115 Views

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. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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 • 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 )**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**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 . ?**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?**Approximability of almost satisfiable Horn-SAT**• Previously known**Approximability of almost satisfiable Horn-SAT**• Previously known**Approximability of almost satisfiable Horn-SAT**• Our result • Comment. People need UGC to get sharp inapprox. result for most of problems**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 ...**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 ...**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).**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 =**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.**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 ...**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}**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**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) .**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.**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**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**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**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**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**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**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**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]).