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Covering CSPs

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Covering CSPs

GillatKol

joint work with IritDinur

CSP= Constraint Satisfaction Problem

Variables: x1,x2,…,xnin {-1,1}.

Constraints: ((x1=1) v (-x2=1) v (x7=-1)), (x2x5= 1), …

Goal:

Ideally: Find assignment that satisfies all constraints.

NP-hard, so we approximate.

Max-CSP:

Restriction:Use only a single asg.

Optimization Goal: Maximize # satisfied constraints.

Min-Cover-CSP (this paper):

Restriction:Must satisfy all constraints.

Optimization Goal: Minimize # asgs.

You invite friends over for dinner.

Each has diff dietary constraints:

- You want everyone to have at least something to eat.
- But, would like to cook as few dishes as possible.

The Covering Number of a CSP instance C,denoted cover(C), is the smallest number of asgsto the variables s.t. every constraint is “covered” (satisfied by at least one asg).

Covering “Extends” Coloring:

[GHS’02]: (Hyper)graph G naturally induces a NAE-CSP instance CG with chromatic(G) 2cover(CG).

Covering & Coloring

x1 ≠ x2

x2≠ x4

x2≠ x5

x3≠ x4

x4 ≠ x5

CG

x1

G

x2

x5

x4

x3

2asgs

over {+,-}

coloring using

4 colors {++,--,+-,-+}

Covering “Extends” Coloring:

[GHS’02]: (Hyper)graph G naturally induces a NAE-CSP instance CG with chromatic(G) 2cover(CG).

Covering allows us to “increase the number of colors” in any predicate .

Covering & Coloring

- Predicate :{+1,-1}t {+1,-1} (-1 = true, 1 = false).
- -CSP = constraints of the form (x1,…,xt).
- The (c,s)-covering Problem: Given a -CSP instance C, decide between (c < s N):
- cover(C) ≤ c.
- cover(C) s.

- Our Goal: Study ’s covering behavior.
- covering ishardif
constc s.t.consts>c: (c,s)-covering is hard.

- covering ishardif

Observation: If is odd ((x) = -(-x)), then cover() 2.

Proof:asgA, {A, -A} covers.

covering 3LIN is easy.

Observation 2: If Odd*, then cover() 2, where

Odd* ={| “contains” an odd predicate}

={|x:(x)=true or(-x)=true}.

covering 3SAT is easy.

Easy Predicates

The Characterization of Covering-Hard Predicates?

Our Covering Dichotomy Conjecture: covering ishardiffOdd*.

Def: 4LIN(x1,x2,x3,x4) = x1x2x3x4.

Result 1: (2,s)-covering 4LINis NP-hard for anyconsts>2.

The “first” interesting new predicate.

4LIN is easy in the max-CSP sense.

Challenge: Getting perfect completeness with 2asgs.

We “doable” the label cover, and apply correlated noise.

Result 1NP-Hardness for covering 4LIN

Result 2 [a la Austrin-Mossel 2009]:

Under a covering unique games conjecture:

If Odd*, and supports a pairwise independent

distribution, then covering is hard.

Challenge: Analyzing soundness for a general predicate.

Observation: Among predicates Odd*, the predicate =NAE has the “largest” support.

Approximate Coloring Problem: Given an O(1)-colorable (hyper)graph, what is the smallest number of colors needed to color it in polynomial time?

lower bound: polylog(n) (hypergraphs) [Khot’02]

upper bound: n

Approximate Coloring Problem: Given an O(1)-colorable (hyper)graph, what is the smallest number of colors needed to color it in polynomial time?

Result 3 [a la Feige’s R3SAT 2002]:

Hypothesis:s.t. given a -CSP instance C, it is hard to tell if C is a random instance, or if cover(C) = 2.

If the hypothesis holds with sufficiently good parameters (density of C), we get polynomialhardness for hypergraph coloring.

Hardness results for are usually obtained through

a -Dictatorship Test.

f:{+1,-1}R{+1,-1} is a dictator if is.t.f(x) = xi.

A 4LIN-Dict Test for f:{+1,-1}R{+1,-1} is specified by a distribution over 4-tuples x,y,z,w{-1,1}R.

It draws x,y,z,wand acceptsifff(x)f(y)f(z)f(w) = -1.

Completeness:f is a dictator Pr[test accepts] 1-.

Soundness:f is “regular” Pr[test accepts] ½+.

low influences,

“far” from dictator

imperfect completeness

A 4LIN-CoveringDict Test for f:{+1,-1}R{+1,-1} is specified by a distribution over x,y,z,w(as before).

Let C be the 4LIN-CSP instanceinduced by the distribution

(every 4-tuplex,y,z,winduces a constraint, f is an asg).

Covering Completeness of the test c:

c dictators that cover C.

Covering Soundness of the test s:

No “regular set” of s functions covers C.

every product of functions from the set has low influences.

A 4LIN-CoveringDict Test for f:{+1,-1}R{+1,-1} is specified by a distribution over x,y,z,w(as before).

Let C be the 4LIN-CSP instanceinduced by the distribution

(every 4-tuplex,y,z,winduces a constraint, f is an asg).

Covering Completeness of the test c:

c dictators that cover C.

Covering Soundness of the test s:

No “regular set” of s functions covers C.

We want such a testwith covering completeness 2 (and super-const covering soundness).

Hastad’sDict Test uses the distribution:

Choose x,y,z{-1,1}R, independently uniformly at rand.

Choose a noise vector r{-1,1}Rin which each coordinate is independently -1 (noise) w.p. ε.

Set w = -xyzr.

Covering Completeness >const: Let f(x) = x1.

f(x)f(y)f(z)f(w) = x1 y1 z1 w1 = -r1.

Thus, f doesn’t cover constraints with noise on r1 (r1=-1).

No constnum ofdictators cover the test’s constraints!

- New Dict Test: Same distribution with tweak on noise.
- x,y,z random, w = -xyzr.
- Partition the noise vector r into pairs (r1,r2), (r3,r4),… For each pair, w.p. 2ε have noise one exactly one element of the pair. There is never noise on both!

- Covering Completeness = 2: Let f(x) = x1 and g(x) = x2.
- There is never noise on both r1 and r2 (noise = -1).
- Thus, at least one of the following holds:
- f(x)f(y)f(z)f(w) = x1 y1 z1 w1 = -r1 = -1
- g(x)g(y)g(z)g(w) = x2 y2 z2 w2= -r2 = -1

- fandg cover the test’s constraints!

Covering is a natural notion,pretty much any max-CSP question can be considered in the context of covering.

Prove the Covering Dichotomy Conjecture in full.

Quantitative results:

We get 4LIN covering soundness Ω(logloglogn).

Can we get Ω(log n) for some ?

Connecting the covering-UGC to known conjectures

Incomparable to UGC, but implies the UGC with completeness 1/c (instead of 1-ε).

Devise ‘direct’ reductions between covering problems.