1. 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 satisfiable instance satisfying solution "almost" satisfiable instance "almost" satisfying solution robust version (against noise) input output

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 Horn 3-SAT Approx. Alg.1-1/(log 1/ε) [Zwick'98] NP-Hardness1-ε c for some c < 1 [KSTW'00] UG-Hardness

7. Approximability of almost satisfiable Horn-SAT Previously known Horn 3-SATHorn 2-SAT Approx. Alg.1-1/(log 1/ε) [Zwick'98] 1-3ε [KSTW'00] NP-Hardness1-ε c for some c < 1 [KSTW'00] 1-1.36ε from Vertex Cover UG-Hardness1-(2-δ)ε from Vertex Cover

8. Approximability of almost satisfiable Horn-SAT Our result Comment. People need UGC to get sharp inapprox. result for most of problems Horn 3-SATHorn 2-SAT Approx. Alg.1-1/(log 1/ε) [Zwick'98] 1-2ε NP-Hardness1-ε c for some c < 1 [KSTW'00] 1-1.36ε from Vertex Cover UG-Hardness1-1/(log 1/ε)1-(2-δ)ε from Vertex Cover

9. Proof framework of the hardness result c vs. s dictatorship test c vs. s dictatorship test [KKMO'07,Rag'08] c vs. s UG-Hardness for the CSP c vs. s UG-Hardness for the CSP not clear how to construct a dictatorship test for HornSAT 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. MaxCut, Linear Equations, Max-2SAT, Vertex Cover...

10. Proof framework of the hardness result c vs. s dictatorship test c vs. s dictatorship test [KKMO'07,Rag'08] c vs. s UG-Hardness for the CSP c vs. s UG-Hardness for the CSP c vs. s integrality gap for the "canonical SDP" c vs. s integrality gap for the "canonical SDP" [Rag'08] construct an SDP gap instance instead 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-k HittingSet).

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/k 0.999 ) 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/k 0.999 ) 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. c vs. s dictatorship test c vs. s dictatorship test [KKMO'07,Rag'08] c vs. s UG-Hardness for the CSP c vs. s UG-Hardness for the CSP c vs. s integrality gap for the "canonical SDP" c vs. s integrality gap for the "canonical SDP" [Rag'08] MaxCut, Linear Equations, Max-2SAT, Vertex Cover... Horn-3SAT 1-in-k HittingSet The first work (and the only one so far) using Raghavendra's theorem to get sharp hardness result.

15. The canonical SDP: Lifted LP + semidefinite constraints

16. The lifted-LP (in Sherali-Adams system) C : the set of clauses For each Cє C, set up local (integral) prob. distribution π C on all truth-assignments {σ : X C -> {0, 1} } –Variables. π C (σ) >= 0 for each σ : X C -> {0, 1} –Constraints. Σ σ π C (σ) = 1 maximize E C єC [Pr σ~πC [C( σ )=1]] Pr σ~πC [ σ (x i )=b 1 Λ σ(x j )=b 2 ] = X (xi,b1),(xj,b2) for all Cє C ; x i, x j єC; b 1,b 2 є{0, 1} consistency of pairwise margins: consistency of singleton margins: s.t. Pr σ~πC [ σ (x i )=b 1 ] = X (xi,b1),(xi,b1) linear expressions

17. The semidefinite constraints Vectors. Introduce v (x,0) and v (x,1) corresponding to the event x = 0 and x = 1. Constraints. – = 0 -- mutually exclusive events –v (x,0) + v (x,1) = I -- probability adds up to 1 –Pr σ~πC [σ(x i )=b 1 Λ σ(x j )=b 2 ] = -- pairwise marginals must be PSD

18. The gap instance for Horn-3SAT.

19. Instance I k : x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... Observation. I k is not satisfiable. Therefore OPT( I k ) < 1 - Ω(1/k).

20. OPT LP ( I k ) >= 1 - 1/2 k x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... Observation. Clauses in different steps share at most one variable. No worry about pairwise margins between different steps.

21. OPT LP ( I k ) >= 1 - 1/2 k x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... x 0 (y 0 ) 1010 π C (σ) 1-δ δ x 0 Λy 0 ->x 1 (y 1 ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-2δ δ loss = 2δ

22. OPT LP ( I k ) >= 1 - 1/2 k x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... x 0 Λy 0 ->x 1 (y 1 ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-2δ δ x 1 Λy 1 ->x 2 (y 2 ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-4δ 2δ loss = 2δ

23. OPT LP ( I k ) >= 1 - 1/2 k x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... x 2 Λy 2 ->x 3 (y 3 ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-8δ 4δ x 1 Λy 1 ->x 2 (y 2 ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-4δ 2δ loss = 2δ

24. OPT LP ( I k ) >= 1 - 1/2 k x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... x 2 Λy 2 ->x 3 (y 3 ) 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-8δ 4δ x k Λy k ->x k+1 x k Λy k ->y k+1 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-2 k+1 δ 2 k δ... loss = 2δ

25. OPT LP ( I k ) >= 1 - 1/2 k x 0, y 0 x 0 Λ y 0 -> x 1, x 0 Λ y 0 -> y 1 x 1 Λ y 1 -> x 2, x 1 Λ y 1 -> y 2 x 2 Λ y 2 -> x 3, x 2 Λ y 2 -> y 3 x k Λ y k -> x k+1, x k Λ y k -> y k+1 x k+1, y k+1 Step 0: Step 1: Step 2: Step 3: Step k+1: Step k+2:...... x k+1 (y k+1 ) 1010 π C (σ) 1-2 k+1 δ 2 k+1 δ x k Λy k ->x k+1 x k Λy k ->y k+1 1 Λ 1 -> 1 0 Λ 1 -> 0 1 Λ 0 -> 0 π C (σ) 1-2 k+1 δ 2 k δ loss = 2δ + 2(1-2 k+1 )δ = 1/2 k (by taking δ = 1/2 k+1 )

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/k 0.999 ) 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. O k (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?