dictator tests and hardness of approximating max-cut-gain ryan o’donnell carnegie mellon (includes...
TRANSCRIPT
Dictator testsand
Hardness of approximating Max-
Cut-Gain
Ryan O’Donnell
Carnegie Mellon
(includes joint work with Subhash Khot of Georgia Tech)
Talk outline
1. Constraint satisfaction problems and hardness of approximation
2. Dictator Tests & “Slightly Dictator” Tests
3. A new Slightly Dictator Test and hardness of approximation
result for the Max-Cut-Gain problem.
Talk outline
1. Constraint satisfaction problems and hardness of approximation
2. Dictator Tests & “Slightly Dictator” Tests
3. A new Slightly Dictator Test and hardness of approximation
result for the Max-Cut-Gain problem
Constraint Satisfaction Problems
Let be a class of predicates (“constraints”) on a few bits; e.g.,
•
• “ X © Y © Z = b ”
• “ X Y ”
The “Max-” constraint satsifaction problem:
• Given m predicates/constraints over n variables,
find assignment satisfying as many as possible.
Max-3Lin
Max-Cut
Max-2SAT
Approximating CSPs
“A is a (c, s)-approximation algorithm for Max-”:
Given an instance where you can satisfy ¸ c fraction of constraints,
A outputs a solution satisfying ¸ s fraction of constraints.
A should run in polynomial time.
Approximating CSPs
• Gaussian Elimination is a (1, 1)-approximation algorithm for Max-3Lin
• Best known (1 − , s)-approximation for Max-3Lin is a trivial algorithm
with s = ½ – output all 0’s or all 1’s. (A (½ , ½)-approximation.)
• Goemans and Williamson ’95 gave a very famous approximation
algorithm for Max-Cut, which is a -approximation and
also a
(c, s)-approximation for every s < .878c.
• G&W is a (.51, .45)-approximation for Max-Cut, worse than trivial
(the Greedy algorithm is a (½ , ½)-approximation algorithm)
• Charikar and Wirth ’04 gave a ( ½ + , ½ + (/log(1/)) )-
approximation for Max-Cut. (A “Max-Cut-Gain” algorithm.)
Hardness of approximation
PCP (“Probabilistically Checkable Proofs”) technology used to prove
NP-hardness of (c,s)-approximation algorithms.
• Håstad ’97: (1 − , ½ + )-approximating Max-3Lin is NP-hard.
• Håstad ’97: (1, 7/8 + )-approximating Max-3SAT is NP-hard.
• KKMO ’04 + MOO ’05:
Doing any better than the Goemans-Williamson
approximation algorithm is NP-hard*.
* Assuming the “Unique Games Conjecture”.
Hardness of approximation
PCP hardness of approximation rule of thumb:
“To prove hardness of (c, s)-approximating Max-, it suffices to give a
“(c, s)-Slightly-Dictator-Test” where the test is from .”
Talk outline
1. Constraint satisfaction problems and hardness of approximation
2. Dictator Tests & “Slightly Dictator” Tests
3. A new Slightly Dictator Test and hardness of approximation
result for the Max-Cut-Gain problem
Dictators
We will be considering m-bit boolean functions:
Function f is called a “Dictator” if it is projection to one coordinate:
for some 1 · i · m.
(AKA “Singleton” AKA “Long Code”)
Dictator Testing
• In the field of “Property Testing”, unknown f given as a black box.
• Want to determine if f belongs to some class of functions C.
• Want to query f on as few strings as possible. (Constantly many.)
• Clearly, must use randomization, must admit some chance of error.
• For hardness-of-approximation, the relevant C is the class of
all m Dictator functions.
Testing Dictators
A (non-adaptive) Dictator Test:
• Picks x1, … , xq 2 {0,1}m in some random fashion.
• Picks a ‘predicate’ on q bits.
• Queries f (x1), …, f (xq).
• Says “YES” or “NO” according to (f (x1), …, f (xq)).
Each f : {0,1}m ! {0,1} has some probability of “passing” the test.
Hope: probability is large for dictators, and small for non-dictators.
Correlation
If f and g are “highly correlated” – i.e., they agree on almost all
inputs – then the probability they pass will be essentially the same.
So if g is highly correlated with a Dictator, we can’t help but let it pass
with high probability.
(A number between −1 and 1.)
Basic Dictator Testing
• If f is a Dictator, passes with probability 1.
• If f has correlation < 1 − with every Dictator, passes with
probability at most 1 − ().
• Number of queries q should be an absolute constant.
(Like 6 or something.)
(Remark 1: Given such a test, you can get a “standard” Dictator Test
by repeating O(1/) times and saying “YES” iff all tests pass.
Remark 2: ) “Assignment tester” (of exponential length) [Din06])
Examples
• Bellare-Goldreich-Sudan ’95: O(1) queries.
• Håstad ’97 probably gave a 3-query one (he at least could’ve).
