial Circuit

Auxiliary MachineState

WORKING STORAGE

0/1 OUTPUT102 / 163

Then, we need C = f (x)For this, we do:

I n = |x|

NextThen, we constructs a combinatorial circuit C ′ consisting of T (n) copiesof M

The output of the ci circuit finish as input in ci+1.

RemarkThe configuration finishes as values on the wires connecting copies.

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I n = |x|

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I n = |x|

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I n = |x|

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The polynomial time algorithm

What F must do:1 Given x it needs to compute circuit C (x) = f (x).2 Satisfiable ⇐⇒ there exists a certificate y such that A (x, y) = 1.

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The polynomial time algorithm

What F must do:1 Given x it needs to compute circuit C (x) = f (x).2 Satisfiable ⇐⇒ there exists a certificate y such that A (x, y) = 1.

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The F process

Given x:It first computes n = |x|.

NextThen it computes C ′ (a combinatorial circuit) by using T (n) copiesof M .

ThenThen, the initial configuration of C ′ consists in the input A (x, y), theoutput is configuration CT(n)

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The F process

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The F process

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Finally C

We then use C ′ to construct CFirst, F modifies circuit C ′ in the following way:

I It hardwires the inputs to C ′ corresponding to the programfor A:

F The initial program counterF The input xF The initial state of memory

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Finally C

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Finally C

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Finally C

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Finally C

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Further

Something NotableThe only remaining inputs to the circuit correspond to the certificate y.

ThenAll outputs to the circuit are ignored, except the one bit of cT(n)corresponding to a computation on A(x, y).

Because the only free input is the certificate yAh!! We have that C (y) = A (x, y)!!!

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Further

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Further

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What we need to prove

FirstF correctly computes a reduction function f .

I C is satisfiable if and only if there is a certificate y such thatA(x, y) = 1.

SecondWe need to show that F runs in polynomial time.

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What we need to prove

FirstF correctly computes a reduction function f .

I C is satisfiable if and only if there is a certificate y such thatA(x, y) = 1.

SecondWe need to show that F runs in polynomial time.

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First, F correctly computes a reduction function f

We do the followingTo show that F correctly computes a reduction function, let us supposethat there exists a certificate y of length O(nk) such that A(x, y) = 1.

⇐=If we apply the bits of y to the inputs of C , the output of C isC (y) = A (x, y) = 1. Thus, if a certificate exists, then C issatisfiable.

=⇒Now, suppose that C is satisfiable. Hence, there exists an input y toC such that C (y) = 1, from which we conclude that A (x, y) = 1.

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Next, we need to show that F runs in polynomial time

With respect to the polynomial reductionThe length of the input x is n, and the certificate y is O(nk).

NextCircuit M implementing the computer hardware has polynomial size.

PropertiesThe circuit C consists of at most t = O

(nk)copies of M .

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(nk)copies of M .

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(nk)copies of M .

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Finally!!!

In conclusionThe language CIRCUIT-SAT is therefore at least as hard as anylanguage in NP, and since it belongs to NP, it is NP-complete.

TheoremThe circuit satisfiability problem is NP-Complete.

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Finally!!!

In conclusionThe language CIRCUIT-SAT is therefore at least as hard as anylanguage in NP, and since it belongs to NP, it is NP-complete.

TheoremThe circuit satisfiability problem is NP-Complete.

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Proving NP-Complete

Several theorems exist to make our life easierWe have the following

LemmaIf L is a Language such that L′ ≤P L for some L′ ∈ NPC , Then L isNP-Hard. Moreover, if L ∈ NP, then L ∈ NPC .

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Proving NP-Complete

Several theorems exist to make our life easierWe have the following

LemmaIf L is a Language such that L′ ≤P L for some L′ ∈ NPC , Then L isNP-Hard. Moreover, if L ∈ NP, then L ∈ NPC .

