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.)