Covering CSPs

# Covering CSPs

## Covering CSPs

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
##### Presentation Transcript

1. Covering CSPs GillatKol joint work with IritDinur

2. 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.

3. 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.

4. 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.

5. 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).

6. 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 {++,--,+-,-+}

7. 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

8. Our Results

9. 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.

10. 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

11. The Characterization of Covering-Hard Predicates? Our Covering Dichotomy Conjecture: covering  ishardiffOdd*.

12. 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

13. 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.

14. 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

15. 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.

16. 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

17. 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.

18. 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).

19. 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!

20. 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!

21. 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.

22. Thank You!