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

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

Nov 15, 2023 386 likes •957 views

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

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VC-dimension VC-dimension for graphs Conclusion

VC-dimension and Erd˝

  • s-P´
  • sa property

Nicolas Bousquet LIRMM, University Montpellier II

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

1

VC-dimension Packing and transversality VC-dimension Applications

2

VC-dimension for graphs Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

3

Conclusion

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Definition The transversality τ of a hypergraph is the minimum number of vertices which intersect all the hyperedges. Linear Programing Variables : for each vi ∈ V , associate xi a non negative integer. Constraints : for each e ∈ E,

  • vi∈e

xi ≥ 1 Objective function : τ = min(

n

  • i=1

xi)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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 vi ∈ V , associate xi a non negative real number. Constraints : for each e ∈ E,

  • vi∈e

xi ≥ 1 Objective function : τ ∗ = min(

n

  • i=1

xi)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Integrality gap

Integrality gap V = {1, ..., 2n} e ∈ E iff |e| = n τ ∗ = 2 τ = n + 1

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Definition The packing number ν of a hypergraph is the maximum number of disjoint hyperedges.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Definition The packing number ν of a hypergraph is the maximum number of disjoint hyperedges. Linear Programing Variables : for each ei ∈ E, associate xi a non negative integer. Constraints : for each v ∈ V ,

  • ei/v∈e

xi ≤ 1 Objective function : ν = max(

|E|

  • i=1

xi)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Definition The packing number ν of a hypergraph is the maximum number of disjoint hyperedges. Linear Fractional Relaxation Variables : for each ei ∈ E, associate xi a non negative real number. Constraints : for each v ∈ V ,

  • ei/v∈e

xi ≥ 1 Objective function : ν∗ = max(

|E|

  • i=1

xi)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Erd˝

  • s-P´
  • sa property

Duality Theorem of Linear Programing τ ∗ = ν∗

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Erd˝

  • s-P´
  • sa property

Duality Theorem of Linear Programing τ ∗ = ν∗ Inequalities ν ≤ ν∗ = τ ∗ ≤ τ

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Erd˝

  • s-P´
  • sa property

Duality Theorem of Linear Programing τ ∗ = ν∗ Inequalities ν ≤ ν∗ = τ ∗ ≤ τ Erd˝

  • s-P´
  • sa property

A class H of hypergraphs has the Erd˝

  • s-P´
  • sa property iff there

exists a function f such that for all H ∈ H, τ ≤ f (ν). Theorem (Erd˝

  • s-P´
  • sa)

The cycle hypergraph of a graph has the Erd˝

  • s-P´
  • sa property.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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 2VC-dimension of a hypergraph is the maximum size of a 2-shattered set.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Gap between VC-dimension and 2VC-dimension

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Theorem

Theorem (Vapnik, Chervonenkis ’72) For every hypergraph H of VC-dimension d : τ ≤ 20dτ ∗log(τ ∗)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Applications

k-majority tournament V = {1, ..., n}. Let P1, P2k−1 be linear orders on V . The tournament realized by P1, ..., P2k−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 2k − 1 linear orders.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Applications

k-majority tournament V = {1, ..., n}. Let P1, P2k−1 be linear orders on V . The tournament realized by P1, ..., P2k−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 2k − 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension 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 ≤ 20dτ ∗log(τ ∗)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Conjecture

Definition The partial orders P1, ..., Pk cover a graph G iff xi ≻ xj for Pl ⇐ ⇒ (xi, xj) ∈ E.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Packing and transversality VC-dimension Applications

Conjecture

Definition The partial orders P1, ..., Pk cover a graph G iff xi ≻ xj for Pl ⇐ ⇒ (xi, xj) ∈ 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˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

1

VC-dimension Packing and transversality VC-dimension Applications

2

VC-dimension for graphs Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

3

Conclusion

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Dual hypergraph

Bipartite incidence graph A hypergraph H = (V , E) can be seen as a bipartite incidence graph G with vertex set (V , E) where (v, e) in an edge iff v ∈ e.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Dual hypergraph

Bipartite incidence graph A hypergraph H = (V , E) can be seen as a bipartite incidence graph G with vertex set (V , E) where (v, e) in an edge iff v ∈ e. Dual hypergraph The pair (V , E) is oriented : the hypergraph associated to the pair (E, V ) is the dual hypergraph denoted by Hd.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Dual hypergraph

Bipartite incidence graph A hypergraph H = (V , E) can be seen as a bipartite incidence graph G with vertex set (V , E) where (v, e) in an edge iff v ∈ e. Dual hypergraph The pair (V , E) is oriented : the hypergraph associated to the pair (E, V ) is the dual hypergraph denoted by Hd. Definition The dual VC-dimension is the VC-dimension of the dual hypergraph.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Duality gap

VC-dimension and dual VC-dimension VC-dimension and dual VC-dimension are linked by an exponential function. If the VC-dimension is arbitrarily large, you have the bipartite graph you want as an induced subgraph.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Duality gap

VC-dimension and dual VC-dimension VC-dimension and dual VC-dimension are linked by an exponential function. A arbitrarily large gap is possible between the 2VC-dimension and the dual 2-VC-dimension.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Theorem

Theorem (Ding, Seymour, Winkler ’91) Let H be a hypergraph of dual 2VC-dimension d then : τ ≤ 11d2(ν + d + 3) · d + ν d

  • Nicolas Bousquet

VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Covering of a planar graph

Theorem (Chepoi, Estellon, Vax` es ’07) There exists a constant m such that, for every planar graph of diameter 2l, there are m balls of radius l which cover the graph.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

We consider the hypergraph of balls of radius l.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

We consider the hypergraph of balls of radius l. Observation H and Hd are the same : x ∈ B(y, l) iff y ∈ B(x, l).

