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  1. Tight Bounds on the Approximability of Almost-satisfiable Horn SAT and Exact Hitting Set Venkatesan Guruswami (CMU) Yuan Zhou (CMU)

  2. Satisfiable CSPs Theorem [Schaefer'78] Only three nontrivial Boolean CSPs for which satisfiability is poly-time decidable • LIN-mod-2 -- linear equations mod 2 • e • 2-SAT • Horn-SAT -- CNF formula where each clause consists of at most one unnegated literal • e.g. , , , (equivalent to )

  3. Almost satisfiable CSPs -satisfiable instance -- satisfiable by removing fraction of clauses Finding almost satisfying assignments input output satisfiable instance satisfying solution robust version (against noise) "almost" satisfiable instance "almost" satisfying solution

  4. Almost satisfiable CSPs -satisfiable instance -- satisfiable by removing fraction of clauses Finding almost satisfying assignments Given a -satisfiable instance, can we efficiently find an assignment satisfying . constraints, where as . ?

  5. The answer... • No for LIN-mod-2 • vs. is NP-Hard [Håstad'01] • Yes for 2-SAT • SDP-based alg. gives vs [Zwick'98] • Improved to vs [CMM'09] • Tight under Unique Games Conjecture [KKMO'07] • Yes for Horn-SAT • LP-based alg. gives vs [Zwick'98] • For Horn-3SAT, Zwick's alg. gives vs • Exponential loss -- is it tight?

  6. Approximability of almost satisfiable Horn-SAT • Previously known

  7. Approximability of almost satisfiable Horn-SAT • Previously known

  8. Approximability of almost satisfiable Horn-SAT • Our result • Comment. People need UGC to get sharp inapprox. result for most of problems

  9. Proof framework of the hardness result Theorem.[Rag'08] There is a canonical SDP relaxation for SDP(Λ) each CSP Λ, such that c vs. s integrality gap => c-η vs. s+η dictator test. not clear how to construct a dictatorship test for HornSAT c vs. s dictatorship test [KKMO'07,Rag'08] c vs. s UG-Hardness for the CSP MaxCut, Linear Equations, Max-2SAT, Vertex Cover ...

  10. Proof framework of the hardness result construct an SDP gap instance instead c vs. s integrality gap for the "canonical SDP" [Rag'08] c vs. s dictatorship test [KKMO'07,Rag'08] c vs. s UG-Hardness for the CSP MaxCut, Linear Equations, Max-2SAT, Vertex Cover ...

  11. Theorem.[Rag'08] There is a canonical SDP relaxation for SDP(Λ) each CSP Λ, such that c vs. s integrality gap => c-η vs. s+η dictator test. Our Theorem 1. There is a (1-2-k) vs. (1-1/k) gap instance for SDP(Horn-3SAT), for every k > 1. Our Theorem 2. A tight gap instance for SDP(1-in-kHittingSet).

  12. 1-in-k HittingSet • U : universe • C : collection of subsets of U of size <= k • Goal : a subset S of U intersecting maximum number of sets in C at exactly one element • Theorem 2. (1-1/k0.999) vs. 1/log k SDP gap. • Corollary. UG-Hard to approx. within O(1/log k). • 1-in-Exact k HittingSet: • Approximability of 1-in-EkHS: 1/e[GT05] <= • C : collection of subsets of U of size k =

  13. 1-in-k HittingSet • U : universe • C : collection of subsets of U of size <= k • Goal : a subset S of U intersecting maximum number of sets in C at exactly one element • Theorem 2. (1-1/k0.999) vs. 1/log k SDP gap. • Corollary. UG-Hard to approx. within O(1/log k). • Fact. An Ω(1/log k) approx. algorithm. • Theorem 3. A (1-1/2k) vs. 0.1 approx. algorithm.

  14. The first work (and the only one so far) using Raghavendra's theorem to get sharp hardness result. c vs. s integrality gap for the "canonical SDP" [Rag'08] Horn-3SAT 1-in-k HittingSet c vs. s dictatorship test [KKMO'07,Rag'08] c vs. s UG-Hardness for the CSP MaxCut, Linear Equations, Max-2SAT, Vertex Cover ...