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So, we have the following strategy

Proceed as follows for proving that something is NP-Complete1 Prove L ∈ NP.2 Select a known NP-Complete language L′.3 Describe an algorithm that computes function f mapping every

instance x ∈ 0, 1∗ of L′ to an instance f (x) of L.4 Prove that the function f satisfies: x ∈ L′ if and only if f (x) ∈ L L

for all x ∈ 0, 1∗.5 Prove the polynomial time of the algorithm.

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Exercises

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Formula Satisfiability (SAT)

An instance of SAT is a boolean formula φ composed of1 n boolean variables x1, x2, ..., xn .2 m boolean connectives (∧ ,∨, ¬, →, ⇐⇒ ).3 Parentheses.

A small exampleφ = ((x1 → x2) ∨ ¬ ((¬x1 ↔ x3) ∨ x4)) ∧ ¬x2

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The satisfiability problem asks whether a given booleanformula is satisfiable

In formal-language terms

SAT = 〈φ〉 |φ is a satisfiable boolean formula

Example: Given 〈x1 = 0, x2 = 0, x3 = 1, x4 = 1〉

φ = ((0→ 0) ∨ ¬((¬0↔ 1) ∨ 1)) ∧ ¬0= (1 ∨ ¬ (1 ∨ 1)) ∧ 1= (1 ∨ 0) ∧ 1= 1

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φ = ((0→ 0) ∨ ¬((¬0↔ 1) ∨ 1)) ∧ ¬0= (1 ∨ ¬ (1 ∨ 1)) ∧ 1= (1 ∨ 0) ∧ 1= 1

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φ = ((0→ 0) ∨ ¬((¬0↔ 1) ∨ 1)) ∧ ¬0= (1 ∨ ¬ (1 ∨ 1)) ∧ 1= (1 ∨ 0) ∧ 1= 1

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φ = ((0→ 0) ∨ ¬((¬0↔ 1) ∨ 1)) ∧ ¬0= (1 ∨ ¬ (1 ∨ 1)) ∧ 1= (1 ∨ 0) ∧ 1= 1

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φ = ((0→ 0) ∨ ¬((¬0↔ 1) ∨ 1)) ∧ ¬0= (1 ∨ ¬ (1 ∨ 1)) ∧ 1= (1 ∨ 0) ∧ 1= 1

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Formula Satisfiability

TheoremSatisfiability of boolean formulas is NP-Complete.

Proof1 The NP part is easy.2 Now, the mapping from a NPC.

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Showing that SAT belongs to NP

CertificateIt consists of a satisfying assignment for an input formula φ.

Then A does the followingThe verifying algorithm simply replaces each variable in the formula withits responding value and then evaluates the expression.

PropertiesThis task is easy to do in polynomial time.If the expression evaluates to 1, then the algorithm has verified thatthe formula is satisfiable.

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Now, we try the mapping from CIRCUIT-SAT to SAT

Naïve algorithmWe can use induction to express any boolean combinational circuit asa boolean formula.

ThenWe simply look at the gate that produces the circuit output andinductively express each of the gate’s inputs as formulas.

NaivelyWe then obtain the formula for the circuit by writing an expressionthat applies the gate’s function to its inputs’ formulas.

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Problem

PROBLEMWhat happens if the circuit fan out? I.e. shared sub-formulas can makethe expression to grow exponentially!!!

Driving Gate

FAN-OUT=N

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Instead, we use the following strategy

FirstFor each wire xi in the circuit C , the formula has a variable xi

Remember you need the last wire to be true!!!

ThenWe can now express how each gate operates as a small formula involvingthe variables of its incident wires.

Actually, we build a sequence of tautologiesx10 ←→ (x7 ∧ x8 ∧ x9)

We call each of these small formulas a clause.

