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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
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 ?
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,
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
• 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