• A 3-query one; if you know Fourier, proof is easy homework ex.:
• with probability ½ do the BLR test:
• pick x, y uniformly, and set z = x © y
• test that f (x) © f (y) © f (z) = 0
• with probability ½ do the NAE test:
• for each i = 1…m, choose (xi, yi, zi) uniformly from {0,1}3 n { (0,0,0), (1,1,1) }
• test that f (x), f (y), f (z) not all equal
xyz
Hardness of approximation
PCP hardness of approximation rule of thumb:
“To prove hardness of (c, s)-approximating Max-, it suffices to give a
“(c, s)-Slightly-Dictator-Test” where the test is from .”
(c, s)-Slightly-Dictator-Tests
• If f is a Dictator, passes with probability ¸ c.
• If f has correlation < with every Dictator (and Dictator-negation),
then f passes with probability < s + 0,
where 0 ! 0 as ! 0.
(“If f passes with high enough prob., it’s slightly Dictatorial.”)
(For PCP purposes, you can sometimes even get away with
“Very-Slightly-Dictator-Tests”…)
Talk outline
1. Constraint satisfaction problems and hardness of approximation
2. Dictator Tests & Slightly Dictator Tests
3. A new Slightly Dictator Test and hardness of approximation
result for the Max-Cut-Gain problem
Max-Cut Slightly-Dictator-Tests
For Max-Cut, you need a 2-query Slightly-Dictator-Test, where the
tests are of the form “f (x) f (y)”.
KKMO ’04 proposed the Noise Sensitivity test:
• Pick x 2 {0,1}m uniformly, form y 2 {0,1}m by flipping each bit independently with probability .
• Test f (x) f (y).
Theorem (conj’d by KKMO, proved in MOO ’05):
This is a (, arccos(1−2)/)-Very-Slightly-Dictator-Test.
Corollaries
• = 1 − : Gives -hardness* for Max-Cut
• : Gives (, .74)-hardness* for Max-Cut (.878-gap)
• = ½ + : Gives (½ + , ½ + (2/) )-hardness* for Max-Cut
The first two are best possible, as Goemans and Williamson gave
matching algorithms.
Last doesn’t match ( ½ + , ½ + (/log(1/)) )-approximation algorithm
of Charikar and Wirth. Our goal: give matching hardness.
A new result
Subhash Khot and I improved the hardness result to match Charikar and
Wirth, by analyzing a new Dictator Test:
• Do the Noise Sensitivity test some fraction of time with 1, and some
fraction of the time with 2, balanced so that Dictators pass w.p. ½ + .
Gives a ( ½ + , ½ + (/log(1/)) )-Slightly-Dictator-Test using tests.
Bonuses:
• It’s a Slightly-Dictator-Test (not Very-Slightly-).
• Unlike MOO ’05, after doing the usual Fourier analysis stuff, the proof is about 10 lines rather than 10 pages.
Main technical analysis
• First, rename bits to −1 and 1, rather than 0 and 1.
• Next, do the usual Fourier analysis stuff…
Let f : {−1,1}m ! {−1,1} be any function, and say it has correlation
ci with the ith Dictator function, i = 1…m.
Let L : {−1,1}m ! R be the function:
L(x1, …, xm) = c1 ¢ x1 + c2 ¢ x2
+ ¢ ¢ ¢ + cm ¢ xm
This gives the linear polynomial over R that f “looks most like”.
Main technical analysis
L(x1, …, xm) = c1 ¢ x1 + c2 ¢ x2
+ ¢ ¢ ¢ + cm ¢ xm
2 := ci2
(2 roughly measures how Dictatorial f is.)
Probability f : {−1,1}m ! {−1,1} passes the test is (essentially) equal to:
Main technical analysis
L(x1, …, xm) = c1 ¢ x1 + c2 ¢ x2
+ ¢ ¢ ¢ + cm ¢ xm
2 := ci2
Conclusion: If all correlations ci are small, the distribution of L looks
like a Gaussian. With variance = 2 .
Gaussian facts
• The probability that a Gaussian random variable with variance 1
goes above t is about exp(−t2 / 2).
• By scaling, the probability that a Gaussian with variance 2
goes above t is about exp(−t2 / 22).
• So the probability that a Gaussian with variance 2
goes above 2 is about exp(−2/2).
• If 2 ¸ 10/ln(1/), we have Prx [L(x) > 2] ¸ 1/5.
Main technical analysis
L(x1, …, xm) = c1 ¢ x1 + c2 ¢ x2
+ ¢ ¢ ¢ + cm ¢ xm
2 := ci2
If all correlations ci are small, then:
If 2 ¸ 10/ln(1/), we have Prx [L(x) > 2] ¸ 1/5
) ( ½ + , ½ + (/log(1/)) )-Slightly-Dictator-Test
Open problem
• Suppose you want a
3-query (1, s)-(Very)-Slightly-Dictator-Test
• Till recently, best s was Håstad’s 3/4.
• Khot & Saket ’06 got s down to 20/27.
• Conjectured (by Zwick) best s: 5/8 (!).
• I’m pretty sure I know the test, but I can’t analyze it…