Approximation Algorithms

Transcription

1 Approximation Algorithms Chapter 26 Semidefinite Programming Zacharias Pitouras 1

2 Introduction LP place a good lower bound on OPT for NP-hard problems Are there other ways of doing this? Vector programs provide another class of relaxations For problems expressed as strict quadratic programs Vector programs are equivalent to semidefinite programs Semidefinite programs can be solved in time polynomial in n and log(1/ε) A factor algorithm for MAX-CUT 2

3 Contents Strict quadratic programs and vector programs Properties of positive semidefinite matrices The semidefinite programming problem Randomized rounding algorithm Improving the guarantee for MAX-2SAT Notes 3

4 The maximum cut problem MAX-CUT Given an undirected graph G=(V,E), with edge weights w: E Q +, find a partition ( SS, ) of V so as to maximize the total weight of edges in this cut, i.e., edges that have one endpoint in S and one endpoint in S. 4

5 Strict quadratic programs and vector programs (1) A quadratic program is the problem of optimizing a quadratic function of integer valued variables, subject to quadratic constraints on these variables. Strict quadratic program: monomials of degree 0 or 2. Strict quadratic program for MAX-CUT: y i an indicator variable for vertex υ i with values +1 or -1. Partition, S = { υi yi = 1, } S = { υi yi = 1} If υ i and υ j on opposite sides, then yy i j = 1 and edge contributes w ij to objective function On the other hand, edge makes no contribution. 5

6 Strict quadratic programs and vector programs (2) An optimal solution to this program is a maximum cut in G. 1 maximize w 1 2 < 1 ( yy ) ij i j i j n 2 subject to yi 1, υi = V y i Z, υ V i 6

7 Strict quadratic programs and vector programs (3) This program relaxes to a vector program A vector program is defined over n vector variables in R n, say υ 1, υ 2,, υ n, and is the problem of optimizing a linear function of the inner products υ i. υ j for 1 i j n, subject to linear constraints on these inner products A vector program is obtained from a linear program by replacing each variable with an inner product of a pair of these vectors 7

8 Strict quadratic programs and vector programs (4) A strict quadratic program over n integer variables defines a vector program over n vector variables in R n Establish a correspondence between the n integer variables and the n vector variables, and replace each degree 2 term with the corresponding inner product y i. y j is replaced with υ i. υ j 8

9 Strict quadratic programs and vector programs (5) Vector program for MAX-CUT 1 maximize w < i j n ( υυ ) ij i j subject to υ υ = 1, υ i i i V n υi R, υi V 9

10 Strict quadratic programs and vector programs (6) Because of the constraint υ i. υ j =1, the vectors υ 1, υ 2,, υ n are constrained to lie on the n- dimensional sphere S n-1 Any feasible solution to the strict quadratic program of MAX-CUT yields a solution to the vector program ( y ) i,0,...,0 is assigned to υ The vector program corresponding to a strict quadratic program is a relaxation of the quadratic program providing an upper bound on OPT Vector programs are approximable to any desired degree of accuracy in polynomial time i 10

11 Properties of positive semidefinite matrices (1) Let A be a real, symmetric n n matrix Then A has real eigenvalues and has n linearly independent eigenvectors A is positive semidefinite if n T x R, x Ax 0 Theorem 26.3 Let A be a real symmetric n n matrix. Then the following are equivalent: n T 1. x R, x Ax 0 2. All eigenvalues of A are nonnegative 3. There is an n n real matrix W, such that A = W T W 11

12 Properties of positive semidefinite matrices (2) 12

13 Properties of positive semidefinite matrices (3) With Cholesky decomposition a real symmetric matrix can be decomposed in polynomial time as A = UΛU T, where Λ is a diagonal matrix whose diagonal entries are the eigenvalues of A A is positive semidefinite if all the entries of Λ are nonnegative giving a polynomial time test for positive semidefiniteness The decomposition WW T is not polynomial time computable, but can be approcimated The sum of two positive semidefinite matrices is also positive semidefinite Convex combination is also positive semidefinite 13

14 The semidefinite programming problem (1) Let Y be an n n matrix of real valued variables with y ij entry The semidefinite programming problem is the problem of maximizing a linear function of the y ij s, subject to linear constraints on them, and the additional constraint that Y be symmetric and positive semidefinite 14

15 The semidefinite programming problem (2) Denote by R n n the space of n n real matrices The trace of a matrix A is the sum of its diagonal entries and is denoted by tr(a) The Frobenius inner product of matrices A, B is defined to be T A B= tr( AB) = ab ij i= 1 j= 1 M n denotes the cone of symmetric n n real matrices For A M A denotes that A is pos. sem. n, 0 n n ij 15

16 The semidefinite programming problem (3) The general semidefinite programming problem, S: max imize C Y subject to D Y = d, 1 i k Y i 0, Y M n i 16

17 The semidefinite programming problem (4) A matrix satisfying all the constraints is a feasible solution and so is any convex combination of these solutions Let A be an infeasible point. Let C be another point. A hyperplane C Y b is called a seperating hyperplane for A if all feasible points satisfy it and point A does not Theorem 26.4 Let S be a semidefinite programming problem, and A be a point in R n n. We can determine in polynomial time, whether A is feasible for S and, if it is not, find a separating hyperplane. 17

18 The semidefinite programming problem (4) Proof: Testing for feasibility involves ensuring that A is symmetric and positive semidefinite and satisfies all the constraints. This can be done in polynomial time. If A is infeasible, a separating hyperplane is obtained as follows. If A is not symmetric, α ij > α ji for some i,j. Then y ij y ji is a separating hyperplane If A is not positive semidefinite, then it has a negative eigenvalue, say λ, and υ the corresponding eigenvector. Then, ( T) T υυ Y= υ Yυ 0 is a separating hyperplane. If any of the linear constraints is violated, it directly yields a separating hyperplane 18

