A semidefinite programming hierarchy for geometric packing problems - PowerPoint PPT Presentation

A semidefinite programming hierarchy for geometric packing problems David de Laat Joint work with Fernando M. de Oliveira Filho and Frank Vallentin DIAMANT Symposium – November 2012

Polydisperse spherical cap packings How can one pack spherical caps of sizes α 1 , . . . , α N on the unit sphere as densely as possible? x α

Maximal stable set problem Simple graph G Stability number: α ( G ) = 3

Maximal weighted stable set problem 0 . 2 0 . 2 Simple weighted graph G 0 . 5 Weighted stability number: α w ( G ) = 0 . 9 0 . 1 0 . 7

Bounds for the maximal stable set problem ◮ Computing α ( G ) is NP-hard

Bounds for the maximal stable set problem ◮ Computing α ( G ) is NP-hard ◮ Any stable set provides a lower bound

Bounds for the maximal stable set problem ◮ Computing α ( G ) is NP-hard ◮ Any stable set provides a lower bound ◮ The theta number provides an upper bound: α ( G ) ≤ ϑ ( G ) α w ( G ) ≤ ϑ w ( G ) and

Bounds for the maximal stable set problem ◮ Computing α ( G ) is NP-hard ◮ Any stable set provides a lower bound ◮ The theta number provides an upper bound: α ( G ) ≤ ϑ ( G ) α w ( G ) ≤ ϑ w ( G ) and ◮ Hierarchy of upper bounds: α ( G ) ≤ . . . ≤ ϑ 6 ( G ) ≤ ϑ 4 ( G ) ≤ ϑ 2 ( G ) = ϑ ( G )

Bounds for the maximal stable set problem ◮ Computing α ( G ) is NP-hard ◮ Any stable set provides a lower bound ◮ The theta number provides an upper bound: α ( G ) ≤ ϑ ( G ) α w ( G ) ≤ ϑ w ( G ) and ◮ Hierarchy of upper bounds: α ( G ) ≤ . . . ≤ ϑ 6 ( G ) ≤ ϑ 4 ( G ) ≤ ϑ 2 ( G ) = ϑ ( G )

Spherical cap packing graph    G :  

Spherical cap packing graph V = S n − 1 × { 1 , . . . , N }    G :  

Spherical cap packing graph V = S n − 1 × { 1 , . . . , N }    G : ( x, i ) ∼ ( y, j ) ⇔ cos( α i + α j ) < x · y and ( x, i ) � = ( y, j )  

Spherical cap packing graph V = S n − 1 × { 1 , . . . , N }    G : ( x, i ) ∼ ( y, j ) ⇔ cos( α i + α j ) < x · y and ( x, i ) � = ( y, j )  w ( x, i ) = normalized area of a cap with angle α i 

Spherical cap packing graph V = S n − 1 × { 1 , . . . , N }    G : ( x, i ) ∼ ( y, j ) ⇔ cos( α i + α j ) < x · y and ( x, i ) � = ( y, j )  w ( x, i ) = normalized area of a cap with angle α i  Stable sets correspond to spherical cap packings

Spherical cap packing graph V = S n − 1 × { 1 , . . . , N }    G : ( x, i ) ∼ ( y, j ) ⇔ cos( α i + α j ) < x · y and ( x, i ) � = ( y, j )  w ( x, i ) = normalized area of a cap with angle α i  Stable sets correspond to spherical cap packings α w ( G ) gives the optimal packing density

The theta number for the spherical cap packing graph ϑ w ( G ) = inf M : K − √ w ⊗ √ w ∈ C ( V × V ) � 0 , K ( u, u ) ≤ M for all u ∈ V, K ( u, v ) ≤ 0 for all { u, v } �∈ E where u � = v.

The theta number for the spherical cap packing graph ϑ w ( G ) = inf M : K − √ w ⊗ √ w ∈ C ( V × V ) � 0 , K ( u, u ) ≤ M for all u ∈ V, K ( u, v ) ≤ 0 for all { u, v } �∈ E where u � = v. V = S n − 1 × { 1 , . . . , N }

The theta number for the spherical cap packing graph ϑ w ( G ) = inf M : K − √ w ⊗ √ w ∈ C ( V × V ) � 0 , K ( u, u ) ≤ M for all u ∈ V, K ( u, v ) ≤ 0 for all { u, v } �∈ E where u � = v. V = S n − 1 × { 1 , . . . , N } Group action: O ( n ) × V → V, A ( x, i ) = ( Ax, i )

The theta number for the spherical cap packing graph ϑ w ( G ) = inf M : K − √ w ⊗ √ w ∈ C ( V × V ) � 0 , K ( u, u ) ≤ M for all u ∈ V, K ( u, v ) ≤ 0 for all { u, v } �∈ E where u � = v. V = S n − 1 × { 1 , . . . , N } Group action: O ( n ) × V → V, A ( x, i ) = ( Ax, i ) By averaging a feasible solution under the group action, we observe that we can restrict to O ( n ) invariant kernels.

The theta number for the spherical cap packing graph ϑ w ( G ) = inf M : K − √ w ⊗ √ w ∈ C ( V × V ) O ( n ) � 0 , K ( u, u ) ≤ M for all u ∈ V, K ( u, v ) ≤ 0 for all { u, v } �∈ E where u � = v. V = S n − 1 × { 1 , . . . , N } Group action: O ( n ) × V → V, A ( x, i ) = ( Ax, i ) By averaging a feasible solution under the group action, we observe that we can restrict to O ( n ) invariant kernels.