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Use CIRCUIT-SAT

We a circuit C ∈CIRCUIT-SAT

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Thus, we have the following clauses

The new boolean formula

φ = x10 ∧ (x4 ↔ ¬x3)∧ (x5 ↔ (x1 ∨ x2))∧ (x6 ↔ ¬x4)∧ (x7 ↔ (x1 ∧ x2 ∧ x4))∧ (x8 ↔ (x5 ∨ x6))∧ (x9 ↔ (x6 ∨ x7))∧ (x10 ↔ (x7 ∧ x8 ∧ x9))

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Furthermore

Something NotableGiven that the circuit C is polynomial in size:

it is straightforward to produce such a formula φ in polynomial time.

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What do we want to prove?

ThenWe want the following

=⇒If C has a satisfying assignment then φ is satisfiable.

⇐=If some assignment causes φ to evaluate to 1 then C is satisfiable.

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SatisfiableIf C has a satisfying assignment, then each wire of the circuit has awell-defined value, and the output of the circuit is 1.

MeaningTherefore, when we assign wire values to variables in φ, each clause of φevaluates to 1, and thus the conjunction of all evaluates to 1.

ConverselyConversely, if some assignment causes φ to evaluate to 1, the circuit C issatisfiable by an analogous argument.

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We have proved thatCIRCUIT − SAT ≤p SAT

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However!

However!Problem: SAT is still too complex.Solution: Use 3-CNF

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However!

However!Problem: SAT is still too complex.Solution: Use 3-CNF

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Definition

FirstA literal in a boolean formula is an occurrence of a variable or itsnegation.

SecondA boolean formula is in conjunctive normal form, or CNF, if it isexpressed as an AND of clauses, each of which is the OR of one ormore literals.

ThirdA boolean formula is in 3-Conjunctive normal form, or 3-CNF, if eachclause has exactly three distinct literals.

(x1 ∨ ¬ ∨ ¬x2) ∧ (x3 ∨ x2 ∨ x4) ∧ (¬x1 ∨ ¬x3 ∨ ¬x4)

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Definition

(x1 ∨ ¬ ∨ ¬x2) ∧ (x3 ∨ x2 ∨ x4) ∧ (¬x1 ∨ ¬x3 ∨ ¬x4)

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Definition

(x1 ∨ ¬ ∨ ¬x2) ∧ (x3 ∨ x2 ∨ x4) ∧ (¬x1 ∨ ¬x3 ∨ ¬x4)

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3-CNF is NP-Complete

TheoremSatisfiability of boolean formulas in 3-Conjunctive normal form isNP-Complete.

ProofThe NP part is similar to the previous theorem.The interesting part is proving that SAT≤p 3-CNF

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Proof NP-Complete of 3-CNF

Parse the formulaExample: φ = ((x1 → x2) ∨ ¬ ((¬x1 ↔ x3) ∨ x4)) ∧ ¬x2

We can use ideas from parsing to create a syntax tree

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Proof NP-Complete of 3-CNF

Parse the formulaExample: φ = ((x1 → x2) ∨ ¬ ((¬x1 ↔ x3) ∨ x4)) ∧ ¬x2

We can use ideas from parsing to create a syntax tree

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Parsing allows us to parse the formula into φ′

This can be done by naming the nodes in the tree

φ′ = y1 ∧ (y1 ↔ (y2 ∧ ¬x2))∧ (y2 ↔ (y3 ∨ y4))∧ (y3 ↔ (x1 → x2))∧ (y4 ↔ ¬y5)∧ (y5 ↔ (y6 ∨ x4))∧ (y6 ↔ (¬x1 ↔ x3))

ProblemWe still not have the disjunctive parts... What can we do?

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We can do the followingWe can build the truth table of each clause φ′i !

For example, the truth table of φ′1 = y1 ↔ (y2 ∧ ¬x2)y1 y2 x3

1 1 11 1 01 0 11 0 00 1 10 1 00 0 10 0 0

y1 ↔ (y2 ∧ ¬x2)01001011

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We can do the followingWe can build the truth table of each clause φ′i !

