The b -bibranching Problem: TDI system, Packing, and Discrete - PowerPoint PPT Presentation

The b -bibranching Problem: TDI system, Packing, and Discrete Convexity Kenjiro Takazawa Hosei University, JPN S T ISMP 2018 @ Bordeaux July 1-6, 2018

Outline 2 (3) b -bibranching Our result ➢ TDI system ➢ Packing ➢ M-convex submodular flow formulation (2-2) Bibranching ➢ TDI system [Schrijver 82] ➢ Packing [Schrijver 82] ➢ M-convex submodular flow formulation [ T . 12] (2-1) b -branching ➢ TDI system [Kakimura, Kamiyama, T . 18] ➢ Packing [Kakimura, Kamiyama, T . 18] Counterpart of b -matching (1) Branching ➢ TDI system [Edmonds 70] ➢ Packing [Edmonds 73]

Branching 3 ◆ Digraph ( V , A ) Definition ⚫ B ⊆ A is a branching ⇔ (i) indeg B ( u ) ≤ 1 ( u ∈ V ) (ii) No undirected cycle Fact Branching: A special case of matroid intersection δ in ( u ) u ➢ (i): Partition matroid ✓ A = δ in ( u 1 )∪δ in ( u 2 )∪∙∙∙∪δ in ( u n ) ✓ | B ∩δ in ( u i )| ≤ 1 ( i =1,2,...., n ) ➢ (ii): Graphic matroid U ✓ | B [ U ]| ≤ | U | - 1 (∅≠ U ⊆ V )

Important Results on Branchings 4 Result (A) TDI linear system ➢ Follows from TDIness of matroid intersection [Edmonds 70] Result (B) Multi-phase greedy algorithm for max weight branching [Chu-Liu 65, Edmonds 67, Bock 71, Fulkerson 74] ➢ NOT true for bipartite matching (!) Result (C) Packing theorem [Edmonds 73] ➢ Also holds for Bipartite matching ( Kőnig’s theorem) • Strongly base orderable matroid intersection [Davies, McDiarmid 76] • Matroids without ( k + 1)-spanned elements [Kotlar, Ziv 05] • [ T . , Yokoi 18]

(A) TDI System for Branchings 5 Linear Program (P) in variable x ∈ R A max. ∑ w ( a ) x ( a ) δ( u ) u s.t. x ( δ in ( u )) ≤ 1 ( u ∈ V ) x ( A [ U ]) ≤ | U | - 1 (∅≠ U ⊆ V ) x ( a ) ≥ 0 ( a ∈ A ) TDI Theorem [Edmonds 70] This linear system is TDI , i.e., ➢ (P) has an integer optimal solution U ➢ If w is integer, the dual program also has an integer optimal solution ➢ Holds for any matroid intersection [Edmonds 70]

(B) Multi-phase Greedy Algorithm 6 [Chu-Liu 65, Edmonds 67, Bock 71, Fulkerson 74] 4 9 4 9 2 1 7 7 7 7 10 10 6 6 9 6 9 3 6 6 6 10 10 10 2 2 = 7 + 4 – 9 1 3 = 9 + 4 – 10 1 = 7 + 4 – 10 3 4 9 2 1 7 7 10 6 9 6 3 6 6 10 10

(C) Packing Disjoint Branchings 7 Theorem [Edmonds 67, Bock 71, Fulkerson 74] Digraph D has one r -arborescence ⇔ |δ in ( U )| ≥ 1 (∅≠ U ⊆ V∖ { r }) U r Disjoint Arborescences Theorem [Edmonds 73] Digraph D has k arc-disjoint r -arborescence ⇔ |δ in ( U )| ≥ k (∅≠ U ⊆ V ∖{ r })

Outline 8 Our result (3) b -bibranching ➢ TDI system ➢ Packing ➢ M-convex submodular flow formulation (2-2) Bibranching ➢ TDI system [Schrijver 82] ➢ Packing [Schrijver 82] ➢ M-convex submodular flow formulation [ T . 12] (2-1) b -branching ➢ TDI system [Kakimura, Kamiyama, T . 18] ➢ Packing [Kakimura, Kamiyama, T . 18] (1) Branching ➢ TDI system [Edmonds 70] ➢ Packing [Edmonds 73]

b -branching 9 2 3 ◆ Digraph ( V , A ) １ 2 ◆ Positive integer vector b ∈ Z V on V Def. [Kakimura, Kamiyama, T . 18] 2 2 U ⚫ B ⊆ A is a b -branching ⇔ ✓ | B [ U ]|= 7 , b ( U )-1= 8 (i) indeg B ( u ) ≤ b ( u ) ( u ∈ V ) 3 2 (ii) | B [ U ]| ≤ b ( U ) – 1 (∅≠ U ⊆ V ) １ 2 Branching: b ( u )=1 • 2 U 2 ➢ (i): Direct sum of uniform matroids × | B ( U )|= 6 , b ( U )-1= ５ ➢ (ii): Sparsity matroid Sparsity matroid [cf. Frank 11] Graph G =( V , E ), Vector b ∈ Z V , Integer k ≥0 ➢ { B ⊆ E : | B [ U ]| ≤ b ( U ) - k } is an independent set family of a matroid k disjoint branchings: indeg( u ) ≤ k ( u ∈ V ) ✓ | B [ U ]| ≤ k | U | – k (∅≠ U ⊆ V )

