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Slides by Asaf Shapira & Michael Lewin & Boaz Slides by Asaf Shapira & Michael Lewin & Boaz Klartag & Oded Schwartz.Klartag & Oded Schwartz.

Adapted from things beyond us.Adapted from things beyond us.

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IntroductionIntroductionIn this lecture we’ll cover:

Definition of PCP Prove some classical inapproximabillity results.

Give a review on some other recent ones.

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Review: Decision, Optimization Review: Decision, Optimization ProblemsProblems A decision problem is

a Boolean function ƒ(X), or alternativelya language L {0, 1}* comprising all strings for

which ƒ is TRUE: L = { X {0, 1}* | ƒ(X) } An optimization problem is

a function ƒ(X, Y) which, given X, is to be maximized (or minimized) over all possible Y’s: maxy[ ƒ(X, Y) ]

A threshold version of max-ƒ(X, Y) isthe language Lt of all strings X for which there

exists Y such that ƒ(X, Y) t(transforming an optimization problem into transforming an optimization problem into

decisiondecision)

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Review: The Class NPReview: The Class NP

The classical definition of the class NP is as follows:We say that a language L {0, 1}* belongs to the

class NP, if there exists a Turing machine VL [referred to as a verifier] such thatX L there exists a witness Y such that

VL(X, Y) accepts, in time |X|O(1)

That is, VL can verify a membership-proof of X in L in time polynomial in the length of X

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Review: NP-Hardness Review: NP-Hardness

A language L is said to be NP-hard if an efficient (polynomial-time) procedure for L can be utilized to obtain an efficient procedure for any NP-language

This definition allows efficient reduction that use the more general, Cook reduction. An efficient algorithm, translating any NP problem to a single instance of L - thereby showing that L NP-hard - is referred to as Karp reduction.

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Review: Characterizing NPReview: Characterizing NP

Thm [Cook, Levin]: For any L NP there is an algorithm that, on input X, constructs in time |X|O(1), a set of Boolean functions, local-tests

L,X = { 1l }

over variables y1,...,ym s.t.: each of 1l depends on o(1) variables and X L there exists an assignment

A: { y1, ..., ym } { 0, 1 } satisfying all

l

[ note that m and l must be at most polynomial in |X| ]

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Approximation - Some Approximation - Some definitionsdefinitionsDefinition: g-approximation A g-approximation of a maximization (similar for

minimization) function f, is an algorithm that on input X, outputs f’(X) such that:

f’(X) f(X)/g(|X|).

Definition: PTAS (polynomial time approximation scheme) We say that a maximization function f, has a PTAS, if for

every g, there is a polynomial pg and a g-

approximation for f, whose running time is pg(|X|).

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Approximation - NP-hard?Approximation - NP-hard?

We know that by using Cook/Karp reductions, we can show many decision problems to be NP-hard.

Can an approximation problem be NP-Hard?

One can easily show, that if there is g,for which there is a g-approximating for TSP, P=NP.

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Strong, PCP Characterizations of NPStrong, PCP Characterizations of NP

Thm[AS,ALMSS]: For any L NP there is a polynomial-time algorithm that, on input X, outputs L,X = { l }

over y1,...,ym s.t.

each of l depends on O(1) variables

X L assignment A: { y1, ..., ym } { 0, 1 } satisfying all L,X

X L assignment A: { y1, ..., ym } { 0, 1 } satisfies < ½ fraction of L,X

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Probabilistically-Checkable-Probabilistically-Checkable-ProofsProofs Hence, Cook-Levin theorem states that a verifier

can efficiently verify membership-proofs for any NP language

PCP characterization of NP, in contrast, states that a membership-proof can be verified probabilistically– by choosing randomly one local-test,– accessing the small set of variables it

depends on,– accept or reject accordingly

erroneously accepting a non-member only with small probability

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Gap ProblemsGap Problems

A gap-problem is a maximization (or minimization) problem ƒ(X, Y), and two thresholds t1 > t2

X must be accepted if maxY[ ƒ(X, Y) ] t1

X must be rejected if maxY[ ƒ(X, Y) ] t2

other X’s may be accepted or rejected (don’t care)

(almost a decision problem, relates to approximation)

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Reducing gap-Problems to Reducing gap-Problems to Approximation ProblemsApproximation Problems

Using an efficient approximation algorithm for ƒ(X, Y) to within a factor g,one can efficiently solve the corresponding gap problem gap-ƒ(X, Y), as long as t1 / t2 > g2

Simply run the approximation algorithm.The outcome clearly determines which side of the gap the given input falls in.

