Old and new results on packing arborescences Zolt an Szigeti - - PowerPoint PPT Presentation

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Old and new results on packing arborescences Zolt an Szigeti Equipe Optimisation Combinatoire Laboratoire G-SCOP INP Grenoble, France 11 juin 2015 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 1 / 26 Outline

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Old and new results on packing arborescences

Zolt´ an Szigeti

´ Equipe Optimisation Combinatoire Laboratoire G-SCOP INP Grenoble, France

11 juin 2015

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 1 / 26

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Outline

Old results

Digraphs

Packing spanning arborescences Packing maximal arborescences

Dypergraphs

Packing spanning hyper-arborescences Packing maximal hyper-arborescences

Matroid-based rooted-digraphs

Matroid-based packing of rooted-arborescences Maximal-rank packing of rooted-arborescences

New results

Matroid-based rooted-dypergraph

Matroid-based packing of rooted-hyper-arborescences Maximal-rank packing of rooted-hyper-arborescences

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 2 / 26

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Reachability in digraph

Definition

Let G = (V , A) be a digraph and X ⊆ V .

1 ρA(X) is the number of arcs entering X, 2 PA(X) is the set of vertices from which X can be reached in

G,

3 QA(X) is the set of vertices that can be reached from X in

G.

X V \ X ρA(X) = 2

  • G

X

PA(X) QA(X) u v x x′

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 3 / 26

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Arborescences

Definition

Let G = (V , A) be a digraph and r ∈ V .

1 A subgraph

T = (U, B) of G is an r-arborescence if

1

r ∈ U with ρB(r) = 0,

2

ρB(u) = 1 for all u ∈ U \ r and

3

ρB(X) ≥ 1 for all X ⊆ V \ r, X ∩ U = ∅.

2 An r-arborescence

T is

1

spanning if U = V ,

2

maximal if U = QA(r).

3 Packing of arborescences is a set of

pairwise arc-disjoint arborescences.

r2 r1

  • T2
  • T1
  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 4 / 26

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Packing spanning arborescences

Theorem (Edmonds 1973)

Let G = (V , A) be a digraph, r ∈ V and k a positive integer.

1 There exists a packing of k spanning r-arborescences

⇐ ⇒

2 ρA(X) ≥ k for all ∅ = X ⊆ V \ r.

r

  • T1
  • T2
  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 5 / 26

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Packing maximal arborescences

Definition

Let G = (V , A) be a digraph and (r1, . . . , rt) ∈ V t.

1 A packing of maximal arborescences is a set {

T1, . . . , Tt} of pairwise arc-disjoint maximal ri-arborescences Ti in G; that is for every v ∈ V , {ri : v ∈ V ( Ti)} = {ri ∈ PA(v)}.

2 For X ⊆ V , pA(X) = |{ri ∈ PA(X) \ X}|. r2 r1

  • T2
  • T1

r2 r1

X

pA(X) = 2

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 6 / 26

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Packing maximal arborescences

Theorem (Kamiyama, Katoh, Takizawa 2009)

Let G = (V , A) be a digraph and (r1, . . . , rt) ∈ V t.

1 There exists a packing of maximal arborescences

⇐ ⇒

2 ρA(X) ≥ pA(X) for all X ⊆ V .

Remark

It implies Edmonds’ theorem.

1 Let r1 = · · · = rk = r. 2 ρA(X) ≥ k for all ∅ = X ⊆ V \ r implies the above condition and that

each vertex is reachable from r.

3 Hence there exists a packing of maximal r-arborescences that is a

packing of spanning r-arborescences.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 7 / 26

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Dypergraphs

Definition

1 Directed hypergraph (shortly dypergraph) is

G= (V , A), where

V denotes the set of vertices and A denotes the set of hyperarcs of G.

2 Hyperarc is a pair (Z, z) such that z ∈ Z ⊆ V , where

z is the head of the hyperarc (Z, z) and the elements of Z \ z = ∅ are the tails of the hyperarc (Z, z).

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 8 / 26

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Reachability in dypergraph

Definition

Let G = (V , A) be a dypergraph and X ⊆ V .

1 Hyperarc (Z, z) enters X if z ∈ X and (Z \ z) ∩ (V \ X) = ∅, 2 ρA(X) is the number of hyperarcs entering X, 3 Path from u to x in

G is v1(= u), (Z1, v2), v2, . . . , vi, (Zi, vi+1), vi+1, . . . , vj(= x) such that vi is a tail of (Zi, vi+1).

4 PA(X) is the set of vertices from which X can be reached in

G,

5 QA(X) is the set of vertices that can be reached from X in

G.

X V \ X ρA(X) = 2

  • G

X

PA(X) u v x x′ QA(X)

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 9 / 26

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Trimming

Definition

Trimming the dypergraph G means replacing each hyperarc (K, v) of G by an arc uv where u is one of the tails of the hyperarc (K, v).

trimming

Definition

h is supermodular : h(X) + h(Y ) ≤ h(X ∩ Y ) + h(X ∪ Y ) ∀ X, Y ⊆ V .

