Counting minimal transversals in -acyclic Hypergraph Benjamin - - PowerPoint PPT Presentation

Counting minimal transversals in β-acyclic Hypergraph

Benjamin Bergougnoux, Florent Capelli†, Mamadou M. Kanté⋆

⋆ LIMOS, CNRS, Université Clermont Auvergne † Université Lille 3, CRIStAL, CNRS, INRIA

Algorithm Seminar, Bergen, November 22, 2019

Hypergraphs

1 2 3 4 5 6 7

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Definitions

Transversal

A set of vertices intersecting all hyperedges. ◮ Transversals of graph: Vertex Cover. ◮ Minimal w.r.t. inclusion. ◮ Every vertex has a private hyperedge. 1 2 3 4 5 6 7

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Definitions

Transversal

A set of vertices intersecting all hyperedges. ◮ Transversals of graph: Vertex Cover. ◮ Minimal w.r.t. inclusion. ◮ Every vertex has a private hyperedge. 1 2 3 4 5 6 7 2

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Motivations

◮ Graphs: vertex covers, dominating sets (transversals of closed neighborhoods),... ◮ AI: models of monotone CNF-formula. Counting and enumerating them have a lot of applications ◮ Graphs Theory, A.I. (robustness), Datamining, Model-checking,...

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Counting

Goal

Find the number of solutions to some problem. ◮ An analogous hierarchy to P ⊆ NP.

◮ Polynomial ◮ #P ◮ #P-hard / complete

◮ Counting can be much harder than finding:

◮ Minimal Transversal, Perfect Matching, 2-SAT,... ◮ k-path, k-cycle!

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Minimality matters

Theorem [Okamoto, Uno, Uehara 2005]

In chordal graphs: ◮ Counting transversals is doable in polynomial time. ◮ Counting minimal transversals is #P-complete. ◮ Chordal graph: no induced cycle of size ≤ 4.

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β-acyclic

Definition

A hypergraph is β-acyclic if it does not have β-cycles. ◮ The incidence graph (vertices/hyperedges) is Bipartite Chordal.

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β-acyclic

Definition bis

A hypergraph is β-acyclic if ◮ Strong Elimination ordering on V (H) : β-leaf

e1 e2 e3

e1 ⊆ e2 ⊆ e3 ⊆ . . . ⊆ eℓ

eℓ

◮ Not related to the x-width of the incidence graph.

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Results on β-acyclic hypergraph

Theorem [Brault-Baron, Capelli, Mengel 2017]

We can count the transversals of a β-acyclic hypergraph in polynomial time.

Theorem [B., Capelli, Kanté 2017]

We can count the minimal transversals of a β-acyclic hypergraph in polynomial time. ◮ The ingredients: recursive decomposition and “blocked transversals”.

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Basic facts

x

H

Fact

The nb. of minimal transversals of H not containing x is the nb. of minimal transversals of H − x.

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Basic facts

H − x

Fact

The nb. of minimal transversals of H not containing x is the nb. of minimal transversals of H − x.

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Basic facts

x

H

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Basic facts

H \ H(x)

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Basic facts

H \ H(x)

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Basic facts

x

H

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Basic facts

x

H

Lemma

T ∪ x is a min. transversal of H iff: ◮ T is a min. transversal of H \ H(x) ◮ T is not a transversal of H

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Formula

Corollary

The following values are equal: ◮ The nb. of minimal transversals of H containing x. ◮ The nb. of minimal transversals of H − H(x) that are not transversals of H. ◮ The nb. of minimal transversals of H − H(x) minus the nb. of minimal transversals of H − H(x) that are transversals of H.

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Blocked-Transversals

S: a subset of vertices.

Definition

S-blocked transversal of H (denoted by S-btr(H)): ◮ Minimal transversals of H and of H \ H(S) ◮ ∅-btr(H) = min. transversals of H. ◮ Private hyperedges cannot contain a vertex of S.

x

H

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Blocked Transversal

Lemma (reformulation)

The nb. of minimal transversals of H =

• nb. of min. transversals of H excluding x

+ nb. of min. transversals of H \ H(x) − nb. of min. transversals of H \ H(x) and of H.

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Blocked Transversal

Lemma (reformulation)

The nb. of minimal transversals of H = #∅-btr(H − x) + nb. of min. transversals of H \ H(x) − nb. of min. transversals of H \ H(x) and of H.

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Blocked Transversal

Lemma (reformulation)

The nb. of minimal transversals of H = #∅-btr(H − x) + #∅-btr(H \ H(x)) − nb. of min. transversals of H \ H(x) and of H.

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Blocked Transversal

Lemma (reformulation)

The nb. of minimal transversals of H = #∅-btr(H − x) + #∅-btr(H \ H(x)) − #x-btr(H).

