Lecture 21 (Oct. 24): Max Cut SDP Gap and Max 2-SAT

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1 CMPUT 67: Approximation Algorithms Fall 014 Lecture 1 Oct. 4): Max Cut SDP Gap and Max -SAT Lecturer: Zachary Friggstad Scribe: Chris Martin 1.1 Near-Tight Analysis of the Max Cut SDP Recall the Max Cut problem. Given an undirected graph G V, E) with edge weights we), e E, find S V that maximizes e δs) we). The following strict quadratic program was presented for this problem in the previous lecture. For each i V, we use the interpretation that v i 1 means i S and v i 1 means i S. maximize: i,j) E wi, j) 1 v iv j MAXCUT-QP) subject to: v i v i 1, i V 1.1) We relaxed this to an SDP by replacing each v i with a vector v i R n and replacing the multiplication of variables with the dot product. maximize: i,j) E wi, j) 1 v i v j MAXCUT-SDP) subject to: v i v i 1, i V 1.) We saw a simple randomized rounding algorithm that rounds an optimal solution to MAXCUT-SDP) to an integer solution with expected value at least α OP T SDP where α : min 0 θ θ 1 cosθ) There is a simple example for Max Cut that shows our analysis is nearly tight. Consider G C, the cycle on nodes {0, 1,, 3, 4}, with all edge weights being 1. The optimum solution is 4 with an optimum cut being, say, S {1, 3} with δs) {0, 1), 1, ),, 3), 3, 4)} we can do no better because C is not bipartite). Next we define a feasible SDP solution with value greater than 4. Let θ 4/. For 0 i 4, let v i cosiθ), siniθ), 0, 0, 0). Note that v i v i 1 for all i, since cos iθ) + sin iθ) 1. So, this is a feasible solution to MAXCUT-SDP. The angle between any two nodes corresponding to an edge i, i + 1) is θ, so v i v i+1 v i v i+1 cosθ) cosθ). Thus, 4 1 cosθ) OP T SDP i0 4 1 cos ) )

2 1- Lecture 1: Max Cut SDP Gap and Max -SAT Therefore, OP T OP T SDP 4 ) ) cos which is very close to α. So, we know that our analysis of the integrality gap is nearly tight. In fact, our initial integrality gap analysis is tight. There are more sophisticated examples showing that for any constant c > α that the integrality gap is at most c [FS0]. 1. Max -SAT The Max -SAT problem is as follows. Given -CNF clauses C 1,, C m over variables x 1, x n, and weights for each clause wc), choose a boolean assignment of the variables that maximizes the total weight of satisfied clauses. An interesting aspect of this problem is that there is a polynomial-time algorithm that determines if all clauses can be satisfied [APT79]. However, if an instance is not satisfiable then the algorithm in [APT79] does not give us much of an idea on the maximum number that can be satisfied; this remains NP-hard. Converting this problem into a strict quadratic program takes a bit of insight. For example, we cannot simply associate a variable v i for each x i and say v i 1 corresponds to x i True. This is because a quadratic term of the form v i v j cannot distinguish between the value v i v j or v i ) v j ). This was not a problem with Max Cut because, intuitively, negating all variables simply swapped the solution S to V S which cut the same edges). We will add a special variable which we call v 0 and define a truth assignment relative to v 0. That is, in any setting of the values v 0, v 1,..., v n { 1, +1} we will set x i True iff v 0 v i. Thus, negating all v i variables does not change the associated truth assignment. With the understanding that each v i will take { 1, +1} values, we can model the clause x 1 x ) with the following expression: 1 1 v 0v v 0v 1.3) Note that this is 1 iff v 0 v 1 v 0 v ), as desired. Expanding this gives the following: v 0v 1 + v 0 v v 0v 1 v ) 1.4) v 0v 1 + v 0 v v 1 v ) 1.) v 0v 1 ) v 0v ) v 1v ). 1.6) Note that 1.) holds because v 0 {1, 1}. Summing 1.6) over all clauses some of the ± signs in front of the quadratic terms may differ, depending on whether the literals in the clause appear positively or negatively) with weights multiplied through gives the weight of satisfied clauses being expressed by a ij 1 + v i v j ) + b ij 1 v i v j ) where the values a ij, b ij are constants 0. This will become our objective function for the quadratic program. maximize: a ij 1 + v i v j ) + b ij 1 v i v j ) MAX-SAT-QP) subject to: v i v i 1, 0 i n 1.7)

