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Packing arborescences : a survey Zolt an Szigeti Combinatorial Optimization Group, G-SCOP Univ. Grenoble Alpes, Grenoble INP, CNRS, France 2017 April 20 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 1 / 1 Outline

  1. Applications : Secret agency network From each agent to any other agent some secret channels exist. Some messages were created and assigned to agents : each message was assigned to one agent and an agent could have been assigned to zero, one or more messages. The messages can then be propagated through the network : any agent may send any message they know to any of their contacts. Can each agent receive each message ? Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 5 / 1

  2. Applications : Secret agency network From each agent to any other agent some secret channels exist. Some messages were created and assigned to agents : each message was assigned to one agent and an agent could have been assigned to zero, one or more messages. The messages can then be propagated through the network : any agent may send any message they know to any of their contacts. Can each agent receive each message ? Today the security rules changed : the transmission of at most one message is allowed via any channel. Can now each agent receive each message ? and the messages that they could have received before ? Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 5 / 1

  3. Applications : Secret agency network The created messages were not independent : it is possible that given a subset of messages, one would get no extra information by adding another message to the set. Can now each agent receive only independent messages that contain all the information ? and all information they could have received before ? Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 6 / 1

  4. Applications : Secret agency network The created messages were not independent : it is possible that given a subset of messages, one would get no extra information by adding another message to the set. Can now each agent receive only independent messages that contain all the information ? and all information they could have received before ? For each channel, one must decide which message is sent (if any). The minimal set of channels through which the same message is sent forms an arborescence. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 6 / 1

  5. Applications : Rigidity Body-Bar Framework Theorem (Tay 1984) ”Rigidity” of a Body-Bar Framework can be characterized by the existence of a spanning tree decomposition. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 7 / 1

  6. Applications : 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 Theorem (Katoh, Tanigawa 2013) with Bar-Boundary ”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) Packing of arborescences 2017 April 20 7 / 1

  7. Packing spanning r -arborescences Theorem (Edmonds 1973) Let � G = ( V , A ) be a digraph, r ∈ V and k a positive integer. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 8 / 1

  8. Packing spanning r -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 ⇐ ⇒ � � T 1 T 2 r Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 8 / 1

  9. Packing spanning r -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 | ∂ ( X ) | ≥ k for all ∅ � = X ⊆ V \ r . � � T 1 T 2 r Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 8 / 1

  10. Packing spanning r i -arborescences Theorem (Edmonds 1973) Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 9 / 1

  11. Packing spanning r i -arborescences Theorem (Edmonds 1973) Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . 1 There exists a packing of spanning r i -arborescences ⇐ ⇒ � � T 1 T 2 r 1 r 2 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 9 / 1

  12. Packing spanning r i -arborescences Theorem (Edmonds 1973) Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . 1 There exists a packing of spanning r i -arborescences ⇐ ⇒ 2 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| for all ∅ � = X ⊆ V . � � T 1 T 2 r 1 r 2 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 9 / 1

  13. Packing reachability arborescences Definition Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 10 / 1

  14. Packing reachability arborescences Definition Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . 1 A packing of reachability arborescences is a set { � T 1 , . . . , � T t } of pairwise arc-disjoint reachability r i -arborescences � T i in � G ; r 1 � T 1 � T 2 r 2 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 10 / 1

  15. Packing reachability arborescences Definition Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . 1 A packing of reachability arborescences is a set { � T 1 , . . . , � T t } of pairwise arc-disjoint reachability r i -arborescences � T i in � G ; that is for every v ∈ V , { r i : v ∈ V ( � T i ) } = { r i ∈ P ( v ) } . r 1 � T 1 � T 2 r 2 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 10 / 1

