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The Max-Cut problem:
Election recounts?
Majorityvs.
Electoral College?
7812
The 2-Lin(mod 3) problem:
Simultaneously satisfy as many as you can.
2-Variable Constraint Satisfaction Problems(“2-CSPs”)
Variables: x1, x2, x3, … , xn
Label Set: (= allowed values for the variables)
Input: Constraints 1, 2, …, N on pairs of variables.
Output: Assignment satisfying as many constrs. as possible.
“Optimization”, “Approximation Algorithms”
Graph version
x1x2 x3 x5x4 x6 x7 x8 x9 x10
= { , }
2-CSP examples
Max-Cut: = {0,1}, ’s of the form “ xi xj ”
2-Lin(mod 3): = 3 , ’s of the form “xi = xj + c”
2-SAT: = {0,1}, ’s are
Vertex-Cover:Input: A graph.Goal: Select as few vertices as possible s.t. all edges are “covered”.
Coloring 3-colorable graphs:Input: a 3-colorable graph. Goal: Legally color it using as few colors as possible.
(running example)
,
, ,
1. Egg on our face re complexity
of algorithms for 2–CSPs.
Story of the talk
2. Efficient “Property Testing” algs.
) Hardness for CSPs
3. Remarkably efficient (2–query!)
Property Testing algs. exist.
Complexity theory dictum
“Essentially every natural algorithmic problem has been shown to be
in P (polynomial time) or NP-hard.”
(Exceptions: Factoring, Graph-Isomorphism.)
This is a lie.
• Given a graph, find a cut achieving at least 90% of the max cut.
• Given a 3-colorable graph, color it using at most 100 colors.
• Find a vertex cover at most 1.99 times the minimum.
• Find a 2-SAT assignment satisfying 95% of the maximum.
• Given (1−)-satisfiable 2-Lin(mod p) system, satisfy (1/p)/2 fraction.
• Find a cut within factor log log log n of the sparsest cut.
• (1 − 1/2k)-approximate Max-k-Cut
• …
We gotta do something about this!
1. Prove problems are in P.
Seems we need a radically new algorithmic idea.
Max-Cut: Most recent working algorithmic idea was from ’89–’92…
Goemans-Williamson ’94 proved it always finds a cut achieving
¸ 87 . 8567 % of the optimum.
2. Prove new NP-hardness results.
Even after much effort, only some success. Not much for 2-CSPs.(“PCP Theorem” [AS’92, ALMSS’92] + “Parallel Repetition Theorem” [Raz’95])
“Unique Games Conjecture” [Khot’02]2-CSPs?
[Khot’02]
[KR’03]
[KKMO’04][MOO’05] [DMR’06]
([MOO’05])
[KV’05]
Max-k-CSP
Max-3-CSP
[ST’06]
[O’0?]
[KO’06]
2-Lin(mod 2)
Vertex-Cover
Max-Cut
2-SAT2-Lin(mod p)
Coloring 3-Colorable
Graphs Sparsest Cut
Max-Cut-Gain( 87 . 8567 % )
A general theory is developing.
2. Efficient “Property Testing” algs.
) Hardness for CSPs
Story of the talk
3. Remarkably efficient (2–query!)
Property Testing algs. exist.
1. Egg on our face re complexity
of algorithms for 2–CSPs.
Property Testing
= “Constant Time Algorithms”= “The art of uninformed decisions”
Input: A “huge” object: e.g., truth table f : m ! .
Output: YES or NO, depending on whether it has property P.
Caveat: You want to answer in constant time.
What you can do:
• Read f(x) for a few random x, say f(x1 ), …, f(xk ).
• Apply a “test”, ( f(x1 ), …, f(xk ) ) ) YES / NO.