19 The semidefinite programming problem (5) Let V be a vector program on n n-dimensional vector variables υ 1, υ 2,, υ n. Define the corresponding semidefinite program, S, over n 2 variables y ij, for 1 i,j n as follows: Replace each inner product υ i. υ j by the variable y ij. Require that matrix Y is symmetric and positive semidefinite Lemma 26.5 Vector program V is equivalent to semidefinite program S 19

20 The semidefinite programming problem (6) Proof: One must show that corresponding to each feasible solution to V, there is a feasible solution to S of the same objective function value and vice versa Let α 1,, α n be a feasible solution to V. Let W be the matrix whose columns are α 1,, α n Then it is easy to see that A=W T W is a feasible solution to S having the same objective function value Let A be a feasible solution to S. By theorem 26.3 there is a n n matrix W such that A=W T W. Let α 1,, α n be the columns of W. Then it is easy to see that α 1,, α n is a feasible solution to V having the same objective function value 20

21 The semidefinite programming problem (7) The semidefinite programming relaxation to MAX-CUT is: 1 maximize w 1 2 < 1 ( yy ) ij i j i j n 2 subject to yi 1, υi = V Y Y 0, M n 21

22 Randomized rounding algorithm (1) Assume we have an optimal solution to the vector program Let α 1,, α n be an optimal solution and let OPT υ denote its objective function value These vectors lie on the n-dimensional unit sphere S n-1 We need a cut ( SS, ) whose weight is a large fraction of OPT υ 22

23 Randomized rounding algorithm (2) Let θ ij denote the angle between vectors α i and α j. The contribution of this pair of vectors to OPT υ is, w ij 2 (1 cos θ ) ij The closer θ ij is to π, the larger the contribution will be Pick r to be a uniformly distributed vector on the unit sphere and let S = { υ a r 0} i i 23

24 Randomized rounding algorithm (3) Lemma 26.6 Pr[ υ i and υ j are separated]= θ ij /π Proof: Project r onto the plane containing α i and α j Now, vertices υ i and υ j will be separated iff the projection lies in one of the two arcs of angle θ ij. Since r has been picked from a spherically symmetric distribution, its projection will be a random direction in the plane. The lemma follows. 24

25 Randomized rounding algorithm (4) Lemma 26.7 Let x 1,, x n be picked independently from the normal distribution with mean 0 and unit standard 2 2 deviation. Let d = x xn Then, ( x /,..., / ) is a random vector on the unit 1 d xn d sphere. Algorithm 26.8 (MAX-CUT) 1. Solve vector program V. Let α 1,, α n be an optimal solution. 2. Pick r to be a uniformly distributed vector on the unit sphere. 3. Let S = υ a r 0 { } i i 25

26 Randomized rounding algorithm (5) Lemma 26.9 E[W] α*opt υ Corollary The integrality gap for vector relaxation is at least a> Theorem There is a randomized approximation algorithm for MAX-CUT achieving an approximation factor of

27 Improving the guarantee for MAX- 2SAT (1) MAX-2SAT is the restriction of MAX- SAT to formulae in which each clause contains at most two literals Already obtained a ¾ algorithm for that Semidefinite programming gives an improved algorithm Idea: convert the obvious quadratic program into a strict quadratic program 27

28 Improving the guarantee for MAX- 2SAT (2) To each Boolean variable x i introduce variable y i which is constrained to be either +1 or -1, for 1 i n. In addition introduce another variable y 0 which is also constrained to be either +1 or -1. x i is true if y i = y 0 and false otherwise The value υ(c) of clause C is defined to be 1 if C is satisfied and 0 otherwise For clauses containing only one literal 1+ yy 0 i 1 yy 0 υ( xi) = and υ( xi) = 2 2 i 28

29 Improving the guarantee for MAX- 2SAT (3) For a clause with 2 literals 1 yy1 yy 0 i 0 υ( xi xj) = 1 υ( xi) υ( xj) = ( 3 2 = + yy ) 0 i + yy 0 j yyy 0 i j 4 1+ yy 1+ yy 1 yy 0 i 0 j i j = j 29

30 Improving the guarantee for MAX- 2SAT (4) MAX-2SAT can be written as a strict quadratic program 0 < i j n ( + yy ) + b ( yy ) maximize α 1 1 ij i j ij i j 2 subject to i 1, 0 y = i n y Z, 0 i n i 30

31 Improving the guarantee for MAX- 2SAT (5) Corresponding vector program relaxation where vector variable υ i corresponds to y i 0 < i j n ( + ) + b ( ) maximize α 1 υ υ 1 υ υ ij i j ij i j subject to υi υj = 1, 0 i n n+ υ 1 i R, 0 i n 31

32 Improving the guarantee for MAX- 2SAT (6) The algorithm is similar to that for MAX- CUT Let α 0,, α n be an optimal solution. Pick a vector r uniformly distributed on the unit sphere in (n+1) dimensions and let y i = 1 iff r a 0 for 0 i n i This gives a truth assignment for the Boolean variables Let W be the random variable denoting the weight of this truth assignment 32

33 Improving the guarantee for MAX- 2SAT (7) Lemma E[W] α * OPT υ Proof: E[ ] W = 2 aij Pr[ yi = yj ] + bij Pr[ yi yj ] 0 < i j n let θ ij denote the angle between a i and a j θij a Pr[ y ] ( 1 cos ) i yj = θij and π 2 θij a Pr[ y ] 1 ( 1 cos ) i = yj = + θij π 2 Therefore, [ ] ( θ ) ( θ ) E W a aij 1+ cos ij + bij 1 cos ij = a OPTυ 0 < i j n 33

34 The END 34