Theorem (Frank 2011)

Let G = (V , A) be a dypergraph and h an integer-valued, intersecting supermodular function on V such that h(∅) = 0 = h(V ). If ρA(X) ≥ h(X) for all X ⊆ V , then G can be trimmed to a digraph G such that ρA(X) ≥ h(X) for all X ⊆ V .

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 10 / 26

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Hyper-arborescences

Definition

Let G = (V , A) be a dypergraph and r ∈ V .

1 A subgraph

T = (U, B) of G is an r-hyper-arborescence if it can be trimmed to an r-arborescence on U∗ ∪ r, where U∗ = {u : ρB(u) = 0} ; that is

1

r ∈ U \ U∗,

2

ρB(u) = 1 for all u ∈ U∗ and

3

ρB(X) ≥ 1 for all X ⊆ V \ r, X ∩ U∗ = ∅.

2 The r-hyper-arborescence

T is

1

spanning if U∗ = V \ r,

2

maximal if U∗ = QA(r) \ r.

U∗

2

  • T2

r2 r1

  • T1

r2 r1

  • T2
  • T1
  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 11 / 26

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Packing spanning hyper-arborescences

Theorem (Frank, T. Kir´ aly, Kriesell 2003)

Let G = (V , A) be a dypergraph, r ∈ V and k a positive integer.

1 There exists a packing of k spanning r-hyper-arborescences

⇐ ⇒

2 ρA(X) ≥ k for all ∅ = X ⊆ V \ r.

Remark

1 It is proved easily by trimming and Edmonds’ theorem. 2 It implies Edmonds’ theorem if

G is a digraph.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 12 / 26

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Packing maximal hyper-arborescences

Theorem (B´ erczi, Frank 2008)

Let G = (V , A) be a dypergraph and (r1, . . . , rt) ∈ V t.

1 There exists a packing of maximal hyper-arborescences

⇐ ⇒

2 ρA(X) ≥ pA(X) for all X ⊆ V .

Remark

1 It is proved not easily by trimming and Kamiyama, Katoh, Takizawa’s

theorem since pA(X) is not intersecting supermodular.

2 It implies 1

Frank, T. Kir´ aly, Kriesell’s theorem if r1 = · · · = rk = r and ρA(X) ≥ k for all ∅ = X ⊆ V \ r,

2

Kamiyama, Katoh, Takizawa’s theorem if G is a digraph.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 13 / 26

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Matroids

Definition

For I ⊆ 2S, M = (S, I) is a matroid if

1 I = ∅, 2 If X ⊆ Y ∈ I then X ∈ I, 3 If X, Y ∈ I with |X| < |Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I.

Examples

1 Sets of linearly independent vectors in a vector space, 2 Edge-sets of forests of a graph, 3 Un,k= {X ⊆ S : |X| ≤ k} where |S| = n, free matroid = Un,n.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 14 / 26

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Matroids

Notion

1 independent sets = I, 1

any subset of an independent set is independent,

2 base = maximal independent set, 1

all basis are of the same size,

3 rank function : r(X) = max{|Y | : Y ∈ I, Y ⊆ X}. 1

non-decreasing,

2

submodular (that is −r is supermodular),

3

X ∈ I if and only if r(X) = |X|.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 15 / 26

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Matroid-based rooted-digraphs

Definition

A matroid-based rooted-digraph is a quadruple ( G, M, S, π) :

1

  • G = (V , A) is a digraph,

2 M is a matroid on a set S = {s1, . . . , st}. 3 π is a placement of the elements of S at vertices of V such that

Sv ∈ I for every v ∈ V , where SX= π−1(X), the elements of S placed at X.

π(s1) π(s3) π(s2)

  • G

M = U3,2 S = {s1, s2, s3}

π(s1) π(s3) π(s2)

X SX = {s1, s2}

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 16 / 26

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Matroid-based packing of rooted-arborescences

Definition

A rooted-arborescence is a pair ( T, s) where

1

  • T is an r-arborescence for some vertex r,

2 s ∈ S, placed at r. π(s1) π(s3) π(s2)

T1 T2 T3

Definition

A packing {( T1, s1), . . . , ( T|S|, s|S|)} of rooted-arborescences is matroid-based if {si ∈ S : v ∈ V ( Ti)} forms a base of S for every v ∈ V .

Remark

For the free matroid M with all k roots at a vertex r,

1 matroid-based packing of rooted-arborescences

⇐ ⇒

2 packing of k spanning r-arborescences.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 17 / 26

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Matroid-based packing of rooted-arborescences

Theorem (Durand de Gevigney, Nguyen, Szigeti 2013)

Let ( G, M, S, π) be a matroid-based rooted-digraph.

1 There is a matroid-based packing of rooted-arborescences

⇐ ⇒

2 ρA(X) ≥ rM(S) − rM(SX) for all ∅ = X ⊆ V .