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

We consider the hypergraph of balls of radius l. Observation H and Hd are the same : x ∈ B(y, l) iff y ∈ B(x, l). (p, q)-property A hypergraph has the (p, q)-property iff for each set of p hyperedges, q of them share a vertex.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

We consider the hypergraph of balls of radius l. Observation H and Hd are the same : x ∈ B(y, l) iff y ∈ B(x, l). (p, q)-property A hypergraph has the (p, q)-property iff for each set of p hyperedges, q of them share a vertex. Theorem (Matousek) Let H be a hypergraph of dual VC-dimension d. There exists a function f such that if H has the (p, d + 1) property then τ ≤ f (p, d).

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

Proof Verify that the dual VC-dimension of H is bounded (by 4). Verify that H has the (p, 6)-property for a constant p. Assume that the dual VC-dimension is at least 5.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

Proof Verify that the dual VC-dimension of H is bounded (by 4). Verify that H has the (p, 6)-property for a constant p. Assume that the dual VC-dimension is at least 5.

  • Nicolas Bousquet

VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Proof

Proof Verify that the dual VC-dimension of H is bounded (by 4). Verify that H has the (p, 6)-property for a constant p. Theorem (Matousek) There exists a function f such that if H has the (p, 6) property then τ ≤ f (p, 6). Hence there exists a fixed number of balls which cover all the vertices of the graph.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Another proof

The dual 2VC-dimension is bounded by 4. The packing number is equal to 1.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Another proof

The dual 2VC-dimension is bounded by 4. The packing number is equal to 1. Ding, Seymour, Winkler Let H be a hypergraph of dual 2VC-dimension d then : τ ≤ 11d2(ν + d + 3) · d + ν d

  • Then τ is bounded by a constant (880 000).

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Conjecture

Conjecture (Chepoi, Estellon, Vax` es) There exists a constant c such that the hypergraph of the balls of radius l of a planar graph satisfies : τ ≤ c · ν

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Definition

Definition We denote by B(x, l) the set of the vertices at distance at most l from x in the graph. The hypergraph of iterated neighborhoods of G is the hypergraph

  • n V with hyperedges B(x, l) for all x and l.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Definition

Definition The VC-dimension of a graph is equal to the VC-dimension of the hypergraph of iterated neighborhoods.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Definition

Definition The VC-dimension of a graph is equal to the maximum VC-dimension of the hypergraph of iterated neighborhoods for all induced subgraph.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Graphs of bounded VC-dimension

Theorem (B.,Thomass´ e) The planar graphs have VC-dimension at most 4. The Kn minor free graphs have VC-dimension at most n − 1.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Graphs of bounded VC-dimension

Theorem (B.,Thomass´ e) The planar graphs have VC-dimension at most 4. The Kn minor free graphs have VC-dimension at most n − 1. The graphs of rankwidth k have VC-dimension at most 3 · 22k+1 + 2.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Erd˝

  • s-P´
  • sa property

Theorem There exists a function f such that if G has VC-dimension d then, the hypergraph of the balls of radius l satisfies : τ ≤ f (ν, d)

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

Erd˝

  • s-P´
  • sa property

Theorem There exists a function f such that if G has VC-dimension d then, the hypergraph of the balls of radius l satisfies : τ ≤ f (ν, d) Theorem (Matousek) Let H be a hypergraph of dual VC-dimension d. There exists a function f such that if H has the (p, d + 1) property then τ ≤ f (p, d). The VC-dimension is bounded. We have to verify that the (p, d + 1) property is verified for p which depends only of ν and d.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion

1

VC-dimension Packing and transversality VC-dimension Applications

2

VC-dimension for graphs Dual VC-dimension VC-dimension of graphs Classes of graphs of bounded VC-dimension Erd˝

  • s-P´
  • sa property

3

Conclusion

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion

Open problem Circle graphs without triangle have bounded VC-dimension.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion

Open problem Circle graphs without triangle have bounded VC-dimension. Open problems A class of graphs of bounded VC-dimension is χ-bounded.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion

Open problem Circle graphs without triangle have bounded VC-dimension. Open problems A class of graphs of bounded VC-dimension is χ-bounded. A class of graphs of bounded 2VC-dimension is χ-bounded.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property
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VC-dimension VC-dimension for graphs Conclusion

Open problem Circle graphs without triangle have bounded VC-dimension. Open problems A class of graphs of bounded VC-dimension is χ-bounded. A class of graphs of bounded 2VC-dimension is χ-bounded. A class of triangle free graphs of bounded 2VC-dimension has chromatic number at most c.

Nicolas Bousquet VC-dimension and Erd˝

  • s-P´
  • sa property