NP, NP-hardness Approximation PCP

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INTRODUCTION

• NP, NP-hardnessApproximationPCP

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Decision, Optimization Problems

• A decision problem is
• a Boolean function ƒ(X), or alternatively
• a 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 Ys
  maxy ƒ(X, Y)
• A threshold version of max-ƒ(X, Y) is
• the language Lt of all strings X for which there
  exists Y such that ƒ(X, Y) ? t
• (transforming an optimization problem into
  decision)

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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 that
• X ? L ? there exists a witness Y such that
  VL(X, Y) accepts, in time XO(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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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 L NP-hard -- is referred to as
  Karp reduction.

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Motivation

• Proving a set L to be NP-hard
• implies that L, for all practical purposes, is
  unsolvable

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Characterizing NP

• Thm Cook, Levin For any L ? NP there is an
  algorithm that, on input X, constructs in time
  XO(1), a set of Boolean functions,
  local-tests ?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 j1, ..., jl
• note that m and l must be at most polynomial in
  X

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NP-hard Problems

• Def MAX-CLIQUE of a graph G - denoted w(G) - is
  the size of the largest complete-subgraph of G
• ThmCook, Karp MAX-CLIQUE is NP-hard
• Proof Given ? above we construct a graph G?
  whose MAX-CLIQUE determines whether ? is
  satisfiable or not

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G?

• has one vertex for each ji ? ? and an assignment
  satisfying ji

j1 . ji . . jl
? ? ? ? ? ? ? ?
All assignmentsto jis variables
? ? ? ? ? ? ? ?
Not satisfying ji
Satisfying ji
? ? ? ? ? ? ? ?
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G?

• two vertexes connected if assignments consistent

j1 . . . ji . . jl
? ? ? ? ? ? ? ?
Consistent values
? ? ? ? ? ? ? ?
Different value same variable
? ? ? ? ? ? ? ?
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• Lemma w(G?) l ? X ? L
• Consider an assignment A satisfying For each i
  consider A's restriction to jis variables The
  corresponding l vertexes form a clique in G?
• Any clique of size m in G? implies an assignment
  satisfying m of j1, ..., jl

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Approximation

• Weve just seen that computing MAX-CLIQUE is
  NP-hard
• How about approximating it ??
• I.e., coming up with a number that deviate from
  it by at most some specified (small as possible)
  factor

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Strong, PCP Characterizations of NP

• ThmAS,ALMSS For any L ? NP there is a
  polynomial-time algorithm that, on input X,
  outputs ?L,X j1, ..., jl over y1,...,ym
  s.t.
• each of j1, ..., jl depends on O(1) variables
• X ? L ? assignment A y1, ..., ym a 0, 1
  satisfying all ?L,X
• X ? L ? " assignment A y1, ..., ym a 0, 1
  satisfies lt ½ fraction of ?L,X

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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-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 Problems

• A gap-problem is
• a maximization can be defined similarly for
  minimization problem ƒ(X, Y), and two thresholds
  t1 gt t2
• X must be accepted if maxY ƒ(X, Y) ? t1
• X must be rejected if maxY ƒ(X, Y) ? t2
• other Xs may be accepted or rejected dont
  care
• almost a decision problem, relates to
  approximation

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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 gt 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-SATD, v, ? is as follows
• instance a set ? j1, ..., jl of
  Boolean-functions (local-tests) over variables
  y1,...,ym of range 2V
• locality each of j1, ..., jl depends on at
  most D variables
• Maximum-Satisfied-Fraction is the fraction of ?
  satisfied by an assignment A y1, ..., ym a
  2vif this fraction
• 1 ? accept
• lt ? ? reject
• D, v and ? may be a function of l

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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 gt 0
• Thm DFKRS NP ? PCP O(1), m, 2-m for m ?
  logc n for any c gt 0
• Conjecture BGLR NP ? PCP O(1), m, 2-m for m
  ? log n

Focus of the rest of this presentation
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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

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Approximating MAX-CLIQUE is NP-hard

• Apply the same reduction above from SAT to
  MAX-CLIQUE (yielding G?), to the gap-SAT of any
  of the above theorems
• The outcome is a gap-CLIQUE problem,which is
  therefore NP-hard for the corresponding factor
• We may conclude that
• approximating MAX-CLIQUE is NP-hard