(Hence, proving a gap problem NP-hard translates to its approximation version, for appropriate factors )

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gap-SAT

Def: gap-SAT[D, v, ] is as follows:

– instance: a set = { l } of Boolean-functions (local-tests) over variables y1,...,ym of range 2V

– locality: each of 1l depends on at most D variables

– Maximum-Satisfied-Fraction is the fraction of satisfied by an assignment A: { y1, ..., ym } 2v

if this fraction = 1 accept < reject

D, v and may be a function of l

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The PCP HierarchyThe PCP Hierarchy

Def: L PCP[ D, V, ] if L is efficiently reducible to gap-SAT[ D, V, ]

– Thm [AS,ALMSS] NP PCP[ O(1), 1, ½] [ The PCP characterization theorem

above ]

– Thm [ RaSa ] NP PCP[ O(1), m, 2-m ] for m logc n for some c > 0

– Thm [ DFKRS ] NP PCP[ O(1), m, 2-m ] for m logc n for any c < 1

– Conjecture [BGLR] NP PCP[ O(1), m, 2-m ] for m log n

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Optimal CharacterizationOptimal Characterization

One cannot expect the error-probability to be less than exponentially small in the number of bits each local-test looks at– since a random assignment would make such

a fraction of the local-tests satisfied One cannot hope for smaller than polynomially small

error-probability– since it would imply less than one local-test

satisfied, hence each local-test, being rather easy to compute, determines completely the outcome

[ the BGLR conjecture is hence optimal in that respect]

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Approximating MAX-CLIQUE is NP-hardApproximating MAX-CLIQUE is NP-hard

We will reduce gap-SAT to gap -CLIQUE.

Given an expression = { l } of Boolean-functions over variables y1,...,ym of range 2V, Each of 1l depends on at most D variables, We must determine whether all the functions can be satisfied or only a fraction less than .

We will construct a graph, G , such that it has a clique

of size r there exists an assignment, satisfying r of the functions y1,...,ym.

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Definition of Definition of GG

For each i , G has a vertex for every satisfying assignment of i

1

..

i

.. .. l

All assignmentsAll assignments

to to i’s variables’s variables

Not satisfying Not satisfying iSatisfying Satisfying

i

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Definition of Definition of GG

Two vertices are connected if the assignments are consistent

1

..

i

.. .. l

Consistent valuesConsistent values

NOT Consistent NOT Consistent Different values Different values of same variableof same variable

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Lemma:

(G) = l X L

Consider an assignment A satisfying

For each i consider A's restriction to i‘s

variables The corresponding l vertexes form a clique

in G

Any clique of size m in G implies an

assignment satisfying m of 1l

Properties of Properties of GG

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Each of the following theorems gives a hardness of approximation result of Max-Clique:

– Thm [AS,ALMSS] NP PCP[ O(1), 1, ½]

– Thm [ RaSa ] NP PCP[ O(1), m, 2-m ] for m logc n for some c > 0

– Thm [ DFKRS ] NP PCP[ O(1), m, 2-m ] for m logc n for any c > 0

– Conjecture [BGLR] NP PCP[ O(1), m, 2-m ] for m log n

Hardness of approximation of Max-CliqueHardness of approximation of Max-Clique

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We will show that if Life Is Meaningful (PNP) Max-3Sat does not have a PTAS.

Given an instance of gap-SAT, = { l } , we will

transform each of the i‘s into a 3-SAT expression i.

As each of the i‘s depends on up to D variables. The equivalent i expressions require exp(D) clauses. Since D = O(1) we still remain with a blow up of O(1)

We define the equivalent 3-SAT expression to be: =

The number of clauses in exp(D) l

Hardness of approximation of Max-3SATHardness of approximation of Max-3SAT

iliψ

1

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If X L then there is an assignment satisfying all l Boolean functions of . Such an assignment satisfies all clauses of .

If X L then no assignment satisfies more then l Boolean functions of . Therefore no assignment satisfies more than || - l.

Therefore solving Gap-3SAT with thresholds t1 = 1 and t2 = 1 - l/|| 1 - /exp(D) is NP-Hard.

We conclude that there can be no PTAS for Max-3SAT.

Gap-3SAT is NP-Hard with thresholds 1 and 7/8+. Can be solved with thresholds 1 and 7/8.

Hardness of approximation of Max-3SATHardness of approximation of Max-3SAT

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The PCP theorem has ushered in a new era of hardness of approximation results. Here we list a few:

We showed that Max-Clique ( and equivalently Max-Independent-Set ) do not has a PTAS. It is known in addition, that to approximate it with a factor of n1- is hard unless co-RP = NP.

Chromatic Number - It is NP-Hard to approximate it within a factor of n1- unless co-RP = NP. There is a simple reduction from Max-Clique which shows that it is NP-Hard to approximate with factor n.

Chromatic Number for 3-colorable graph - NP-Hard to approximate with factor 5/3- (i.e. to differentiate between 4 and 3). Can be approximated within O(nlogO(1) n).

More Results Related to PCPMore Results Related to PCP

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Set Cover - NP-Hard to approximate it within a factor of ln n. Cannot be approximated within factor (1-)ln n unless NP Dtime(nloglogn).

More Results Related to PCPMore Results Related to PCP

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Maximum Satisfying Linear Sub-System - The problem: Given a linear system Ax=b (A is n x m matrix ) in field F, find the largest number of equations that can be satisfied by some x.

– If all equations can be satisfied the problem is in P.– If F=Q NP-Hard to approximate by factor m. Can be

approximated in O(m/logm).– If F=GF(q) can be approximated by factor q (even a

random assignment gives such a factor). NP-Hard to approximate within q-. Also NP-Hard for equations with only 3 variables.

– For equations with only 2 variables. NP-Hard to approximated within 1.0909 but can be approximated within 1.383

More Results Related to PCPMore Results Related to PCP