PCP and Inapproximability

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PCP and Inapproximability

• Irit Dinur
• NEC

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Given any function.. How complicated is it to
compute it?

• Example the Minimum Vertex Cover function
• Facts 1. Best algorithm runs in time (1.21)n
  Robson 86
• 2. VC is NP-hard. Karp 72
• What about approximation.. Output a vertex cover
  thats nearly minimal!

Minimum Vertex Cover
Vertex-Cover Given a network of roads. Each road
requires a toll payment. Goal Put up the
smallest number of toll booths
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What do we mean by approximation? Each instance
has many solutions, each has a value. In
optimization, we are seeking the minimal.
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Approximation
An approximation algorithm finds a solution
within a certain neighborhood of MIN

• Example An algorithm for Approximating Vertex
  Cover
• Given G, find a maximal set of edges that do not
  touch each
  other.
• Add both vertices of each edge to the vertex
  cover.

MIN
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Approximation
This is a solution all edges are covered
How big is it? No more than twice the minimum!
An approximation algorithm finds a solution
within a certain neighborhood of MIN

• Example An algorithm for Approximating Vertex
  Cover
• Given G, find a maximal set of edges that do not
  touch each
  other.
• Add both vertices of each edge to the vertex
  cover.

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Approximation
How big is it? No more than twice the minimum!
An approximation algorithm finds a solution
within a certain neighborhood of MIN

• Example An algorithm for Approximating Vertex
  Cover
• Given G, find a maximal set of edges that do not
  touch each
  other.
• Add both vertices of each edge to the vertex
  cover.

Weve seen an approximation algorithm for
Vertex-Cover, with approximation factor 2.
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Approximation
x 2
x 3/2
x 4/3
x 1.99
Weve seen a factor 2 algorithm. Q Is there a
factor 1.99 algorithm? 3/2 ? 4/3 ?
Not unless PNP, AS,ALMSS,BGS,Raz,Hastad,DS
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Finding the Approximation Threshold

• Rich variety of approximation algorithms.
• Various problems with vastly differing
  approximation ratios.
• PCP theorem breakthrough, we can prove hardness
  of approximation.
• Moreover, its within reach to find the exact
  threshold where a problem changes from being easy
  to hard. (theoretic and practical importance)
• Vertex Cover
• Upper bound 2-o(1) BYE, MS, Hal
• Best hardness result
• Thm DS02 NP-hard to approximate to within
  1.36.
• Conjecture NP-hard to within 2- ? ??gt0

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How to show hardness?
Take any NP-hard problem, say SAT, and
reduce Translating a formula ? into a graph G,
s.t.
? is unSAT
VC(G) gt k
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The gap-version of the problem
MIN k

• Given a 1c approximation algorithm,
• If MIN k then Approx lt (1c)?k.
• A 1c approximation algorithm could decide
  whether the minimum is
• below k or
• above (1c)?k

In other words, it would solve the gap
problem Given a graph G, decide if the minimum
VC is
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How to show hardness of approximation?
Take any NP-hard problem, say SAT, and
reduce Translating a formula ? into a graph G,
s.t. Decades, complexity of approximations
remains mysterious Rich variety of algorithms,
hardly any lower-bounds Why?
?
? is unSAT
VC(G) gt k
VC(G) gt (1c)k
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Why is it hard to prove?
?(x1,x2,,xn)
0 1 0
Simple Every SAT assignment translates to a
size-k VC for G.
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Why is it hard to prove?
?(x1,x2,,xn)
0 1 0
?
? is unSAT
VC(G) gt k
VC(G) gt (1c)k
Simple Every SAT assignment translates to a
size-k VC for G. And Every size-k VC
translates to a SAT assignment. But, far from
easy Given a VC of size lt (1c)k, how to decode
it into a SAT assignment?
Combinatorial Decoding
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Probabilistically Checkable Proofs
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PCP a referees dream

• Background context Interactive Proofs
• Say you get a paper for refereeing.
• Select a few random lemmas and verify only them.
• Will this work? in general, of course not!
• PCP Theorem Every proof can be compiled into
  PCP-language so that by reading only a few bits
  of the new PCP proof, correct verification can be
  achieved with high probability.
• Why is this amazing?
• One tiny bug or hole or gap can cause much agony
• In a proof usually the correctness of each step
  depends on much of what happened before it
• It seems that if you only read a few bits, you
  can easily be cheated!

