1 / 29

Venkatesan Guruswami (CMU) Yuan Zhou (CMU)

Tight Bounds on the Approximability of Almost-satisfiable Horn SAT and Exact Hitting Set. Venkatesan Guruswami (CMU) Yuan Zhou (CMU). Satisfiable CSPs. Theorem [Schaefer'78]. Only three nontrivial Boolean CSPs for which satisfiability is poly-time decidable.

victor-mcpherson
Download Presentation

Venkatesan Guruswami (CMU) Yuan Zhou (CMU)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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?

More Related