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Covering CSPs. Gillat Kol joint work with Irit Dinur. Constraint Satisfaction Problem. CSP = Constraint S atisfaction P roblem Variables : x 1 ,x 2 ,…, x n in {-1,1 } . Constraints: ((x 1 =1) v (-x 2 =1) v (x 7 =-1)) , (x 2  x 5 = 1 ) , … Goal :

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Covering CSPs

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Covering CSPs

GillatKol

joint work with IritDinur


Constraint Satisfaction Problem

CSP= Constraint Satisfaction Problem

Variables: x1,x2,…,xnin {-1,1}.

Constraints: ((x1=1) v (-x2=1) v (x7=-1)), (x2x5= 1), …

Goal:

Ideally: Find assignment that satisfies all constraints.

NP-hard, so we approximate.


Optimization Notions

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.


Example: The Dinner Party Problem

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.


Covering Number

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


Our Results


Covering 

  • 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.


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  ishardiffOdd*.


Def: 4LIN(x1,x2,x3,x4) = x1x2x3x4.

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 2Partial Proof for the Dichotomy Conjecture

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.


Result 3Connecting covering  and NAE

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


Result 3Connecting covering  and NAE

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.


Covering Dictatorship Test for 4LIN(part of the proof of Result 1)


Dictatorship Test

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


Covering Dictatorship Test

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.


Covering Dictatorship Test

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’s Dictatorship Test

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!


Getting Perfect Completeness

  • 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!


Many Open Problems

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.


Thank You!


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