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PCP and Inapproximability

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How big is it? No more than twice the minimum! This is a solution: all edges are covered ... Why is this amazing? One tiny bug or hole or gap can cause much agony... – PowerPoint PPT presentation

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Title: PCP and Inapproximability


1
PCP and Inapproximability
  • Irit Dinur
  • NEC

2
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
3
What do we mean by approximation? Each instance
has many solutions, each has a value. In
optimization, we are seeking the minimal.
4
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
5
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.

6
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.
7
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
8
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

9
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
10
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
11
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
12
Why is it hard to prove?
?(x1,x2,,xn)
0 1 0
Simple Every SAT assignment translates to a
size-k VC for G.
13
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
14
Probabilistically Checkable Proofs
15
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!

16
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.

17
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

18
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.

19
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.

20
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

21
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
22
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

23
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

24
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

25
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
26
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
27
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
28
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.

29
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.

30
Long-Code of R, LCR?0,12R
  • One bit for every subset of R

1
2
R
. . .
31
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
32
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.

33
The Disjointness Graph of the Biased Long-Code
34
1
2
. . .
R
What is a codeword?
35
(No Transcript)
36
  • 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.

37
(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

38
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.

39
Thanks
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