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More Concretely
(x1vx3vx12)(x1vx2vx8)(x7vxnvx10)(x1vx15vx14)
1
0
1
1
1
0
0
0
1
0
1
1
0
0
1
0
0
1
0
1

• An NP statement can be written as a 3SAT formula
• A proof for its satisfiability is an assignment
  of 0/1
• We can verify it clause by clause.

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More Concretely
(x1vx3vx12)(x1vx2vx8)(x7vxnvx10)(x1vx15vx14)
x1
x5
xn-3
x9
x13
x2
x6
xn-2
x10
x14

x3
x7
xn-1
x11
x15
x4
x8
xn
x12
x16

• An NP statement can be written as a 3SAT formula
• A proof for its satisfiability is an assignment
  of 0/1
• We can verify it clause by clause.
• Murphys law! we detect an error only on the
  last clause

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More Concretely
?(y1v y13v y2)(y15v y19v y29)(y22v y13v
y21)(y4v y31v y24)

• ?(x1vx3vx12)(x1vx15vx14)

PCP

• The PCP theorem, gives a compiler translating
  ? into ? s.t. ? admits super-efficient
  verification.
• If ? is satisfiable, then so is ?
• If ? is not satisfiable, then every assignment
  satisfies at
• most 99 of ?s clauses.

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More Concretely
?(y1v y13v y2)(y15v y19v y29)(y22v y13v
y21)(y4v y31v y24)

• ?(x1vx3vx12)(x1vx15vx14)

PCP

• Conclusion AS,ALMSS gap-SAT is NP-hard
• Given a 3-SAT formula ? it is NP-hard to
  decide whether
• ? is satisfiable
• Every assignment satisfies at most 99 of ?s
  clauses.

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This is an Inapproximability Result!
PCP

• Hardness for gap-3SAT, shows that it is NP-hard
  to approximate the following optimization
  problem
• Max-3-SAT Given a 3SAT formula,
  ?(y1vy3vy12)(y1vy15vy14),
• find the maximal fraction of satisfiable
  clauses.
• A random assignment easily satisfies 7/8
• Håstad proved theres no better algorithm

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Some Brief History

• Context Interactive proofs
• Connection to Approximation FGLSS 91
• PCP Theorem AS, ALMSS 92
• Immediately for many problems, e.g. vertex cover,
  max-cut, metric-TSP, max-3SAT, bounded-degree-cliq
  ue,

?
? is unSAT
? is lt 99 SAT
VC(G) gt (1c)k
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Some Brief History

• Context Interactive proofs
• Connection to Approximation FGLSS 91
• PCP Theorem AS, ALMSS 92
• Immediately for many problems, e.g. vertex cover,
  max-cut, metric-TSP, max-3SAT, bounded-degree-cliq
  ue, PY 91
• Boom of hardness of approximation results
• Emphasis on
• Better PCP parameters
• Tight inapproximability results

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Tighter Results

• ,BGLR,FK,BS Better reductions with explicit
  constants
• BGS 95 Introduced the Long-Code.
• e.g. for VertexCover 1.068, for Max-CUT 1.014
• Håstad 96-97 Clique is NP-hard to approximate
  to within n1-? Optimal gap for 3-SAT and for
  Linear equations.
• Using Fourier analysis of the marvelous
  Long-Code, and a Stronger PCP Raz 95
• Hardness factor for VertexCover 1.166, for
  Max-CUT 1.062

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My Work

• The Biased Long-Code a generalization of the
  Long-Code. D.,Safra 02
• New perspectives on the Long Code yielding
  powerful new techniques.
• Analysis of influence of variables on Boolean
  Functions
• Extremal Set Combinatorics.
• New stronger enhancements of the PCP theorem,
  e.g. the Layered PCP in DGKR 02, DRS 02
• Leading to best-known inapproximability results
  for
• Vertex-Cover Hardness for vertex cover 1.367.
  D.,Safra 02
• Approximate Hypergraph Coloring Approximate
  hypergraph coloring. D.,Regev,Smyth 02
• Hypergraph Vertex Cover D. D.,Guruswami,Khot
  D.,Guruswami,Khot,Regev 02

