VC-dimension and Erd os-P osa property Nicolas Bousquet LIRMM, - PowerPoint PPT Presentation

VC-dimension VC-dimension for graphs Conclusion VC-dimension and Erd˝ os-P´ osa property Nicolas Bousquet LIRMM, University Montpellier II Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications VC-dimension 1 Packing and transversality VC-dimension Applications VC-dimension for graphs 2 Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝ os-P´ osa property Conclusion 3 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition A hypergraph is a pair ( V , E ) where V is a set of vertices and E is a set of hyperedges (subsets of vertices). Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition The transversality τ of a hypergraph is the minimum number of vertices which intersect all the hyperedges. �� �� �� �� �� �� �� �� Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition The transversality τ of a hypergraph is the minimum number of vertices which intersect all the hyperedges. Linear Programing Variables : for each v i ∈ V , associate x i a non negative integer. Constraints : for each e ∈ E , � x i ≥ 1 v i ∈ e Objective function : n � τ = min ( x i ) i =1 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition The transversality τ of a hypergraph is the minimum number of vertices which intersect all the hyperedges. Linear Fractional Relaxation Variables : for each v i ∈ V , associate x i a non negative real number. Constraints : for each e ∈ E , � x i ≥ 1 v i ∈ e Objective function : n τ ∗ = min ( � x i ) i =1 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Integrality gap Integrality gap V = { 1 , ..., 2 n } e ∈ E iff | e | = n τ ∗ = 2 τ = n + 1 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition The packing number ν of a hypergraph is the maximum number of disjoint hyperedges. Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition The packing number ν of a hypergraph is the maximum number of disjoint hyperedges. Linear Programing Variables : for each e i ∈ E , associate x i a non negative integer. Constraints : for each v ∈ V , � x i ≤ 1 e i / v ∈ e Objective function : | E | � ν = max ( x i ) i =1 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Definition The packing number ν of a hypergraph is the maximum number of disjoint hyperedges. Linear Fractional Relaxation Variables : for each e i ∈ E , associate x i a non negative real number. Constraints : for each v ∈ V , � x i ≥ 1 e i / v ∈ e Objective function : | E | ν ∗ = max ( � x i ) i =1 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Integrality gap Integrality gap The vertices of H are the edges of a clique on n vertices. The hyperedges are the maximum stars of the clique. ν = 1 ν ∗ = n / 2 Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Erd˝ os-P´ osa property Duality Theorem of Linear Programing τ ∗ = ν ∗ Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Erd˝ os-P´ osa property Duality Theorem of Linear Programing τ ∗ = ν ∗ Inequalities ν ≤ ν ∗ = τ ∗ ≤ τ Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Erd˝ os-P´ osa property Duality Theorem of Linear Programing τ ∗ = ν ∗ Inequalities ν ≤ ν ∗ = τ ∗ ≤ τ Erd˝ os-P´ osa property A class H of hypergraphs has the Erd˝ os-P´ osa property iff there exists a function f such that for all H ∈ H , τ ≤ f ( ν ). Theorem (Erd˝ os-P´ osa) The cycle hypergraph of a graph has the Erd˝ os-P´ osa property. Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications VC-dimension Definition A set X ⊆ V is shattered iff for all Y ⊆ X , there exists e ∈ E such that e ∩ X = Y . The VC-dimension of a hypergraph is the maximum size of a shattered set. Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications 2VC-dimension Definition A set X ⊆ V is 2 -shattered iff for all Y ⊆ X with | Y | = 2, it exists e ∈ E such that e ∩ X = Y . The 2 VC-dimension of a hypergraph is the maximum size of a 2-shattered set. Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Gap between VC-dimension and 2VC-dimension Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Theorem Theorem (Vapnik, Chervonenkis ’72) For every hypergraph H of VC-dimension d : τ ≤ 20 d τ ∗ log ( τ ∗ ) Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Applications k -majority tournament V = { 1 , ..., n } . Let P 1 , P 2 k − 1 be linear orders on V . The tournament realized by P 1 , ..., P 2 k − 1 has an edge from i to j iff i ≻ j in at least k orders. A tournament is a k-majority tournament iff it can be realized by 2 k − 1 linear orders. Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Applications k -majority tournament V = { 1 , ..., n } . Let P 1 , P 2 k − 1 be linear orders on V . The tournament realized by P 1 , ..., P 2 k − 1 has an edge from i to j iff i ≻ j in at least k orders. A tournament is a k-majority tournament iff it can be realized by 2 k − 1 linear orders. Theorem (Alon, Brightwell, Kierstead, Kotochka, Winkler ’04) Each k -majority tournament has a dominating set of size O ( k · log ( k )). Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Proof Consider the hypergraph H with hyperedges the in-neighborhoods of the vertices of T : a transversal of H is a dominating set of T . Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Proof Consider the hypergraph H with hyperedges the in-neighborhoods of the vertices of T : a transversal of H is a dominating set of T . τ ∗ is bounded (by 2). Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Proof Consider the hypergraph H with hyperedges the in-neighborhoods of the vertices of T : a transversal of H is a dominating set of T . τ ∗ is bounded (by 2). The VC-dimension is bounded (by O ( k · log ( k ))). Theorem (Vapnik, Chervonenkis ’72) For every hypergraph H of VC-dimension d : τ ≤ 20 d τ ∗ log ( τ ∗ ) Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Conjecture Definition The partial orders P 1 , ..., P k cover a graph G iff x i ≻ x j for P l ⇐ ⇒ ( x i , x j ) ∈ E . Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property

VC-dimension Packing and transversality VC-dimension for graphs VC-dimension Conclusion Applications Conjecture Definition The partial orders P 1 , ..., P k cover a graph G iff x i ≻ x j for P l ⇐ ⇒ ( x i , x j ) ∈ E . Conjecture A tournament covered by at most k partial orders have a dominating set of size at most f ( k ). Nicolas Bousquet VC-dimension and Erd˝ os-P´ osa property