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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)), (x2x5= 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**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 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 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.