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

A semidefinite programming hierarchy for packing problems in discrete geometry David de Laat (TU Delft) Joint work with Frank Vallentin (Universit¨ at zu K¨ oln) Applications of Real Algebraic Geometry Aalto University – February 28, 2014

Contents 1. Modeling geometric packing problems 2. Convergence to the optimal density 3. Duality theory 4. Harmonic analysis on subset spaces 5. Reduction to semidefinite programs

Packing problems in discrete geometry

Packing problems in discrete geometry ◮ These problems can be modeled as maximum independent set problems in graphs on infinitely many vertices

Packing problems in discrete geometry ◮ These problems can be modeled as maximum independent set problems in graphs on infinitely many vertices Spherical cap packings What is the maximum number of spherical caps of size t in S n − 1 such that no two caps intersect in their interiors? V = S n − 1 , G = ( V, E ) , E = {{ x, y } : x · y ∈ ( t, 1) }

Packing problems in discrete geometry ◮ These problems can be modeled as maximum independent set problems in graphs on infinitely many vertices Spherical cap packings What is the maximum number of spherical caps of size t in S n − 1 such that no two caps intersect in their interiors? V = S n − 1 , G = ( V, E ) , E = {{ x, y } : x · y ∈ ( t, 1) } ◮ Independent sets correspond to valid packings

The Lasserre hierarchy for finite graphs ◮ Maximum independent set problem for a finite graph as a 0/1 polynomial optimization problem: � � � α ( G ) = max x v : x v ∈ { 0 , 1 } for v ∈ V, x u + x v ≤ 1 for { u, v } ∈ E v ∈ V

The Lasserre hierarchy for finite graphs ◮ Maximum independent set problem for a finite graph as a 0/1 polynomial optimization problem: � � � α ( G ) = max x v : x v ∈ { 0 , 1 } for v ∈ V, x u + x v ≤ 1 for { u, v } ∈ E v ∈ V ◮ The Lasserre hierarchy for this problem (Laurent, 2003): � � � y { x } : y ∈ R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) � 0 x ∈ V

The Lasserre hierarchy for finite graphs ◮ Maximum independent set problem for a finite graph as a 0/1 polynomial optimization problem: � � � α ( G ) = max x v : x v ∈ { 0 , 1 } for v ∈ V, x u + x v ≤ 1 for { u, v } ∈ E v ∈ V ◮ The Lasserre hierarchy for this problem (Laurent, 2003): � � � y { x } : y ∈ R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) � 0 x ∈ V ◮ I 2 t is the set of independent sets of cardinality at most 2 t

The Lasserre hierarchy for finite graphs ◮ Maximum independent set problem for a finite graph as a 0/1 polynomial optimization problem: � � � α ( G ) = max x v : x v ∈ { 0 , 1 } for v ∈ V, x u + x v ≤ 1 for { u, v } ∈ E v ∈ V ◮ The Lasserre hierarchy for this problem (Laurent, 2003): � � � y { x } : y ∈ R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) � 0 x ∈ V ◮ I 2 t is the set of independent sets of cardinality at most 2 t ◮ M t ( y ) is the matrix with rows and columns indexed by I t and if J ∪ J ′ ∈ I 2 t , � y J ∪ J ′ M t ( y ) J,J ′ = 0 otherwise

The Lasserre hierarchy for finite graphs ◮ Maximum independent set problem for a finite graph as a 0/1 polynomial optimization problem: � � � α ( G ) = max x v : x v ∈ { 0 , 1 } for v ∈ V, x u + x v ≤ 1 for { u, v } ∈ E v ∈ V ◮ The Lasserre hierarchy for this problem (Laurent, 2003): � � � y { x } : y ∈ R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) � 0 x ∈ V ◮ I 2 t is the set of independent sets of cardinality at most 2 t ◮ M t ( y ) is the matrix with rows and columns indexed by I t and if J ∪ J ′ ∈ I 2 t , � y J ∪ J ′ M t ( y ) J,J ′ = 0 otherwise ◮ ϑ ′ ( G ) = las 1 ( G ) ≥ las 2 ( G ) ≥ . . . ≥ las α ( G ) ( G ) = α ( G )