  15. The canonical SDP:Lifted LP + semidefinite constraints

  16. The lifted-LP(in Sherali-Adams system) • C: the set of clauses • For each CєC, set up local (integral) prob. distributionπC on all truth-assignments {σ : XC -> {0, 1} } • Variables. πC(σ) >= 0 for each σ : XC -> {0, 1} • Constraints. Σσ πC(σ) = 1 maximize ECєC[Prσ~πC[C(σ)=1]] linear expressions consistency of singleton margins: s.t. Prσ~πC[σ(xi)=b1] = X(xi,b1),(xi,b1) consistency of pairwise margins: Prσ~πC[σ(xi)=b1 Λ σ(xj)=b2] = X(xi,b1),(xj,b2) for all CєC; xi, xjєC; b1,b2є{0, 1}

  17. The semidefinite constraints • Vectors. Introduce v(x,0) and v(x,1) corresponding to the event x = 0 and x = 1. • Constraints. • <v(x,0), v(x,1)> = 0 -- mutually exclusive events • v(x,0) + v(x,1) = I -- probability adds up to 1 • Prσ~πC[σ(xi)=b1Λ σ(xj)=b2] = <v(xi,b1),v(xj,b2)> -- pairwise marginals must be PSD

  18. The gap instance for Horn-3SAT.

  19. Instance Ik: Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 Observation.Ik is not satisfiable. Therefore OPT(Ik) < 1 - Ω(1/k) .

  20. OPTLP(Ik) >= 1 - 1/2k Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 Observation. Clauses in different steps share at most one variable. No worry about pairwise margins between different steps.

  21. x0Λy0->x1(y1) πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-2δ δ δ OPTLP(Ik) >= 1 - 1/2k Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 loss = 2δ x0(y0) πC(σ) 1 0 1-δ δ

  22. x1Λy1->x2(y2) πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-4δ 2δ 2δ OPTLP(Ik) >= 1 - 1/2k Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 loss = 2δ x0Λy0->x1(y1) πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-2δ δ δ

  23. x2Λy2->x3(y3) πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-8δ 4δ 4δ OPTLP(Ik) >= 1 - 1/2k Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 loss = 2δ x1Λy1->x2(y2) πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-4δ 2δ 2δ

  24. xkΛyk->xk+1 xkΛyk->yk+1 πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-2k+1δ 2kδ 2kδ OPTLP(Ik) >= 1 - 1/2k Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 loss = 2δ x2Λy2->x3(y3) πC(σ) ... 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-8δ 4δ 4δ

  25. xk+1(yk+1) πC(σ) 1 0 1-2k+1δ 2k+1δ OPTLP(Ik) >= 1 - 1/2k Step 0: x0, y0 Step 1: x0Λ y0 -> x1, x0Λ y0 -> y1 Step 2: x1Λ y1 -> x2, x1Λ y1 -> y2 Step 3: x2Λ y2 -> x3, x2Λ y2 -> y3 ... ... ... ... Step k+1: xkΛ yk -> xk+1, xkΛ yk -> yk+1 Step k+2: xk+1, yk+1 loss = 2δ + 2(1-2k+1)δ = 1/2k (by taking δ = 1/2k+1) xkΛyk->xk+1 xkΛyk->yk+1 πC(σ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 1-2k+1δ 2kδ 2kδ

  26. Getting a good SDP solution • No vectors corresponding to the previous LP solution • Because of the extra semidefinite constraints • Solution: twist the LP solution in several ways

  27. Summary of our results • (1 - ε) vs (1 - 1/(log 1/ε))UG-Hardness for Horn-3SAT • (1 - 1/k0.999) vs 1/log k UG-Hardness for 1-in-k HittingSet • (1 - ε) vs (1 - 2ε) algorithm for Horn-2SAT • (1 - 1/2k) vs 0.1 approximation algorithm for 1-in-k HittingSet

  28. Open directions • NP-Hardness for approximating 1-in-k HittingSet. Ok(1)? • For which CSPs does it suffice to show an LP integrality gap? • Study finding almost satisfiable solutions for non-Boolean CSPs. • Conjecture. There are poly-time algorithms for almost satisfiable CSPs that cannot express linear equations (i.e. "bounded width" CSPs, by [Barto-Kozik'09]).

  29. The End.Any questions?