  16. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  17. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  18. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 2 | ∂ ( X ) | ≥ |{ r i ∈ P ( X ) \ X }| for all X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  19. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 2 | ∂ ( X ) | ≥ |{ r i ∈ P ( X ) \ X }| for all X ⊆ V . Remark It implies Edmonds’ theorem. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  20. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 2 | ∂ ( X ) | ≥ |{ r i ∈ P ( X ) \ X }| for all X ⊆ V . Remark It implies Edmonds’ theorem. 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| for all ∅ � = X ⊆ V implies the above condition Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  21. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 2 | ∂ ( X ) | ≥ |{ r i ∈ P ( X ) \ X }| for all X ⊆ V . Remark It implies Edmonds’ theorem. 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| for all ∅ � = X ⊆ V implies the above condition and that each vertex is reachable from each r i . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  22. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 2 | ∂ ( X ) | ≥ |{ r i ∈ P ( X ) \ X }| for all X ⊆ V . Remark It implies Edmonds’ theorem. 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| for all ∅ � = X ⊆ V implies the above condition and that each vertex is reachable from each r i . 2 Thus there exists a packing of reachability r i -arborescences Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  23. Packing reachability arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) r 1 Let � G = ( V , A ) be a digraph, ( r 1 , . . . , r t ) ∈ V t . � T 1 1 ∃ a packing of reachability arborescences � T 2 ⇐ ⇒ r 2 2 | ∂ ( X ) | ≥ |{ r i ∈ P ( X ) \ X }| for all X ⊆ V . Remark It implies Edmonds’ theorem. 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| for all ∅ � = X ⊆ V implies the above condition and that each vertex is reachable from each r i . 2 Thus there exists a packing of reachability r i -arborescences and hence spanning r i -arborescences. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 11 / 1

  24. Matroids Definition For I ⊆ 2 E (independent sets), M = ( E , 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 . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 12 / 1

  25. Matroids Definition For I ⊆ 2 E (independent sets), M = ( E , 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 for matroids 1 Linear : Sets of linearly independent vectors in a vector space, 2 Graphic : Edge-sets of forests of a graph, 3 Uniform : U n , k = { X ⊆ E : | X | ≤ k } where | E | = n , 4 Free : U n , n , 5 Transversal : end-vertices in S of matchings of bipartite graph ( S , T ; E ) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 12 / 1

  26. Matroids Notion 1 base : maximal independent set, 2 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } , submodular Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 13 / 1

  27. Matroids Notion 1 base : maximal independent set, 2 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } , submodular Theorem (Gr¨ otschel, Lov´ asz, Schrijver 1981 ; Iwata, Fleischer, Fujishige 2001 ; Schrijver 2000) The minimum of a submodular function can be found in polynomial time. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 13 / 1

  28. Matroids Notion 1 base : maximal independent set, 2 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } , submodular Theorem (Gr¨ otschel, Lov´ asz, Schrijver 1981 ; Iwata, Fleischer, Fujishige 2001 ; Schrijver 2000) The minimum of a submodular function can be found in polynomial time. Theorem (Edmonds 1970,1979) Let M 1 = ( E , r 1 ) , M 2 = ( E , r 2 ) be matroids on E, k ∈ Z + , w : E → R . 1 M 1 and M 2 have a common independent set of size k ⇐ ⇒ r 1 ( X ) + r 2 ( E \ X ) ≥ k ∀ X ⊆ E . 2 A common base of M 1 and M 2 of minimum weight can be found in polynomial time. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 13 / 1

  29. Matroid-based rooted-digraphs Definition A matroid-based rooted-digraph is a quadruple ( � G , M , S , π ) : � G = ( V , A ) is a digraph, 1 2 M is a matroid on a set S = { s 1 , . . . , s t } . 3 π is a placement of the elements of S at vertices of V such that S v ∈ I for every v ∈ V , where S X = π − 1 ( X ) , the elements of S placed at X . π ( s 1 ) π ( s 1 ) X π ( s 2 ) π ( s 2 ) � S X = { s 1 , s 2 } G S = { s 1 , s 2 , s 3 } M = U 3 , 2 π ( s 3 ) π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 14 / 1