The Lasserre hierarchy for finite graphs ◮ Maximum independent set problem for a finite graph as a 0/1 polynomial optimization problem: � � � α ( G ) = max x v : x v ∈ { 0 , 1 } for v ∈ V, x u + x v ≤ 1 for { u, v } ∈ E v ∈ V ◮ The Lasserre hierarchy for this problem (Laurent, 2003): � � � y { x } : y ∈ R I 2 t las t ( G ) = max ≥ 0 , y ∅ = 1 , M t ( y ) � 0 x ∈ V ◮ I 2 t is the set of independent sets of cardinality at most 2 t ◮ M t ( y ) is the matrix with rows and columns indexed by I t and if J ∪ J ′ ∈ I 2 t , � y J ∪ J ′ M t ( y ) J,J ′ = 0 otherwise ◮ ϑ ′ ( G ) = las 1 ( G ) ≥ las 2 ( G ) ≥ . . . ≥ las α ( G ) ( G ) = α ( G ) ◮ ϑ ( G ) is the Lov´ asz ϑ -number which specializes to the Delsarte LP-bound when G is the binary code graph

Generalization to infinite graphs ◮ Linear programming bound for spherical cap packings (Delsarte, 1977 / Kabatiansky, Levenshtein, 1978)

Generalization to infinite graphs ◮ Linear programming bound for spherical cap packings (Delsarte, 1977 / Kabatiansky, Levenshtein, 1978) ◮ Generalization of the ϑ -number (Bachoc, Nebe, de Oliveira, Vallentin, 2009)

Topological packing graphs ◮ We consider graphs where ◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations

Topological packing graphs ◮ We consider graphs where ◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations Definition A topological packing graph is a graph where - the vertex set is a Hausdorff topological space - each finite clique is contained in an open clique

Topological packing graphs ◮ We consider graphs where ◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations Definition A topological packing graph is a graph where - the vertex set is a Hausdorff topological space - each finite clique is contained in an open clique ◮ We consider compact topological packing graphs

Topological packing graphs ◮ We consider graphs where ◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations Definition A topological packing graph is a graph where - the vertex set is a Hausdorff topological space - each finite clique is contained in an open clique ◮ We consider compact topological packing graphs ◮ These graphs have finite independence number

Generalization for compact topological packing graphs � las t ( G ) = sup : , , �

Generalization for compact topological packing graphs � las t ( G ) = sup : λ ∈ M ( I 2 t ) ≥ 0 , , �

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , , �

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , �

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0 ◮ V t is the set of subsets of V of cardinality ≤ t

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0 ◮ V t is the set of subsets of V of cardinality ≤ t ◮ Quotient map: q : V t → V t \ {∅} , ( v 1 , . . . , v t ) �→ { v 1 , . . . , v t }

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0 ◮ V t is the set of subsets of V of cardinality ≤ t ◮ Quotient map: q : V t → V t \ {∅} , ( v 1 , . . . , v t ) �→ { v 1 , . . . , v t } ◮ V t \ {∅} is equipped with the quotient topology

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0 ◮ V t is the set of subsets of V of cardinality ≤ t ◮ Quotient map: q : V t → V t \ {∅} , ( v 1 , . . . , v t ) �→ { v 1 , . . . , v t } ◮ V t \ {∅} is equipped with the quotient topology ◮ V t is the disjoint union of V t \ {∅} with {∅}

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0 ◮ V t is the set of subsets of V of cardinality ≤ t ◮ Quotient map: q : V t → V t \ {∅} , ( v 1 , . . . , v t ) �→ { v 1 , . . . , v t } ◮ V t \ {∅} is equipped with the quotient topology ◮ V t is the disjoint union of V t \ {∅} with {∅} ◮ I t gets its topology as a subset of V t

Generalization for compact topological packing graphs � las t ( G ) = sup λ ( I =1 ) : λ ∈ M ( I 2 t ) ≥ 0 , λ ( {∅} ) = 1 , � A ∗ t λ ∈ M ( V t × V t ) � 0