A characterization of closed under intersection binary hypergraphs by a sequence of trees

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Submitted on 14 May 2018

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A characterization of closed under intersection binary hypergraphs by a sequence of trees

Célia Châtel, François Brucker, Pascal Prea

To cite this version:

Célia Châtel, François Brucker, Pascal Prea. A characterization of closed under intersection binary hypergraphs by a sequence of trees. Journées nationales GDR Informatique Mathématique, Apr 2018, Palaiseau, France. �hal-01790946�

A characterization of closed under intersection

binary hypergraphs by a sequence of trees

C´elia Chˆ

atel, Fran¸cois Brucker & Pascal Pr´ea

Laboratoire d’Informatique et Syst`emes (UMR 7020), Marseille

A characterization of closed under intersection

binary hypergraphs by a sequence of trees

C´elia Chˆ

atel, Fran¸cois Brucker & Pascal Pr´ea

Laboratoire d’Informatique et Syst`emes (UMR 7020), Marseille

Context

In 1983, Lehel proposed a characterization of totally balanced hypergraphs by a sequence of trees based on the contraction of all the edges of each tree.

1 2 3 4 5 6 14 24 34 45 56 124 245 234 456 1245 2456 2345 12456 12345 123456

In machine learning, decision trees are wide used binary systems based on yes/no ques-tions. Why not decision lattices ?

X1 ≤ α α < X1

prop1 ∧ prop2

prop1 prop2

Binary hypergraphs

A hypergraph H = (V, E) is said to be binary if:

∀v ∈ E, v 6= ∅ (

|{w ∈ E, v ≺ w}| ≤ 2 |{u ∈ E, u ≺ v}| ≤ 2

Binary hypergraph Non binary hypergraph

The Hasse diagram of the partially ordered set of the hypergraph’s hyperedges is a well adapted representation. A Hasse diagram is such that :

• hyperedges of the hypergraph are vertices of the diagram

• an edge goes from x to y iff x ≺ y (i.e x ( y, @z, x ⊆ z ⊆ y)

1 2 3 4

Hasse diagram of a binary hypergraph

1 2 3 4

Hasse diagram of a non binary hypergraph

Totally balanced and binarizable

hypergraphs

H = (V, E) totally balanced iff it has no

spe-cial cycle i.e (v₀, e₀, . . . v_n, e_n) such that,

• ∀i, vi ∈ ei

• ∀i, vi ∈ ei_{−1 mod n}

• ∀j 6= i, j 6= i − 1 mod n, vi ∈ e/ j

4-cycle H = (V, E) is binarizable iff

∃H0 = (V 0, E0) binary hypergraph such that V ⊆ V 0 and E ⊆ E0

Non binary hypergraph Binarized hypergraph

Theorem

A closed under intersection hypergraph is binarizable iff it is totally balanced.

Algorithm

Theorem

A hypergraph closed under intersection is binary iff its hyperedges are the vertices of a sequence of trees constructed by the algorithm.

Example

1 2 3 4 5 6 3 45 12 56 345 456 12345 1 2 3 5 6 45 45 12 56 345 12345 3456 3 5 6 45 12 12 345 456 123456 Example of construction Hypergraph obtained 1 2 3 4 5 6

Hasse diagram of the hypergraph

Conclusion

We give a characterization by a sequence of trees of binary closed under intersection hyper-graphs similar to Lehel’s characterization of totally balanced hyperhyper-graphs. This construction can also be extended to

• build a top down or bottom up binary hypergraph, • binarize totally balanced non binary hypergraphs,