Relax and do Something Random: The MAXCUT Approximation of Goemans and Williamson

Transcription

1 Relax and do Something Random: The MAXCUT Approximation of Goemans and Williamson Michael McCoy January 29, 2010

2 What is a cut? Given a graph (V, E) with edge weights w ij 0, w 13 1 w 12 2 w 25 3 w 34 4 w 24 5 a cut S is a subset of the vertices S V. The weight of the cut ω(s) is the sum of the weights of the edges that cross the cut : ω(s) = i S, j / S w ij

3 Cut Example Here, S = {1, 4, 5}. 1 w 12 w 13 2 w 25 w w 34 4 The weight of the cut ω({1, 4, 5}) is ω({1, 4, 5}) = w 12 + w 13 + w 24 + w 25 + w 34. (1)

4 MAXCUT Determining a subset S V that maximizes ω(s) is the MAXCUT problem: maximize subject to ω(s) S V (MAXCUT) Equiavelently, we can write MAXCUT as 1 maximize 4 i,j w ij(1 σ i σ j ) subject to σ i = ±1 for all i V (MAXCUT ) Equivalent by setting i S σ i = +1.

5 MAXCUT Determining a subset S V that maximizes ω(s) is the MAXCUT problem: maximize subject to ω(s) S V (MAXCUT) Equiavelently, we can write MAXCUT as 1 maximize 4 i,j w ij(1 σ i σ j ) subject to σ i = ±1 for all i V (MAXCUT ) Equivalent by setting i S σ i = +1. MAXCUT is known to be NP-complete.

6 Relax Key Idea: Replace integers σ i = 1 with norm-1 vectors u i = 1, and scalar multiplication with vector multiplication. 1 maximize 4 i,j w ij(1 u i, u j ) subject to u i = 1 for all i in V (RELAX) This is a relaxation of MAXCUT since the original problem is contained in this problem, e.g., take u i = (±1, 0,..., 0). We will show later how to compute the u i s using a semidefinite program.

7 A key result Lemma Let r be a random 1 vector. For any unit vectors u i and u j, P ( sign( u i, r ) sign( u j, r ) ) = arccos( u i, u j ). π 1 By which me mean that r is drawn from a spherically symmetric distribution with zero mass at the origin.

8 A key result Lemma Let r be a random 1 vector. For any unit vectors u i and u j, P ( sign( u i, r ) sign( u j, r ) ) = arccos( u i, u j ). π As an immediate consequence of this Lemma, we have that [ 1 E 2 1 ] 2 sign( u i, r ) sign( u j, r ) = 1 π arccos( u i, u j ). 1 By which me mean that r is drawn from a spherically symmetric distribution with zero mass at the origin.

9 Proof. Using a suitable rotation, we can assume without loss that u i = (1, 0,..., 0) and u j = (a, b, 0,..., 0). (Why?) u j u i, u j θ ij u i

10 Proof. The signs of u i, r and u j, r are different if and only if the tangent plane of r separates u i and u j. u j u i, u j θ ij u i

11 Proof. The signs of u i, r and u j, r are different if and only if the tangent plane of r separates u i and u j. Here, the tangent plane does not separate u i and u j. r

12 Proof. The signs of u i, r and u j, r are different if and only if the tangent plane of r separates u i and u j. Here, the tangent plane separates u i and u j. r

13 Proof. A random line bisects an angle of θ ij with probability θ ij π, u j u i, u j θ ij u i

14 Proof. A random line bisects an angle of θ ij with probability θ ij π, but cos(θ ij ) = u i, u j, u j u i, u j θ ij u i

15 Proof. A random line bisects an angle of θ ij with probability θ ij π, but cos(θ ij ) = u i, u j, so that θ ij π = arccos( u i,u j ) π. u j u i, u j θ ij u i

16 Converting back to scalars Suppose we have the vectors u i that solve RELAX. Then do the following: 1. Choose a random vector r. 2. Set ˆσ i = sign( u i, r ) for all i V. 3. Equivalently, set i Ŝ if sign( u i, r ) 0.

17 Converting back to scalars Theorem Let S be a cut that optimizes MAXCUT. Then where α > E[ω ( Ŝ ) ] α ω(s ),

18 Converting back to scalars Theorem Let S be a cut that optimizes MAXCUT. Then where α > E[ω ( Ŝ ) ] α ω(s ), For the proof, we will use the fact that where arccos(y) π α 1 y 2 for all 1 y 1, 2θ α = min 0 θ π π(1 cos(θ)) > 0.87.

19 Proof. By the corollary to the Lemma and our fact, E[ω ( Ŝ ) ] = 1 2 i,j w ij arccos u i, u j π α w ij (1 u i, u j ). 4 i,j Since u i and u j maximize maximize the right-hand side over the unit sphere, by restriction, we have E[ω ( Ŝ ) ] α w ij (1 σ i σ j ). 4 for any σ i = ±1. In particular, the inequality holds for the maximum possible choice of signs, which by definition is α ω(s ). i,j

20 The step to semidefinite For a square matrix Σ, following are equivalent: 1. Σ 0, Σ ii = 1, and rank(σ) = 1, 2. Σ = σσ t, where σ i = ±1. Setting (W ) ij = w ij, note that w ij σ i σ j = tr(w σσ t ) (2) i,j Thus, MAXCUT is equivalent to 1 maximize 4 i,j w ij 1 4 tr(w Σ) subject to Σ 0, Σ ii = 1, and rank(σ) = 1. (3)

21 Semidefinite relaxation By dropping the rank-1 restriction, (3) becomes a semidefinite program: minimize tr(w { Σ) Σ 0, (4) subject to Σ ii = 1 Setting Σ = U t U via a Cholesky factorization, the restriction Σ ii = 1 implies that U has unit-norm columns. That is, (4) is equivalent to RELAX.

22 How to compute it For large problems, the current state-of-the-art algorithm for computing the solution to these semidefinite programs is available in Burer and Montiero [2]. For moderately sized problems, use Matlab s CVX package [3].

23 How to compute it For large problems, the current state-of-the-art algorithm for computing the solution to these semidefinite programs is available in Burer and Montiero [2]. For moderately sized problems, use Matlab s CVX package [3]. The entire code using CVX:... % define W, N cvx_begin sdp variable Sigma(N,N) symmetric minimize trace(w*sigma) subject to Sigma >= 0; diag(sigma) == ones(n,1); cvx_end U = chol(sigma); sigma = sign(u*randn(n,1));

24 How to compute it For large problems, the current state-of-the-art algorithm for computing the solution to these semidefinite programs is available in Burer and Montiero [2]. For moderately sized problems, use Matlab s CVX package [3]. The entire code using CVX:... % define W, N cvx_begin sdp variable Sigma(N,N) symmetric minimize trace(w*sigma) subject to Sigma >= 0; diag(sigma) == ones(n,1); cvx_end U = chol(sigma); % May fail occasionally due to numerical inaccuracy sigma = sign(u*randn(n,1));

25 For More Details M.X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1145, S. Burer and R. Monteiro. Local Minima and Convergence in Low-Rank Semidefinite Programming. Mathematical Programming, 103(3): , December M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming (web page and software)