  30. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 � T 1 � T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  31. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  32. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � � T is a π ( s )-arborescence. T 3 2 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  33. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � � T is a π ( s )-arborescence. T 3 2 π ( s 3 ) Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is matroid-based if Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  34. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � � T is a π ( s )-arborescence. T 3 2 π ( s 3 ) Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is matroid-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S for every v ∈ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  35. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � � T is a π ( s )-arborescence. T 3 2 π ( s 3 ) Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is matroid-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S for every v ∈ V . Remark For the free matroid M , Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  36. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � � T is a π ( s )-arborescence. T 3 2 π ( s 3 ) Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is matroid-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S for every v ∈ V . Remark For the free matroid M , 1 matroid-based packing of rooted-arborescences ⇐ ⇒ Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  37. Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where � T 2 1 s ∈ S, � T 1 � � T is a π ( s )-arborescence. T 3 2 π ( s 3 ) Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is matroid-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S for every v ∈ V . Remark For the free matroid M , 1 matroid-based packing of rooted-arborescences ⇐ ⇒ 2 packing of spanning π ( s i )-arborescences. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 15 / 1

  38. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  39. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) π ( s 1 ) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. π ( s 2 ) � 1 ∃ matroid-based packing of rooted-arborescences T 2 � T 1 ⇐ ⇒ � T 3 π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  40. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) π ( s 1 ) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. π ( s 2 ) � 1 ∃ matroid-based packing of rooted-arborescences T 2 � T 1 ⇐ ⇒ � T 3 π ( s 3 ) 2 | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  41. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) π ( s 1 ) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. π ( s 2 ) � 1 ∃ matroid-based packing of rooted-arborescences T 2 � T 1 ⇐ ⇒ � T 3 π ( s 3 ) 2 | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V . Remark It implies Edmonds’ theorem if M is the free matroid and π ( s i ) = r i . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  42. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) π ( s 1 ) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. π ( s 2 ) � 1 ∃ matroid-based packing of rooted-arborescences T 2 � T 1 ⇐ ⇒ � T 3 π ( s 3 ) 2 | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V . Remark It implies Edmonds’ theorem if M is the free matroid and π ( s i ) = r i . 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| = r M (S) − r M (S X ) for all ∅ � = X ⊆ V implies the above condition. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  43. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) π ( s 1 ) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. π ( s 2 ) � 1 ∃ matroid-based packing of rooted-arborescences T 2 � T 1 ⇐ ⇒ � T 3 π ( s 3 ) 2 | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V . Remark It implies Edmonds’ theorem if M is the free matroid and π ( s i ) = r i . 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| = r M (S) − r M (S X ) for all ∅ � = X ⊆ V implies the above condition. 2 Thus there exists a matroid-based packing of rooted-arborescences Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  44. Matroid-based packing of rooted-arborescences Theorem ( Durand de Gevigney, Nguyen, Szigeti 2013) π ( s 1 ) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. π ( s 2 ) � 1 ∃ matroid-based packing of rooted-arborescences T 2 � T 1 ⇐ ⇒ � T 3 π ( s 3 ) 2 | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V . Remark It implies Edmonds’ theorem if M is the free matroid and π ( s i ) = r i . 1 | ∂ ( X ) | ≥ |{ r i ∈ V \ X }| = r M (S) − r M (S X ) for all ∅ � = X ⊆ V implies the above condition. 2 Thus there exists a matroid-based packing of rooted-arborescences and, by Remark, a packing of spanning r i -arborescences. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 16 / 1

  45. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  46. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S P ( v ) for every v ∈ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  47. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S P ( v ) for every v ∈ V . Theorem (Cs. Kir´ aly 2016) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  48. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S P ( v ) for every v ∈ V . Theorem (Cs. Kir´ aly 2016) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. 1 There exists a reachability-based packing of rooted-arborescences ⇐ ⇒ Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  49. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S P ( v ) for every v ∈ V . Theorem (Cs. Kir´ aly 2016) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. 1 There exists a reachability-based packing of rooted-arborescences ⇐ ⇒ 2 | ∂ ( X ) | ≥ r M (S P ( X ) ) − r M (S X ) for all X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  50. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S P ( v ) for every v ∈ V . Theorem (Cs. Kir´ aly 2016) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. 1 There exists a reachability-based packing of rooted-arborescences ⇐ ⇒ 2 | ∂ ( X ) | ≥ r M (S P ( X ) ) − r M (S X ) for all X ⊆ V . Remark 1 It implies DdG-N-Sz’ theorem if | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V , 1 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  51. Reachability-based packing of rooted-arborescences Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is reachabi- lity-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S P ( v ) for every v ∈ V . Theorem (Cs. Kir´ aly 2016) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. 1 There exists a reachability-based packing of rooted-arborescences ⇐ ⇒ 2 | ∂ ( X ) | ≥ r M (S P ( X ) ) − r M (S X ) for all X ⊆ V . Remark 1 It implies DdG-N-Sz’ theorem if | ∂ ( X ) | ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V , 1 Kamiyama, Katoh, Takizawa’s theorem if M is the free matroid. 2 Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 17 / 1

