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Introduction to PCP. Introduction. In this lecture we’ll cover: Definition of PCP Prove some classical hardness of approximation results Review some recent ones. Review: Decision, Optimization Problems. A decision problem is a Boolean function ƒ(X) , or alternatively

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Introduction to PCP

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Introduction

In this lecture we’ll cover:

• Definition of PCP
• Prove some classical hardness of approximation results
• Review some recent ones
Review: Decision, Optimization Problems
• A decision problem is

a Boolean function ƒ(X), or alternatively

a languageL  {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) is

the languageLt of all strings X for which there exists Y such that ƒ(X, Y)  t

[transforming an optimization problem into decision]

Review: The Class NP

The classical definition of the class NP is:

A language L  {0, 1}* belongs to the classNP, if there exists a Turing machine VL[referred to as a verifier] such that

X  L there exists a witnessY such thatVL(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

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 LNP-hard - is referred to as Karp reduction
Review: Characterizing NP

Thm[Cook,Levin]:For L NP there’s an algorithm that, on input X, constructs, in time |X|O(1), a set of local-constraints (Boolean functions)L,X = { j1, ..., jl }over variables y1,...,ym s.t.:

• each of j1, ..., jl depends on o(1) variables
• X  L there exists an assignment A: { y1, ..., ym } a { 0, 1 }satisfying all L,X

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

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NP characterization

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IfX  L,all of

the local tests are

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Approximation - Some Definitions

Def: g-approximation

A g-approximationof 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|).

Def: PTAS (poly-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-approximationfor f, whose running time is pg(|X|)

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.

PCP

AS,ALMSS

X  L assignment A: { y1, ..., ym }  { 0, 1 }satisfies < ½ fraction of L,X

Characterization of NP

Thm[Cook,Levin]:For L NP there’s an algorithm that, on input X, constructs, in time |X|O(1), a set of local-constraints (Boolean functions)L,X = { j1, ..., jl }over variables y1,...,ym s.t.:

• each of j1, ..., jl depends on o(1) variables
• X  L there exists an assignment A: { y1, ..., ym } a { 0, 1 }satisfying all L,X

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PCP NP characterization

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Probabilistically Checkable Proofs
• 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-constraint,
• accessing the small set of variables it depends on,
• accept or reject accordingly
• erroneously accepting a non-member only with small probability
Gap 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)

Reducing gap-Problems to Approximation 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 )
gap-SAT
• Def: gap-SAT[D, v, ] is as follows:
• Instance: a set  = { j1, ..., jl }of Boolean-functions (local-constraints)over variables y1,...,ym of range 2V
• Locality: each of j1, ..., jl depends on at mostD variables
• Maximum-Satisfied-Fraction is the fraction of  satisfied by an assignment A: { y1, ..., ym } a 2vif this fraction
• = 1  accept
• < reject
• D, v and  may be a function of l
The 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
Optimal 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]

Approximating MAX-IS is NP-hard

We will reduce gap-SAT to gap –Independent-Set.

Given an expression  = { j1, ..., jl }of Boolean-functions over variables y1,...,ym of range 2V, Each of j1, ..., jl depends on at mostD variables, we must determine whether all the functions can be satisfied or only a fraction less than .

We will construct a graph, G , that has an independent set of size r there exists an assignment, satisfying r of the local-constraints y1,...,ym.

q

(q,r)-co-partite Graph G=(QR, E)
• Comprise q=|Q| cliques of size r=|R|:E {(<i,j1>, <i,j2>) | iQ, j1,j2 R}

q

Thm:IS( r,  ) is NP-hard as long as r  ( 1 / )cfor some constant c

Gap Independent-Set

Instance: an (q,r)-co-partite graph G=(qR, E)

Problem: distinguish between

• Good: IS(G) = q
• Bad: every set I  V s.t. |I|> q contains an edge

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gap-SAT  gap-IS

Construct a graphG that has 1 clique i  ,

in which 1 vertex  satisfying assignment for i

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gap-SAT  gap-IS

Two vertices are connected if the assignments they represent are inconsistent

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gap-SAT  gap-IS

Lemma:a(G) = k (independent set of size k) X  L (There is an assignment that satisfies k clauses)

• Consider an assignment A satisfying k clauses. For each clause i consider A's restriction to ji‘s variables The corresponding k vertexes form an independent set in G
• Any independent set of size k in G implies an assignment satisfying k of j1, ..., jl

Hence: Gap-IS is NP hard, and IS is NP-hard to approximate!

Hardness of approximation of Max-IS

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

• 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 forMax-3SAT

Assuming the PCP theorem, we will show that if PNP,Max-3Sat does not have a PTAS:

Theorem: There is a constant C>0 so that computing (1+c) approximations to Max-3Sat is NP-hard

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Hardness of approximation forMax-3SAT

SAT formula

Equivalent 3SAT formula

variables

Given an instance of gap-SAT,  = { j1, ..., jl }, we will

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

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Hardness of approximation forMax-3SAT
• Hence each function can be represented as a CNF formula i:

Given an instance of gap-SAT,  = { j1, ..., jl }, there are O(n) functions ji . Each of the ji‘s depends on up to D=O(1) variables.

(a conjunction of 2^D clauses, each of size at most D)

Note that the number of clauses is still constant.

Overall, we build a CNF formula: a conjunction of i (one for or each local test).

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Hardness of approximation forMax-3SAT

Now rewrite every D-clause as a group of 3-clauses to obtain a 3-CNF:

Note that this is still a constant blow up in the number of clauses.

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Hardness of approximation forMax-3SAT

In case  is NOT satisfyable, some constant fraction of the j1 are not satisfied, and for each, at least one clause in i isn’t satisfied.

Hardness of approximation forMax-3SAT

Conclusion:

In case the original SAT formula  isn’t satisfied, a constant number of 3SAT formula i are not satisfied, and for each at least one clause isn’t satisfied.

Because each i contains a constant number of clauses, altogether a constant number of clauses in the resulting 3SAT aren’t satisfied.

This provides a gap, and hence 3SAT cannot be approximated to within some constant unless P=NP!

More Results Related to PCP

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(nlogO(1) n).
More Results Related to PCP
• Vertex Cover – Very easy to approximate within a factor of 2. NP-Hard to approximate it within a factor of 4/3.
• Max-3-Sat – Known to be approximable within a factor of 8/7. NP-Hard to approximate within a factor of 8/7- for every >0
• 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 PCP
• 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