Rounding procedures in Semidefinite Programming

Transcription

1 Rounding procedures in Semidefinite Programming Philippe Meurdesoif Université Bordeaux 1 & INRIA-Bordeaux CIMINLP, March 19, 2009.

2 1 Presentation 2 Roundings for MaxCut problems 3 Randomized roundings from (0,1) formulations 4 Heavily constrained problems 5 Conclusions

3 1 Presentation Semidefinite Programming Properties of X What input? What output? 2 Roundings for MaxCut problems 3 Randomized roundings from (0,1) formulations 4 Heavily constrained problems 5 Conclusions

4 Presentation Semidefinite Programming Linear Program in the elements of a psd matrix Useful tool : stronger relaxations than LP Numerous applications : Stable set (Lovász 79), MaxCut (Goemans, Williamson 95), Vertex coloring (Karger, Motwani, Sudan 98) Generic tool : "recipe" LP0-1 SDP (Poljak, Rendl, Wolkowicz 95, Lemaréchal, Oustry 01) 4 / 25

5 Presentation Semidefinite Programming st min c T x st Ax b = 0 Ax b = 0 (A T A, X) 2b T Ax + b T b = 0 x i {0, 1} X ii = x i X xx T = 0 min c T x SDP relaxation : X xx T 0 reformulated via the Schur lemma (-1,1)-programming : x i { 1, 1} X i i = 1. 5 / 25

6 Presentation Properties of X In an ideal world : Xij {0, 1} Typically : Xij [0, 1] In an ideal world : rank X =1 Typically : rank X n In an ideal world : (x i ) {0, 1}s.t.X ij = x ix j Typically : (v i ) R p s.t.x ij = vt i v j So, need for a rounding procedure. 6 / 25

7 Presentation What input? X ii in recipe, play the same role as x i in an LP relaxation. X ij indicates "proximity" between v i and v j v i R n obtained by Cholesky factorization (not unique) Any modification of X or v i s (convex combination, rotation, projection...) 7 / 25

8 Presentation What output? x i {0, 1} or{ 1, 1}, 8 / 25

9 Presentation What output? x i {0, 1} or{ 1, 1}, X psd with X ij {0, 1} or{ 1, 1}, "close" to X 8 / 25

10 Presentation What output? x i {0, 1} or{ 1, 1}, X psd with X ij {0, 1} or{ 1, 1}, "close" to X rank-one X psd 8 / 25

11 Presentation What output? x i {0, 1} or{ 1, 1}, X psd with X ij {0, 1} or{ 1, 1}, "close" to X rank-one X psd satisfying constraints 8 / 25

12 Presentation What output? x i {0, 1} or{ 1, 1}, X psd with X ij {0, 1} or{ 1, 1}, "close" to X rank-one X psd satisfying constraints without degrading objective function 8 / 25

13 1 Presentation 2 Roundings for MaxCut problems MaxCut SDP relaxation Goemans-Williamson s algorithm Outward rotation RPR 2 (Feige, Langberg 06) A few conclusions MaxGraphPartitioning (Han, Ye, Zhang 01) Rounding algorithm 3 Randomized roundings from (0,1) formulations 4 Heavily constrained problems 5 Conclusions

14 relaxation from a (-1,1) formulation st min 1 2 wij (1 X ij ) X ij = 1 X 0 X ij = 1 iff i and j in the same set. Can be derandomized in polynomial time (Mahajan-Ramesh 95) 10 / 25

15 Goemans-Williamson s algorithm Max-Cut Hyperplane point of view Vector point of view Covariance point of view 11 / 25

16 Goemans-Williamson s algorithm Max-Cut Hyperplane point of view Randomly generate a hyperplane Separate v i s using the hyperplane cut vertices Vector point of view Covariance point of view 11 / 25

17 Goemans-Williamson s algorithm Max-Cut Hyperplane point of view Randomly generate a hyperplane Separate v i s using the hyperplane cut vertices Vector point of view Randomly generate a unit vector r Separate vertices with respect to sgn(v T i r) Covariance point of view 11 / 25

18 Goemans-Williamson s algorithm Max-Cut Hyperplane point of view Randomly generate a hyperplane Separate v i s using the hyperplane cut vertices Vector point of view Randomly generate a unit vector r Separate vertices with respect to sgn(v T i r) Covariance point of view Randomly generate a vector u using multivariate normal distribution with mean 0 and covariance matrix X Separate vertices with respect to sgn(u) 11 / 25

19 Goemans-Williamson s algorithm Max-Cut Hyperplane point of view Randomly generate a hyperplane Separate v i s using the hyperplane cut vertices Vector point of view Randomly generate a unit vector r Separate vertices with respect to sgn(v T i r) Covariance point of view Randomly generate a vector u using multivariate normal distribution with mean 0 and covariance matrix X Separate vertices with respect to sgn(u) Max-k-Cut, Coloring k hyperplanes defining areas, or k unit vectors (vertex i mapped to the random vector closest to v i ) 11 / 25

20 Roundings for MaxCut problems Outward rotation Zwick 99 Before rounding, modify the v i s v i R n v i R 2n (c j ) orthonormal, orthogonal to v i s define v i = 1 γv i + γc i for someγ [0, 1] γ = 0, v i = v i γ = 1, v i mutually orthogonal Same as taking a combination of X and I. Better guarantees for certain "light" instances of MaxCut 12 / 25

21 Langberg 06) RPR 2 Project v i s on a randomly generated vector r : x i = v T i r For some function f : R [0, 1], set a i = 1 with probability f (x i ). 13 / 25