  52. Packing spanning arborescences with matroid intersection Remark Let � G = ( V + s , A ) and G be the underlying undirected graph of � G . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 18 / 1

  53. Packing spanning arborescences with matroid intersection Remark Let � G = ( V + s , A ) and G be the underlying undirected graph of � G . F ⊆ A is a packing of k spanning s -arborescences of � � G ⇐ ⇒ 1 � � T 1 T 2 r Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 18 / 1

  54. Packing spanning arborescences with matroid intersection Remark Let � G = ( V + s , A ) and G be the underlying undirected graph of � G . F ⊆ A is a packing of k spanning s -arborescences of � � G ⇐ ⇒ 1 2 F is a packing of k spanning trees of G , | ∂ � F ( v ) | = k ∀ v ∈ V ⇐ ⇒ � � T 1 T 2 r Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 18 / 1

  55. Packing spanning arborescences with matroid intersection Remark Let � G = ( V + s , A ) and G be the underlying undirected graph of � G . F ⊆ A is a packing of k spanning s -arborescences of � � G ⇐ ⇒ 1 2 F is a packing of k spanning trees of G , | ∂ � F ( v ) | = k ∀ v ∈ V ⇐ ⇒ 3 F is a common base of M 1 = k -sum of the graphic matroid of G and M 2 = ⊕ v ∈ V U | ∂ ( v ) | , k . � � T 1 T 2 r Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 18 / 1

  56. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  57. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Theorem Given a digraph � G = ( V + s , A ) , k ∈ Z + and a matroid M = ( A , r ) which is the direct sum of the matroids M v = ( ∂ ( v ) , r v ) ∀ v ∈ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  58. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Theorem Given a digraph � G = ( V + s , A ) , k ∈ Z + and a matroid M = ( A , r ) which is the direct sum of the matroids M v = ( ∂ ( v ) , r v ) ∀ v ∈ V . � G has an M -restricted packing of k spanning s-arborescences ⇐ ⇒ Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  59. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Theorem Given a digraph � G = ( V + s , A ) , k ∈ Z + and a matroid M = ( A , r ) which is the direct sum of the matroids M v = ( ∂ ( v ) , r v ) ∀ v ∈ V . � G has an M -restricted packing of k spanning s-arborescences ⇐ ⇒ r ( ∂ ( X )) ≥ k ∀ ∅ � = X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  60. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Theorem Given a digraph � G = ( V + s , A ) , k ∈ Z + and a matroid M = ( A , r ) which is the direct sum of the matroids M v = ( ∂ ( v ) , r v ) ∀ v ∈ V . � G has an M -restricted packing of k spanning s-arborescences ⇐ ⇒ r ( ∂ ( X )) ≥ k ∀ ∅ � = X ⊆ V . Remarks 1 For free matroid, we are back to packing of k spanning s -arborescen. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  61. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Theorem Given a digraph � G = ( V + s , A ) , k ∈ Z + and a matroid M = ( A , r ) which is the direct sum of the matroids M v = ( ∂ ( v ) , r v ) ∀ v ∈ V . � G has an M -restricted packing of k spanning s-arborescences ⇐ ⇒ r ( ∂ ( X )) ≥ k ∀ ∅ � = X ⊆ V . Remarks 1 For free matroid, we are back to packing of k spanning s -arborescen. 2 This problem can also be formulated as matroid intersection. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  62. Matroid-restricted packing of spanning s -arborescences Definition Given a digraph � G = ( V + s , A ) and a matroid M = ( A , I ) , a packing of spanning s -arborescences T 1 , . . . , T k is matroid-restricted if ∪ k 1 A ( T i ) ∈ I . Theorem Given a digraph � G = ( V + s , A ) , k ∈ Z + and a matroid M = ( A , r ) which is the direct sum of the matroids M v = ( ∂ ( v ) , r v ) ∀ v ∈ V . � G has an M -restricted packing of k spanning s-arborescences ⇐ ⇒ r ( ∂ ( X )) ≥ k ∀ ∅ � = X ⊆ V . Remarks 1 For free matroid, we are back to packing of k spanning s -arborescen. 2 This problem can also be formulated as matroid intersection. 3 For general matroid M , the problem is NP-complete, even for k = 1 . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 19 / 1