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Combinatorial Decoding

• Proving hardness for gap-VC, we translate ? into
  G and then prove 2 things
• I.
• II.
• The hard part of the proof is part II, showing
  that
• Every VC in G of size lt (1c)k can be decoded
  into a satisfying assignment for ?.
• In standard coding theory, we encode n bits by m
  bits (mgtn), and are able to recover somewhat
  corrupt codewords.
• Every word, if close enough, we can decode
• In combinatorial decoding, we encode an
  assignment for ? by a vertex cover in G and are
  able to recover somewhat corrupt vertex covers.
  
• Every VC, if small enough, we can decode

? is lt 99 SAT
VC(G) gt (1c)k
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The Underlying Structure

• Starting Point the PCP theorem
• Enhance it
• Apply the Long-Code on small sub-components.
• The hardest part of these works is the interplay
  combining these two parts

PCP
Enhanced PCP
Long-Code
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Vertex Cover

• A very loose outline of the construction
• A satisfying assignment can be encoded into a
  vertex-cover.
• A vertex-cover for the graph is a vertex cover in
  each H .
• Decode each small vertex cover in H into a
  value for the underlying y variable.
• Then, show consistency between these values.
• Combinatorial Question Construct such a graph H
  .

? (y1v y13v y2) (y15v y19v y29) (y22v y13v y21)
(y4v y31v y24)
y1
y2
y3
ym
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Sub-Goal Construct a graph H

• Such that,
• Each value in 1,2,..,R corresponds to a small
  vertex cover for H (i.e. of size k ½V).
• Every vertex cover for H , if smaller than
  (2-?)k roughly corresponds to a single value in
  1,2,..,R.
• Technique
• Biased Long-Code,
• Analysis of influence of variables on Boolean
  functions,
• Erdös-Ko-Rado theorems on intersecting families
  of subsets.

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Long-Code of R

• R elements, can be most concisely
• encoded by log R bits.
• Seeking redundancy properties we use
• many more bits in the encoding.
• The Long-Code is the most redundant
• way, using 2R bits.

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Long-Code of R, LCR?0,12R

• One bit for every subset of R

1
2
R
. . .
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Long-Code of R, LCR?0,12R

• One bit for every subset of R
• How do we encode the element i?R?
• (Whats the value of LC(i)?)

1
2
R
. . .
1
0
0
1
1
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The p-Biased Long-Code

• Endow the bits with the product distribution
• For each subset F, ?p(F) pF(1-p)R\F
• Roughly take only subsets whose size is p?R.

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The Disjointness Graph of the Biased Long-Code
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1
2
. . .
R
What is a codeword?
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(No Transcript)
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• A codeword is a vertex cover
• The complement of a vertex-cover is always an
  independent set.
• In this graph, an independent set is an
  intersecting family of subsets.
• Claim a long-code codeword, i.e. all subsets
  containing i is a largest independent set, and
  its complement, a smallest vertex cover.
• Maximal Intersecting Families of Subsets
  Erdös-Ko-Rado 61
• Lemma The ?p size of an intersecting family is ?
  p (proof using shadows Kruskal 63, Katona
  68)
• Much more difficult to prove Any vertex cover
  whose size is lt 1-p2 is decodable into a value
  in 1,,R. (combinatorial decoding)
• Using the complete characterization of maximal
  intersecting families by Ahlswede and Khachatrian
  97, and Friedguts Theorem on when Boolean
  Functions are Juntas, etc.

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(2-?)k ???

• We constructed a graph s.t.,
• Each value in 1,2,..,R corresponds to a small
  vertex cover for H (i.e. of size k).
• Every vertex cover for H , if smaller than
  (4/3)k roughly corresponds to a single value in
  1,2,..,R.
• Now we can plug it into the whole construction

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Future Directions

• Finding the true threshold (stronger
  combinatorial decoding)
• Factor 2 inapproximability for Vertex Cover
• Other problems approximate coloring, etc.
• Simplification of PCP, locally testable codes.
• Decoding in completely different contexts, with
  applications for database privacy.

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Thanks