Remark

It implies Edmonds’ theorem if M is the free matroid with all k roots at the vertex r.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 18 / 26

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Maximal-rank packing of rooted-arborescences

Definition

A packing {( T1, s1), . . . , ( T|S|, s|S|)} of rooted-arborescences is of maximal rank if {si ∈ S : v ∈ V ( Ti)} forms a base of SPA(v) for every v ∈ V .

Theorem (Cs. Kir´ aly 2013)

Let ( G, M, S, π) be a matroid-based rooted-digraph.

1 There exists a maximal-rank packing of rooted-arborescences

⇐ ⇒

2 ρA(X) ≥ rM(SPA(X)) − rM(SX) for all X ⊆ V .

Remark

1 It implies 1

DdG-N-Sz’ theorem if ρA(X) ≥ rM(S) − rM(SX) for all ∅ = X ⊆ V ,

2

Kamiyama, Katoh, Takizawa’s theorem if M is the free matroid.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 19 / 26

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Matroid-based rooted-dypergraphs

Definition

A matroid-based rooted-dypergraph is a quadruple ( G, M, S, π) :

1

  • G = (V , A) is a dypergraph,

2 M is a matroid on a set S = {s1, . . . , st}. 3 π is a placement of the elements of S at vertices of V such that

Sv ∈ I for every v ∈ V .

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 20 / 26

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Matroid-based packing of rooted-hyper-arborescences

Definition

1 A rooted-hyper-arborescence is a triple (

T , r, s) where T is an r-hyper-arborescence and s is an element of S placed at r.

2 A packing {(

T1, r1, s1), . . . , ( T|S|, r|S|, s|S|)} of rooted-hyper- arborescences is matroid-based if {si ∈ S : v ∈ QA(

Ti)(ri)} forms a

base of S for every v ∈ V .

Theorem (L´ eonard, Szigeti 2013)

Let ( G, M, S, π) be a matroid-based rooted-dypergraph.

1 There is a matroid-based packing of rooted-hyper-arborescences ⇐

⇒

2 ρA(X) ≥ rM(S) − rM(SX) for all ∅ = X ⊆ V .

Remark

1 It is proved easily by trimming and DdG-N-Sz’ theorem.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 21 / 26

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Maximal-rank packing of rooted-hyper-arborescences

Definition

Packing {( T1, r1, s1), . . . , ( T|S|, r|S|, s|S|)} of rooted-hyper-arborescences is

  • f maximal rank if {si ∈ S : v ∈ QA(

Ti )(ri)} forms a base of SPA(v)

∀v ∈ V .

Theorem (Szigeti 2015-)

Let ( G, M, S, π) be a matroid-based rooted-dypergraph.

1 There is a maximal-rank packing of rooted-hyper-arborescences ⇐

⇒

2 ρA(X) ≥ rM(SPA(X)) − rM(SX) for all X ⊆ V .

Remark

1 It is proved not easily by trimming and Cs. Kir´

aly’s theorem since rM(SPA(X)) − rM(SX) is not intersecting supermodular.

2 It implies all the previous results.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 22 / 26

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Proof of necessity

Proof

1 Let {(

T1, r1, s1), . . . , ( T|S|, r|S|, s|S|)} be a maximal-rank packing of rooted-hyper-arborescences in ( G, M, S, π).

2 Let Bv = {si ∈ S : v ∈ QA(

Ti )(ri)} (base of SPA(v)) and X ⊆ V .

3 For each root si ∈

v∈X Bv \ SX, there exists a vertex v ∈ X such

that si ∈ Bv and then since Ti is an ri-hyper-arborescence, ri / ∈ X and v ∈ QA(

Ti)(ri) ∩ X, there exists a hyperarc of

Ti that enters X.

4 Since the hyper-arborescences are arc-disjoint,

ρA(X) ≥ |

v∈X Bv \ SX|

≥ rM(

v∈X Bv \ SX)

≥ rM(

v∈X Bv) − rM(SX)

≥ rM(

v∈X SPA(v)) − rM(SX)

= rM(SPA(X)) − rM(SX).

ri = π(si ) rℓ = π(sℓ) rj = π(sj )

X

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 23 / 26

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Conclusion

mixed hypergraph maximal matroid graph mixed hypergraph mixed mixed hypergraph hypergraph maximal maximal maximal matroid matroid matroid mixed hypergraph maximal mixed hypergraph matroid mixed maximal matroid hypergraph maximal matroid mixed hypergraph maximal matroid

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 24 / 26

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Thank you for your attention !

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 25 / 26

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Motivation : Rigidity

Body-Bar Framework Theorem (Tay 1984)

”Rigidity” of a Body-Bar Framework can be characterized by the existence of a spanning tree decomposition.

Body-Bar Framework with Bar-Boundary Theorem (Katoh, Tanigawa 2013)

”Rigidity” of a Body-Bar Framework with Bar-Boundary can be characterized by the existence of a matroid-based rooted-tree decomposition.

  • Z. Szigeti (G-SCOP, Grenoble)

On packing of arborescences 11 juin 2015 26 / 26