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Blocked Transversal

Lemma (reformulation)

The nb. of minimal transversals of H = #∅-btr(H − x) + #∅-btr(H \ H(x)) − #x-btr(H).

Lemma (generalization)

The nb. of S-blocked transversals of H = #S-btr(H − x) + #S-btr(H \ H(x)) − #S ∪ {x}-btr(H \ (H(x) ∩ H(S))).

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Blocked Transversal

Using the formula leads to a combinatorial explosion in general!

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Decomposition of β-acyclic hypergraph

◮ Strong elimination ordering on V (H) : x1, . . . , xn. ◮ Induced order on the hyperedges : e1, . . . , em.

Definition

H(x, e) ⇒ the hyperedges e connected to e through vertices x .

2 1 3 4 5 6 7 2 1 3

H H(3, {2, 3})

◮ H(xn, em) = H. ◮ Goal is to compute #∅-btr(H(xn, em)) with the help of:

◮ The formula: #S-btr(H) = #S-btr(H − x)+#S-btr(H \ H(x))−#S ∪ {x}-btr(H \ (H(x) ∩ H(S))) ◮ Structural properties of the hypergraphs H(x, e).

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Avoid the explosion

Combinatorial explosion !

To compute #{w}-btr(H) we need to compute: ◮ #{x, w}-btr(H \ (H(x) ∩ H(w))),...

But in β-acyclic hypergraphs

We can compute #{x, w}-btr(H \ (H(x) ∩ H(w))) from ◮ #x-btr(H′). ◮ #w-btr(H′). We can express each H′ as a H(y, f) where y < x.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4 2 1

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4

#∅-btr(H(4, e2)) #∅-btr(H(4, e5)) #5-btr(H(4, e5))

2 1

#∅-btr(H(2, e1))

Each term is computable from smaller terms.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4

#∅-btr(H(4, e2)) #∅-btr(H(4, e5)) #5-btr(H(4, e5))

2 1

#∅-btr(H(3, e2)) #4-btr(H(3, e2)) #∅-btr(H(2, e5)) #4-btr(H(2, e5))

Each term is computable from smaller terms.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4

#∅-btr(H(4, e2)) #∅-btr(H(4, e5)) #5-btr(H(4, e5))

2 1

#∅-btr(H(3, e2)) #4-btr(H(3, e2))

Each term is computable from smaller terms.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4

#∅-btr(H(4, e2)) #∅-btr(H(4, e5)) #5-btr(H(4, e5))

2 1

#∅-btr(H(3, e2)) #5-btr(H(2, e5)) #4-btr(H(3, e2)) #∅-btr(H(2, e1))

Each term is computable from smaller terms.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4 2 1

We end up with a DAG.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4 2 1

Sink are easily computable.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4 2 1

As a salmon, we go back to the source.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4 2 1

As a salmon, we go back to the source.

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Dynamic programming

#∅-btr(H(5, e5))

3 5 4 2 1

Number of terms is polynomial ⇒ Algorithm polynomial.

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Result and Consequences

Theorem [B. , Capelli, Kanté 2017]

We can count the number of minimal transversals of a β-acyclic hypergraph in polynomial time. ◮ Min. dominating sets ⇒ min. transversals of closed neighborhoods. ◮ Strongly chordal graphs ⇔ {N[x] | x ∈ V (G)} is β-acyclic.

Corollary

We can count the number of min. dominating sets of Strongly Chordal graph in polynomial time.

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Minimal dominating sets

◮ Stongly Chordal graphs ⇔ Chordal graphs + k-sun free, for k 3.

3-sun 4-sun 5-sun

[Kanté, Uno 2017]

Counting min. dominating sets of Chordal graphs is #P-complete.

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Minimal dominating sets

◮ Stongly Chordal graphs ⇔ Chordal graphs + k-sun free, for k 3.

3-sun 4-sun 5-sun

[Kanté, Uno 2017]

Counting min. dominating sets of Chordal graphs is #P-complete.

Conjecture [Kanté, Uno 2017]

Counting min. dominating sets of a subclass of Chordal graphs is ◮ doable in polynomial time if this class is k-sun free, for k 4, ◮ #P-complet otherwise.

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Possible generalizations

Theorem [Brault-Baron, Capelli, Mengel 2015]

We can count the Models of a β-acyclic CNF-formula in polynomial time.

Corollary of our result

We can count the minimal models of a monotone β-acyclic CNF-formula in polynomial time. ◮ Can we do it for non-monotone formulas?

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Beyond β-acyclicity

β-hypertree width β-acyclic mim(incidence graph) tw(incidence graph) cw(incidence graph) Hypertree width

Polytime or XP Para-NP-hard

?

point-width

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Thank you!

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