Transcript Constraint Satisfaction, Semidefinite Programming and the

Locally Testable Codes
Analogues to the
Unique Games Conjecture
Do Not Exist
Gillat Kol
joint work with Ran Raz
Summary
• The Unique Games Conjecture (UGC) is an important
open problem in the study of PCPs
• It conjectures the existence of PCPs with special
properties
• Known PCP constructions are based on Locally
Testable Codes (LTCs) with analogues properties
• We show that LTCs with properties analogues to the
UGC do not exist
• Thus, show limitations of some of the current PCP
constructions techniques
The PCP Theorem
The PCP Theorem
• An unbounded prover wants to convince a poly-time
verifier that SAT, by supplying a proof
• The verifier wants to only read constant number of
symbols from the proof
• PCP Thm [BFL,FGLSS,AS,ALMSS ‘92]: This can be done!
‐Completeness: SAT   proof accepted whp
‐Soundness:
SAT   proof rejected whp
• The proof supplied by the prover is called a
probabilistically checkable proof (PCP)
The PCP Theorem
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Probabilistically
Checkable Proof p
(2 queries)
1. Toss coins to get locations i and j
2. Query pi and pj
3. Using pi and pj, decide if to accept
Verifier
The Unique Games Conjecture
Why is the UGC Interesting?
• Almost all hardness of approximation results rely on
the PCP Theorem
• Yet, for many fundamental problems, optimal
hardness results are still not know
• The UGC is a strengthening of the PCP Theorem
shown to imply many improved hardness results
Max-Cut [MOO ‘05, KKMO ‘07],
Vertex-Cover [KR ‘08],
CSPs [Rag ‘08],
…
Unique Tests
• The UGC deals with verifiers V that read 2 locations
and only make unique tests:
i,j queried by V
 permutation ij:    s.t.
V accepts iff ij(pi) = pj
• That is, after reading location i, there exists a unique
value for location j that makes V accept (and vice versa)
The Unique Games Conjecture
Unique Games Conjecture [Khot ‘02]:
,s > 0 consts
 (const size depends on ,s) s.t.
V checking proofs for “SAT” over 
by only performing unique tests
Completeness 1-: SAT   proof accepted wp ≥ 1-
Soundness s:
SAT   proof accepted wp < s
Parallel Repetition Theorem [Raz ‘98]: Such a verifier
exists when uniqueness is relaxed to projection
Locally Testable Codes
Error Correcting Codes
• Hamming Distance:
‐dist(u,w) = frac of coordinates u and w disagree on
‐agree(u,w) = frac of coordinates u and w agree on
• Error Correcting Code: C n
• Relative Distance: C has relative distance 1- if
u  w  C, dist(u,w) ≥ 1-
equiv.
agree(u,w)  
High relative distance  Good error correcting ability
Locally Testable Codes
Locally Testable Code: A code C with a tester (prob algo)
that checks if a given word v is in C
by only reading a constant number of locations
Completeness 1-: vC
 accept wp ≥ 1-
Soundness s, : dist(v,C) > 1-  accept wp < s
equiv.
accept wp  s  uC, agree(u,v)  
PCPs and LTCs
• Both PCP verifiers and LTC testers test if a given string is
“close” to being “good” (good = valid proof /codeword)
by reading only a constant number of locations in it
• Known PCP constructions are based on LTCs with
analogues properties
“LTCs Analogues to the UGC”?
• (,,s,)-LTC:
Relative distance 1- (codewords agree   frac)
Completeness
1- (codewords accepted wp  1-)
Soundness
s,  (dist > 1-  accept wp < s)
• The UGC requires a low-error PCP with unique tests
• Uniqueness: A Unique LTC is an LTC with unique tests
• Low-error: In known PCPs, the error originates from the
completeness, soundness, and distance of the LTC used
Thus, we would have wanted:
 > 0 const, LTC with ,,s <  for some 
Our Results
Our Result
Theorem (Main):
Let C n be an (,,s,)-unique LTC.
Denote c1 = 10-102 and c2 = 1010||/ (consts)
If s  10-5 and ,  c1 then |C|  c2
• I.e., fixing  fixes a const c1, s.t.  and  cannot both be
smaller than c1, unless C is of const size
• Some Tightness:  = {a, b, c, …}, C = {an, bn, cn, …}.
C is a unique-LTC with ==0 (test: vi = vj), and |C|=||
Proof
Constraint Graphs
• Proof by way of contradiction:
Let C be such a unique LTC with tester T
• T can be viewed as a constraint graph G
‐Vertex set = [n]
‐There exist an edge (i,j) if T may query locations (i,j)
‐The edge (i,j) is associated with ij
• A word v satisfies the edge (i,j) if ij(vi) = vj
Step 1 (Main): Decompose G
Decompose G to small connected components by
removing only a small number of edges (obtain G*)
• Each connected component of G* contains  n vertices
• G* contains  210-4e edges (e = #edges in G)
G
G*
 n vertices
 210-4e edges
Step 2: Constructing a “Bad” Word
• Set k  1/ constant
• Partition the connected components of G* to k sets, each
containing  n/k vertices (components of G* are small)
• Let v* be “balanced” hybrid of any k different codewords
(|C| large), agreeing with each on one of the k parts of G*
G*
v* Violates Soundness
• v* is far from the code:
‐v* is a hybrid of codewords
‐Codewords disagree on most coordinates (relative dist)
‐v* cannot agree with either on many coordinates
• v* is accepted with non-negligible prob:
‐On every component of G*, v* agrees with a codeword
‐On this component, v* only violates the edges violated
by the codeword
‐v* satisfies most of the edges in G* (Completeness)
‐v* satisfies many edges (G* contains many edges)
v* violates soundness!
Thank You!