22 Langberg 06) RPR 2 Project v i s on a randomly generated vector r : x i = v T i r For some function f : R [0, 1], set a i = 1 with probability f (x i ). Hyperplane rounding special case of RPR 2 with f = 1 R + Outward rounding special case of RPR 2 (with complicated f ) 13 / 25

23 Roundings for MaxCut problems A few conclusions Very good approximation ratios Find feasible solutions 14 / 25

24 Roundings for MaxCut problems A few conclusions Very good approximation ratios Find feasible solutions Easy to find feasible solutions : no constraints! 14 / 25

25 Roundings for MaxCut problems A few conclusions Very good approximation ratios Find feasible solutions Easy to find feasible solutions : no constraints! What if there are constraints? 14 / 25

26 MaxGraphPartitioning (Han, Ye, Zhang 01) MGP max w(s) Card(S) = k 15 / 25

27 MaxGraphPartitioning (Han, Ye, Zhang 01) MGP max w(s) Card(S) = k Generalizes Max-Cut, Max-k-Cluster, Max-Vertex-Cover / 25

28 MaxGraphPartitioning (Han, Ye, Zhang 01) MGP max w(s) Card(S) = k Generalizes Max-Cut, Max-k-Cluster, Max-Vertex-Cover... with a cardinality constraint 15 / 25

29 MaxGraphPartitioning (Han, Ye, Zhang 01) MGP max w(s) Card(S) = k Generalizes Max-Cut, Max-k-Cluster, Max-Vertex-Cover... with a cardinality constraint (-1,1) formulation with additional row/column for linear terms Variable matrix : X 0i 1 X 0i X 15 / 25

30 algorithm Generate vector u randomly with multivariate normal distribution with mean 0 and covariance matrix where Y =γx + (1 γ)p P = 1 p... p p 1... p 2. p 2... p 2 p p and p = 2k/n 1 Define S ={i : u i 0} Use a combinatorial (greedy?) algorithm to adjust size of S. Generalizes Goemans-Williamson s and Zwick s rounding procedures For p = 2k/n 1, the expected size of S is k. 16 / 25

31 1 Presentation 2 Roundings for MaxCut problems 3 Randomized roundings from (0,1) formulations Projection Rounding Perron-Frobenius rounding 4 Heavily constrained problems 5 Conclusions

32 Randomized roundings from (0,1) formulations So far, we have dealt with (-1,1) formulations. Are there techniques for problems in (0,1) formulations? An example : Side Chain Positioning (in molecules) (Chazelle, Kingsford, Singh 04) Map object i to one element in set V i Variables x iv Constraints v x iv = 1 Quadratic objective function Numerous problems (Bin Packing, Vertex Coloring, Frequency Assignment...) 18 / 25

33 Projection Rounding The SDP relaxation associates vectors v iu to couples (i, u) Let q iu be the squared norm of v iu Constraints imply that u q iu = 1, hence probabilities Projection rounding : for each i choose one u V i with probability q iu (partition [0, 1] into intervals) Rounding performs better if for each i there is one u with high probability wrt other probabilities. 19 / 25

34 Perron-Frobenius rounding Idea : find a rank-one matrix close to X More general than the prefious family of problems. Ideally, a rank-one matrix has only one eigenvalue > 0. Use an eigenvector z associated to this value. By the Perron-Frobenius theorem, there is one z 0 Rounding method : use the elements of (eventually scaled) z as probability for accepting to set x i = 1. Usually outperforms projection rounding. Scaling can be used to verify cardinality constraints : ẑ k/(e T z)z leads to an expected cardinality of k. 20 / 25

35 Perron-Frobenius rounding A new (deterministic) rounding algorithm for MaxStable Solve the Schrijver strengthening of Lovász theta number : max(j, X) s.t. TrX = 1, X 0, X 0. Compute z (nonnegative) eigenvector corresponding to λ max (X ) Create S by sequentially selecting vertices in non-increasing order of their z i s. Seems to work good on some random graphs (but we need more tests to be more precise). Scaling is not important. What if G is an edge? 1/2I is optimal, how do we compute z? 21 / 25

36 1 Presentation 2 Roundings for MaxCut problems 3 Randomized roundings from (0,1) formulations 4 Heavily constrained problems 5 Conclusions

37 Heavily constrained problems What can we do if problems are heavily constrained? "Depends of the model" Vertex Coloring : Extract the edges where there is a conflict after rounding, and recolor the induced subgraph. For a (0,1) formulation : fix fractional X ij to one : merge elements or zero : weaker (in graph coloring, add an edge) Fix X ii = 0 imply fixing all X ij = 0 for this i. Either bound for an enumeration algorithm, or guide through the enumeration tree : heuristics Dukanovic-Rendl heuristic for vertex coloring, for instance : fix to 1 the largest off-diagonal of X. 23 / 25

38 1 Presentation 2 Roundings for MaxCut problems 3 Randomized roundings from (0,1) formulations 4 Heavily constrained problems 5 Conclusions

39 Conclusions There is not a unified technique for rounding SDPs, although there are few parametric frameworks (RPR 2, Han et al.). Random algorithms seem to be the norm (although some deterministic algorithms exist for specific problems) Usually tested on un- or lightly constrained problems. Depends of the original (unrelaxed) formulation Yet (-1,1) and (0,1) are "equivalent" (reformulations between both models exist), so roundings should be "translatable". Only simple constraints can be handled, and combinatorial post-optimization is often needed. 25 / 25