Generalization of Schoenberg’s theorem A kernel K ∈ C ( V × V ) O ( n ) is of the form � 0 ∞ � f ij,k P n k ( x · y ) , K (( x, i ) , ( y, j )) = k =0 where ( f ij,k ) N i,j =1 � 0 for all k

Generalization of Schoenberg’s theorem A kernel K ∈ C ( V × V ) O ( n ) is of the form � 0 ∞ � f ij,k P n k ( x · y ) , K (( x, i ) , ( y, j )) = k =0 where ( f ij,k ) N i,j =1 � 0 for all k ◮ We obtain a program with finitely many variables

Generalization of Schoenberg’s theorem A kernel K ∈ C ( V × V ) O ( n ) is of the form � 0 ∞ � f ij,k P n k ( x · y ) , K (( x, i ) , ( y, j )) = k =0 where ( f ij,k ) N i,j =1 � 0 for all k ◮ We obtain a program with finitely many variables ◮ N = 1 : reduces to Delsarte, Goethels, and Seidel LP bound

Generalization of Schoenberg’s theorem A kernel K ∈ C ( V × V ) O ( n ) is of the form � 0 ∞ � f ij,k P n k ( x · y ) , K (( x, i ) , ( y, j )) = k =0 where ( f ij,k ) N i,j =1 � 0 for all k ◮ We obtain a program with finitely many variables ◮ N = 1 : reduces to Delsarte, Goethels, and Seidel LP bound ◮ Still infinitely many constraints

Generalization of Schoenberg’s theorem A kernel K ∈ C ( V × V ) O ( n ) is of the form � 0 ∞ � f ij,k P n k ( x · y ) , K (( x, i ) , ( y, j )) = k =0 where ( f ij,k ) N i,j =1 � 0 for all k ◮ We obtain a program with finitely many variables ◮ N = 1 : reduces to Delsarte, Goethels, and Seidel LP bound ◮ Still infinitely many constraints ◮ Use a sums of squares characterization

Binary spherical cap packings on the 2 -sphere 0 . 96 1 . 0 0 . 94 0 . 92 0 . 8 0 . 9 0 . 88 0 . 6 0 . 86 0 . 84 0 . 4 0 . 82 0 . 8 0 . 78 0 . 2 0 . 76 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

SDP bound / Geometric bound (Florian 2001) 1 . 0 Geo. 0 . 8 0 . 6 0 . 4 SDP 0 . 2 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

Spherical codes on the 2 -sphere 0 . 90 Icosahedron Octahedron 0 . 88 0 . 86 Simplex 0 . 84 0 . 82 0 . 80 0 . 78 0 . 76 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

The truncated octahedron packing This packing is maximal: ◮ it has density 0 . 9056 . . . ◮ the semidefinite programming bound is 0 . 9079 . . . ◮ the next packing ( 4 big caps, 19 small caps) would have density 0 . 9103 . . .

Packings associated to the n -prism ◮ The geometric bound is sharp for n ≥ 6 ◮ For n = 5 there is a geometrical proof (Florian, Heppes 1999) ◮ The semidefinite programming bound is sharp for n = 5

Packing graphs ◮ We generalize the Lasserre hierarchy to infinite graphs

Packing graphs ◮ We generalize the Lasserre hierarchy to infinite graphs ◮ A packing graph: ◮ The vertex set is a Hausdorff topological space ◮ Each finite clique is contained in an open clique

Packing graphs ◮ We generalize the Lasserre hierarchy to infinite graphs ◮ A packing graph: ◮ The vertex set is a Hausdorff topological space ◮ Each finite clique is contained in an open clique ◮ We consider compact second-countable packing graphs

Packing graphs ◮ We generalize the Lasserre hierarchy to infinite graphs ◮ A packing graph: ◮ The vertex set is a Hausdorff topological space ◮ Each finite clique is contained in an open clique ◮ We consider compact second-countable packing graphs ◮ These graphs have finite stability number

Packing graphs ◮ We generalize the Lasserre hierarchy to infinite graphs ◮ A packing graph: ◮ The vertex set is a Hausdorff topological space ◮ Each finite clique is contained in an open clique ◮ We consider compact second-countable packing graphs ◮ These graphs have finite stability number ◮ Example: graphs where the vertex set is a compact metric space such that x and y are adjacent if d ( x, y ) ∈ (0 , δ )

A semidefinite programming hierarchy ◮ Sub ( V, t ) = set of nonempty subsets of V with ≤ t elements

A semidefinite programming hierarchy ◮ Sub ( V, t ) = set of nonempty subsets of V with ≤ t elements ◮ I 2 t = subcollection of Sub ( V, 2 t ) consisting of stable sets

A semidefinite programming hierarchy ◮ Sub ( V, t ) = set of nonempty subsets of V with ≤ t elements ◮ I 2 t = subcollection of Sub ( V, 2 t ) consisting of stable sets ◮ V t = Sub ( V, t ) ∪ {∅}

A semidefinite programming hierarchy ◮ Sub ( V, t ) = set of nonempty subsets of V with ≤ t elements ◮ I 2 t = subcollection of Sub ( V, 2 t ) consisting of stable sets ◮ V t = Sub ( V, t ) ∪ {∅} ◮ We define the operator A t : C ( V t × V t ) sym → C ( I 2 t ) by � f ( J, J ′ ) A t f ( S ) = J,J ′ ∈ V t : J ∪ J ′ = S