For example, the truth table of φ′1 = y1 ↔ (y2 ∧ ¬x2)y1 y2 x3

1 1 11 1 01 0 11 0 00 1 10 1 00 0 10 0 0

y1 ↔ (y2 ∧ ¬x2)01001011

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From this, we have

Disjunctive normal form (or DNF)In each of the zeros we put a conjunction that evaluate to ONE

y1 y2 x3

1 1 11 1 01 0 11 0 00 1 10 1 00 0 10 0 0

y1 ↔ (y2 ∧ ¬x2)0 y1 ∧ y2 ∧ x31 · · ·0 y1 ∧ ¬y2 ∧ x30 y1 ∧ ¬y2 ∧ ¬x31 · · ·0 ¬y1 ∧ y2 ∧ ¬x31 · · ·1 · · ·

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Then, we use disjunctions to put all them togetherWe have then an OR of AND’s

I = (y1 ∧ y2 ∧ x3) ∨ (y1 ∧ ¬y2 ∧ x3) ∨ (y1 ∧ ¬y2 ∧ ¬x3) ∨ (¬y1 ∧ y2 ∧ ¬x3)

Thus, we have that ¬I ≡ φ′1

y1 y2 x3

1 1 11 1 01 0 11 0 00 1 10 1 00 0 10 0 0

y1 ↔ (y2 ∧ ¬x2) I ¬I0 1 01 0 10 1 00 1 01 0 10 1 01 0 11 0 1

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Then, we use disjunctions to put all them togetherWe have then an OR of AND’s

I = (y1 ∧ y2 ∧ x3) ∨ (y1 ∧ ¬y2 ∧ x3) ∨ (y1 ∧ ¬y2 ∧ ¬x3) ∨ (¬y1 ∧ y2 ∧ ¬x3)

Thus, we have that ¬I ≡ φ′1

y1 y2 x3

1 1 11 1 01 0 11 0 00 1 10 1 00 0 10 0 0

y1 ↔ (y2 ∧ ¬x2) I ¬I0 1 01 0 10 1 00 1 01 0 10 1 01 0 11 0 1

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Using DeMorgan’s laws

We obtainφ′′1 = (¬y1∨¬y2∨¬x2)∧ (¬y1∨y2∨¬x2)∧ (¬y1∨y2∨x2)∧ (y1∨¬y2∨x2)

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Now, we need to include more literals as necessary

Given Ci as a disjunctive part of the previous formula.If Ci has 3 distinct literals, then simply include Ci as a clause of φ.

If Ci has 2 distinct literalsif Ci = (Ii ∨ I2), where I1 and I2 are literals, then include

(I1 ∨ I2 ∨ p) ∧ (I1 ∨ I2 ∨ ¬p)

as clauses of φ.Why?

(I1 ∨ I2 ∨ p) ∧ (I1 ∨ I2 ∨ ¬p) = (I1 ∨ I2) ∨ (p ∧ ¬p) =(I1 ∨ I2) ∨ (F) = I1 ∨ I2

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(I1 ∨ I2 ∨ p) ∧ (I1 ∨ I2 ∨ ¬p)

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(I1 ∨ I2 ∨ p) ∧ (I1 ∨ I2 ∨ ¬p)

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(I1 ∨ I2 ∨ p) ∧ (I1 ∨ I2 ∨ ¬p)

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If Ci has just 1 distinct literal IThen include (I ∨ p ∨ q)∧ (I ∨ p ∨¬q)∧ (I ∨¬p ∨ q)∧ (I ∨¬p ∨¬q)as clauses of φ.Why?

(I ∨ p ∨ q) ∧ (I ∨ p ∨ ¬q)∧...(I ∨ ¬p ∨ q) ∧ (I ∨ ¬p ∨ ¬q) = I ∨ [(p ∨ q) ∧ (p ∨ ¬q) ∧ . . .