enumerating minimal dominating sets in triangle-free graphsodefrain/pdf/stacs2019.pdfenumerating...

Enumerating minimal dominating setsin triangle-free graphs

Oscar Defrain

LIMOS, Clermont-Ferrand.

STACS 2019

March 13–16, TU Berlin.

Joint work with Marthe Bonamy (LaBRI, Bordeaux), Marc Heinrich

(LIRIS, Lyon) and Jean-Florent Raymond (TU Berlin).

Slides of Jean-Florent Raymond.

Introduction to enumeration algorithms

Typical question:

given input I , list all objects of type X in I

• cycles, cliques, stable sets, dominating sets, etc. of a graph

• transversals of a hypergraph

• variable assignments satisfying a formula

• trains to Paris leaving tomorrow before 10:00

• shortest paths from Montpellier to Marseille

• . . .

Remark: possibly many objects!

Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13

Typical question:

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Typical question:

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Typical question:

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Typical question:

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Typical question:

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Typical question:

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Two perspectives about complexity

Input-sensitive: in terms of input size

• [Fomin, Grandoni, and Pyatkin, 2008]:

O (1.7159n)-time for Minimal Dominating Set

n = |V (G )|• upper-bounds the number of objects

Output-sensitive: in terms of input+output size

• [Fredman and Khachiyan, 1996]:

O(N logN

)-time for Minimal Dominating Set

N = |G |+ |D(G )|

n = |V (G )|

• upper-bounds the number of objects

O(N logN

N = |G |+ |D(G )|

O(N logN

N = |G |+ |D(G )|

O(N logN

N = |G |+ |D(G )|

O(N logN

N = |G |+ |D(G )|

“Fast” output-sensitive algorithms

n: input size

s: output size

• output-polynomial: poly(n + s) to get all the solutions

• polynomial delay: poly(n) per solution

Note: output space is not counted.

n: input size

s: output size

n: input size

s: output size

Minimal dominating sets

• N(v): neighbors of a vertex v

• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)

“D can see everybody else”

• minimal dominating set: inclusion-wise minimal DS

• private neighbor of v ∈ D:

vertex that is

{dominated by v, and

not dominated by D \ {v}(possibly v)

• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.

• set of minimal DS denoted by D(G )

Simple observation:

a DS is minimal if and only if it is irredundant

(if all its vertices have a private neighbor)

vertex that is

Simple observation:

vertex that is

Simple observation:

vertex that is

Simple observation:

vertex that is

Simple observation:

vertex that is

Simple observation:

vertex that is

Simple observation:

Enumeration of minimal dominating sets

Problem: enumerating D(G ) given G .

Dream goal: an output-polynomial time algorithm

Running time polynomial in (input size + output size)

General case: open (best is quasi-polynomial)

Known: degenerate graphs, graphs of bounded clique-width or

LMIM-width, split graphs, chordal graph, graphs of girth > 7, etc.

Theorem (Bonamy, D., Heinrich, Raymond, 2018+)

There is an output-polynomial time algorithm enumerating

minimal dominating sets in triangle-free graphs.

Minimal dominating sets in split graphs [KLMN14]

stable set (maximal)clique

D minimal DS in G

mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S

Enumeration: complete every irredundant X ⊆ C in S

linear delay [Kante, Limouzy, Mary, and Nourine, 2014]

D minimal DS in G

Then D ∩ S = {all vertices not dominated by D ∩ C}

D minimal DS in G

Peeling graphs

Goal: enumeration of minimal DS in triangle-free graphs

Peeling graphs

1. given minimal DS of Vi

allowing vertices of G − Vi

2. enumerate those of Vi+1

allowing vertices of G − Vi+1

extend each minimal DS of Vi

to minimal DS of Vi+1

Peeling graphs

Growing partial minimal dominating sets

Important wanted properties:

• no cycle (using a parent relation: lexicographical order)

• no leaf before level p (no exponential blowup)

Computing Vi+1 from Vi

Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1

Observation:

• possibly D dominates Vi+1

• if not then D ∪ {ui+1} does.

candidate extension: minimal X s.t. D ∪ X dominates Vi+1

Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)

Observation:

extension is always possible, hence

Observation:

Which are the candidate extensions of D?

D: minimal DS of Vi

Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1

• if ui+1 dom. by D: they only have to dominate S

→ exactly the minimal DS of Split(C ,S)

• or ui+1 not dom. by D: they should also dom. ui+1

→ irredundant {t} ∪ Q s.t.

t ∈ N(ui+1)

Q ⊆ C minimal DS of Split(C ,S)

D: minimal DS of Vi

N(ui+1) ∩ Vi+1 stable set!

t ∈ N(ui+1)

D: minimal DS of Vi

dominated by D

t ∈ N(ui+1)

D: minimal DS of Vi

dominated by D

SCN(S) \ {ui+1}

t ∈ N(ui+1)

D: minimal DS of Vi

dominated by D

SCN(S) \ {ui+1}

t ∈ N(ui+1)

D: minimal DS of Vi

dominated by D

SCN(S) \ {ui+1}

→ exactly the minimal DS of Split(C , S)

t ∈ N(ui+1)

D: minimal DS of Vi

dominated by D

SCN(S) \ {ui+1}

t ∈ N(ui+1)

D: minimal DS of Vi

dominated by D

SCN(S) \ {ui+1}

t ∈ N(ui+1)

The algorithm

For each D minimal DS of Vi :

• compute all candidate extensions; in time O(poly(n) · |D(G )|)• only keep the X ∪ D’s that are minimal and children of D.

The algorithm

• compute all candidate extensions;

in time O(poly(n) · |D(G )|)• only keep the X ∪ D’s that are minimal and children of D.

The algorithm

• compute all candidate extensions; in time O(poly(n) · |D(G )|)

• only keep the X ∪ D’s that are minimal and children of D.

The algorithm

Results and problems

The set D(G ) of minimal dominating sets of any triangle-free

graph G can be enumerated in time poly(n) · |D(G )|2 and

polynomial space.

Deciding if a vertex set X can be extended into a minimal

dominating set is NP-complete in bipartite graphs.

• complexity improvements?

• extensions to other classes? Kt-free and variants

Thank you!

polynomial space.

Thank you!

polynomial space.

Thank you!

polynomial space.

• extensions to other classes?

Kt-free and variants

Thank you!

polynomial space.

Thank you!

polynomial space.

Thank you!

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