  63. New model : Matroid-rooted digraphs Transformation π ( s 1 ) π ( s 2 ) s 1 s 2 s s 3 π ( s 3 ) Matroid on the vertices Matroid on the arcs leaving s Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 20 / 1

  64. New model : Matroid-rooted digraphs Transformation π ( s 1 ) π ( s 2 ) s 1 s 2 s s 3 π ( s 3 ) Matroid-based packing Matroid-based packing Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 20 / 1

  65. New model : Matroid-rooted digraphs Transformation π ( s 1 ) π ( s 2 ) s 1 s 2 s s 3 π ( s 3 ) Matroid-based packing Matroid-based packing Theorem (Durand de Gevigney, Nguyen, Szigeti 2013) Let ( � G = ( V + s , A ) , M ) be a matroid-rooted digraph. 1 There is a M -based packing of s-arborescences ⇐ ⇒ Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 20 / 1

  66. New model : Matroid-rooted digraphs Transformation π ( s 1 ) π ( s 2 ) s 1 s 2 s s 3 π ( s 3 ) Matroid-based packing Matroid-based packing Theorem (Durand de Gevigney, Nguyen, Szigeti 2013) Let ( � G = ( V + s , A ) , M ) be a matroid-rooted digraph. 1 There is a M -based packing of s-arborescences ⇐ ⇒ 2 r M ( ∂ ( s , X )) + | ∂ ( V \ X , X ) | ≥ r M ( ∂ ( s , V )) for all ∅ � = X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 20 / 1

  67. Matroid-based packing of spanning s -arborescences s 1 s 2 s s 3 Matroid-based packing of s -arborescences Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 21 / 1

  68. Matroid-based packing of spanning s -arborescences s 1 s 2 s s 3 Matroid-based packing of spanning s -arborescences Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 21 / 1

  69. Matroid-based packing of spanning s -arborescences Conjecture (B´ erczi, Frank 2015) Let ( � G = ( V + s , A ) , M ) be a matroid-rooted digraph. � G has an M -based packing of spanning s -arborescences ⇐ ⇒ r M ( ∂ ( s , X )) + | ∂ ( V \ X , X ) | ≥ r M ( ∂ ( s , V )) ∀ X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 21 / 1

  70. Matroid-based packing of spanning s -arborescences Conjecture (B´ erczi, Frank 2015) Let ( � G = ( V + s , A ) , M ) be a matroid-rooted digraph. � G has an M -based packing of spanning s -arborescences ⇐ ⇒ r M ( ∂ ( s , X )) + | ∂ ( V \ X , X ) | ≥ r M ( ∂ ( s , V )) ∀ X ⊆ V . Theorem (Fortier, Cs. Kir´ aly, Szigeti, Tanigawa 2016-) Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 21 / 1

  71. Matroid-based packing of spanning s -arborescences Conjecture (B´ erczi, Frank 2015) Let ( � G = ( V + s , A ) , M ) be a matroid-rooted digraph. � G has an M -based packing of spanning s -arborescences ⇐ ⇒ r M ( ∂ ( s , X )) + | ∂ ( V \ X , X ) | ≥ r M ( ∂ ( s , V )) ∀ X ⊆ V . Theorem (Fortier, Cs. Kir´ aly, Szigeti, Tanigawa 2016-) 1 Conjecture is not true in general. Z. Szigeti (G-SCOP, Grenoble) Packing of arborescences 2017 April 20 21 / 1

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