Testing “Dictatorships”
For CSP hardness reductions, relevant P is being a “Dictatorship”:
f(x) = xi
f a Dictatorship ) test outputs YES with prob. ¸ pYES
f “very non-Dictatorial” ) test outputs YES with prob. · pNO
Testing “Dictatorships”
“ k-query, -based, (pYES , pNO) Dictatorship test ”
(for an unknown f : m ! )
1. x1 , x2, …, xk chosen at random (somehow) from m
2. ( f(x1 ), f(x2 ), …, f(xk ) ) is output, either YES or NO
Requirement: f is a Dictatorship ) Pr[output YES] ¸ pYES
f “very non-Dictatorial” ) Pr[output YES] · pNO
CSP hardness Rule of Thumb
NP-hardness (or “Unique Games Conjecture”) reduction for:
“Satisfying a fraction of the optimum,
given a k-CSP instance with constraints.”
pNO
pYES
Remark: This idea is old: from [BGS’95].
Novelty: 2-query Dictatorship tests exist!
“ k-query, -based, (pYES , pNO) Dictatorship test ”
Rule of Thumb example: Max-Cut
Max-Cut: 2-CSP over {0,1} with constraints of form “ xi xj ”.
2-query, “ ”-based, (90%, 80%) Dictator test for f : {0,1}m ! {0,1}
) “Assuming UGC, it is NP-hard to find cuts
that achieve 88.888 % of the optimal cut.”
1. Pick x, y 2 {0,1}m in some clever random way.
2. Query f(x), f(y) and output YES iff f(x) f(y).
) Dictatorships pass w.p. ¸ 90%,
“Totally not Dictatorships”pass w.p. · 80%.
Story of the talk
1. Egg on our face re complexity
of algorithms for 2–CSPs.
2. Efficient “Property Testing” algs.
) Hardness for CSPs
3. Remarkably efficient (2–query!)
Property Testing algs. exist.
f
2-query, “ ”-based Dictatorship test?
m voters
winner
f : {0,1}m ! {0,1}
Voting: 0 & 1 are two parties. m voters. f is voting rule.
2-query, “ ”-based Dictatorship test?
[KKMO’04] suggestion:
Election #1: Each voter flips a coin.
Election #2: Each voter, with probability 90%, reverses their vote.
Test: Winner #1 Winner #2.
Prob[ Dictatorship passes ]:
Prob[ Majority passes ]:
Prob[ Electoral College passes ]:
¼ 79.5%
90%
¼ 70.1%
Majority Is The Highest
[KKMO’04] conjectured, [MOO’05] proved:
“Majority is the non-Dictator passing the test with highest probability.”
Hence: “ ”-based, ( ,
Hence: UGC-hardness of finding cut within
● [GW’94] is optimal Max Cut alg., assuming UGC
● Resolves conjectures [Kalai’03,’04] in theory of
voting,
also problems [ADFS ’04] in combinatorics.
● Result can be used to prove (sometimes improve)
essentially all known UGC reductions.
Consequences:
79.5%
90%
90% 79.5 % ) Dictatorship test.) Dictatorship test.
of Max Cut. ¼ 88.4 %
“Unique Games Conjecture” [Khot’02]2-CSPs?
[Khot’02]
[KR’03]
[KKMO’04][MOO’05] [DMR’06]
([MOO’05])
[KV’05]
Max-k-CSP
Max-3-CSP
[ST’06]
[O’0?]
[KO’06]
2-Lin(mod 2)
Vertex-Cover
Max-Cut
2-SAT2-Lin(mod p)
Coloring 3-Colorable
Graphs Sparsest Cut
Max-Cut-Gain( 87 . 8567 % )
A general theory is developing. [MOO’05]
The proof that Majority is the highest
1. Generalize Central Limit Theorem.
“Sums of random 0’s and 1’s ! Gaussians.”
“Polynomials of random 0’s and 1’s ! polynomials of Gaussians.”
2. m Gaussians is like uniform distribution on m-dim. sphere.
Problem becomes a cut problem on the sphere.
Specifically “Min-Bisection”.
3. For small noise params (angles),
essentially similar to finding the
blob of half-volume w/ smallest perimeter.
(Connections to Double Bubble problem.)
Open problems I’m thinking about
1. Prove Unique Games Conjecture.
([FKO]: trying to give reduction from Max-Cut hardness.)
2. Analyze various other constant-query Dictatorship tests.
3. Change from the “Dictatorship test f : {0,1}m ! {0,1}” paradigm.
([KO’06] has some partial work on this.)
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