Slides by Asaf Shapira & Michael Lewin & Boaz Klartag & Oded Schwartz.
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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 followsWe 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
That is referred to as 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: -approximation An -approximation of a maximization (similar for
minimization) function f, is a function , g, such that on input X, outputs g(X) such that:
g(X) f(X).
Definition: PTAS (polynomial time approximation scheme) We say that a maximization function f, has a PTAS, if for
every 1 0, there is a polynomial -approximation for f, where the algorithm is polynomial in |X| and .
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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 ,for which there is an -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 > 0
– 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 expresion 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-Independet-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 clogn for some constant c. Can not be approximated within factor (c-)logn 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