3 Lecture 1: Max Cut SDP Gap and Max -SAT 1-3 As in the SDP for Max Cut, we will replace the variables v i with vectors v i R n+1 and replace the products with vector dot products to get the following SDP. maximize: a ij 1 + v i v j ) + b ij 1 v i v j ) MAX-SAT-SDP) subject to: v i v i 1, 0 i n 1.8) We round MAX-SAT-SDP using an algorithm similar to the one presented in the previous lecture for Max Cut. 1. Solve MAX-SAT-SDP to get vectors v i, 0 i n.. Let r R n+1 be a random unit vector. 3. Define values y i, 0 i n by 4. For each 1 i n, set y i { 1 if v i r 0 1 if vi r < 0 { True if y0 y x i : i 1 False if y 0 y i 1 Claim 1 The expected weight of satisfied clauses is α OP T SDP. Proof. A clause of the form, say, x i x j is satisfied if and only if y 0 y i ) y 0 y j ) y i y j ) 1 otherwise the expression is 0) where the y i are the values constructed in the algorithm. Thus, by how we collected the quadratic terms we have that the value of the truth assignment is exactly a ij 1 + y i y j ) + b ij 1 y i y j ). Let θ ij be the angle between vi and v j. From the Max Cut analysis, we know Pr[y i y j 1] θij. This also shows Pr[y i y j 1] 1 θij. Therefore, E[y i y j ] Pr[y i y j 1] 1 + Pr[y i y j 1] 1) 1 θ ) ij 1 + ) θij 1) 1 θ ij. By linearity of expectation, the expected weight of satisfied clauses is exactly E a ij 1 + y i y j ) + b ij 1 y i y j ) a ij 1 + E[y i y j ]) + b ij 1 E[y i y j ]) a ij θ ij a ij 1 θ ij ) + b ij θ ij ) + b ij θij 1.9)

4 1-4 Lecture 1: Max Cut SDP Gap and Max -SAT Recall that for any θ [0, ] we have θ α 1 cosθ) where α > is the constant in the Max Cut analysis. One can also show that for θ [0, ] we also have 1 θ α 1 + cosθ). Using these bounds, the fact that cosθ ij ) vi v j, and that all a ij and b ij coefficients are nonnegative we conclude by bounding expression 1.9) as follows: a ij 1 θ ) ij + b ij θij a ij α 1 + vi vj ) + b ij α 1 vi vj ) α OP T SDP. 1.3 Discussion One can better with Max-SAT. There is a 0.94-approximation via SDP techniques [LLZ0] and this is nearly tight. Under the Unique Games Conjecture, there is no approximation [K+07]. Assuming only P NP, there is no 1/ 0.94 approximation for Max-SAT [H01]. In general, SDPs perform very well for Constraint Satisfiaction Problems where each clause is over only two variables i.e. a -CSP) and where the variables take values from a constant-size domain. For example, Max- SAT and Max Cut are two such problems. Another is the the 3-colouring variant where we must colour each vertex one of three colours and the goal is to maximize the number of edges that whose endpoints are coloured differently. This is a -CSP with domain size 3. A random colouring satisfies each constraint with probability /3, leading to a randomized /3-approximation. However, SDP-based techniques approximate this problem within a constant better than /3. In general, any -CSP over a constant-size domain can be approximated better than the random assignment using SDPs [H08]. Finally, under the Unique Games Conjecture UGC) we have a complete characterization of constraint satisfaction problems of constant constraint size over constant domains including problems like Max-3SAT). Namely, Raghavendra describes an SDP relaxation for such CSPs and proves, in some sense, that the if integrality gap for this relaxation is γ then it is UGC-hard to approximate the problem within any constant better than γ [R08]. He also describes an algorithm that rounds the SDP and obtains an approximation guarantee essentially equal to the integrality gap γ. References APT79 B. Aspvall, M. Plass and R. Tarjan, A linear-time algorithm for testing the truth of certain quantified boolean formulas, Information Processing Letters 83), 11 13, FS0 U. Feige and G. Schechtman, On the optimality of the random hyperplane rounding technique for MAX- CUT, Random Structures and Algorithms, 0: , 00. H01 J. Håstad, Some optimal inapproximability results, Journal of the ACM, 48: , 001. H08 J. Håstad, Every -CSP allows nontrivial approximation, Computational Complexity, 174):49 66, 008.

5 Lecture 1: Max Cut SDP Gap and Max -SAT 1- K+07 S. Khot, G. Kindler, E. Mossel, and R. O Donnell, Optimal inapproximability results for MAX-CUT and Other -Variable CSPs?, SIAM Journal on Computing, 371):319 37, 007. LLZ0 M. Lewin, D. Livnat, and U. Zwick, Improved rounding techniques for the MAX-SAT and MAX-DICUT problems. In Proceedings of IPCO, 00. R08 P. Raghavendra, Optimal algorithms and inapproximability results for every CSP?, In Proceedings of STOC, 008.

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