Results on b -branchings 10 [Kakimura, Kamiyama, T . 18] Result (A) TDI linear system [Schrijver 82] ➢ Holds for any matroid intersection [Edmonds 70] Result (B) Multi-phase greedy algorithm for max weight b -branching ➢ More tractable than bipartite matching Result (C) Packing Theorem

(A) TDI System for b -branchings 11 Linear Program (P) in variable x ∈ R A 2 3 max. ∑ w ( a ) x ( a ) １ 2 s.t. x ( δ in ( u )) ≤ b ( u ) ( u ∈ V ) x ( A [ U ]) ≤ b ( U ) - 1 (∅≠ U ⊆ V ) U 2 2 x ( a ) ≥ 0 ( a ∈ A ) ✓ | B [ U ]|= 7 , b ( U )-1= 8 TDI Theorem 2 3 This linear system is TDI , i.e., １ 2 ➢ (P) has an integer optimal solution 2 ➢ If w is integer, the dual program also U 2 has an integer optimal solution × | B ( U )|= 6 , b ( U )-1= ５ ➢ Holds for any matroid intersection [Edmonds 70]

(B) Multi-phase Greedy Algorithm 12 [Kakimura, Kamiyama, T . 18] b ( u )=2 ( ∀ u ∈ V ) U 11 11 10 10 8 8 5 5 11 7 11 7 5 12 5 12 4 4 9 2 9 2 |B [ U ] |=b ( U ) 8 8 11 2 = 8 + 5 – 11 2 v U 3 10 4 = 4 + 5 – 5 8 5 3 = 5 + 5 – 7 11 7 5 12 4 -1 -1 = 2 + 5 – 8 4 b ( v U ) := 1 9 2 8

(C) Packing Disjoint b -branchings 13 [Kakimura, Kamiyama, T . 18] Theorem 2 3 b -branching with indeg( u )= b ( u ) ( u ∈ V ∖{ r }) ⇔ １ １ ➢ | δ in ( u )| ≥ b ( u ) ( u ∈ V ∖{ r }) ➢ |δ in ( U )| ≥ 1 (∅ ≠ U ⊆ V∖ { r }) r 2 2 U Theorem k disjoint b -branchings with indeg( u )= b ( u ) ( u ∈ V ∖{ r }) ⇔ ➢ | δ in ( u )| ≥ k ∙ b ( u ) ( u ∈ V ∖{ r }) ➢ |δ in ( U )| ≥ k (∅ ≠ U ⊆ V∖ { r }) U Th. [Edmonds 67, Bock 71, Fulkerson 74] 2 2 r -arborescence 1 ⇔ |δ in ( U )| ≥ 1 (∅≠ U ⊆ V∖ { r }) r Th. [Edmonds 73] 2 2 k disjoint r -arborescence ⇔ |δ in ( U )| ≥ k (∅≠ U ⊆ V ∖{ r })

Outline 14 Our result (3) b -bibranching ➢ TDI system ➢ Packing ➢ M-convex submodular flow formulation (2-2) Bibranching ➢ TDI system [Schrijver 82] ➢ Packing [Schrijver 82] ➢ M-convex submodular flow formulation [ T . 12] (2-1) b -branching ➢ TDI system [Kakimura, Kamiyama, T . 18] ➢ Packing [Kakimura, Kamiyama, T . 18] (1) Branching ➢ TDI system [Edmonds 70] ➢ Packing [Edmonds 73]

Bibranching 15 ◆ Digraph ( V , A ) S T ◆ Partition { S , T } of V u Definition [Schrijver 82] ⚫ B ⊆ A is a bibranching ⇔ In ( V , B ), ➢ ∀ v ∈ T is reachable form S v ➢ ∀ u ∈ S reaches T Bibranching B ⊆A ⚫ Can assume A [ T , S ] = ∅ ◆ Motivations: ➢ Generalization of Arborescence Bipartite Edge Cover ➢ Packing theorem is used in a proof of Woodall’s conjecture in source-sink connected digraphs [Schrijver 82]

Special Cases of Bibranchings 16 ◆ Arborescence ◆ Bipartite edge cover ⚫ A [ S ] = A [ T ] = ∅ ⚫ S = { r } T S T { r } Bibranching = Edge cover B ⊆ A is a bibranching Minimal bibranching ⟺ In ( V , B ), = r -aborescence ➢ ∀ v ∈ T is reachable from S ➢ ∀ u ∈ S reaches T

Alternative Definition of Bibranchings 17 Alternative Definition of Bibranchings S T B ⊆ A is a bibranching ⇔ For F = B [ S , T ], ➢ B [ T ] is a branching with R ( B [ T ])= ∂ - F ➢ B [ S ] is a cobranching with R *( B [ S ])= ∂ + F ➢ B ⊆ A is a cobranching ⇔ Reversal of a branching ➢ Root R ( B ) of a branching B = { v ∈ V : | B ∩δ in v| =0} ➢ Root R *( B ) of a cobranching B = { v ∈ V : | B ∩δ +out v| =0} B ⊆ A is a bibranching ⟺ In ( V , B ), ∀ t ∈ T is reachable from S Minimal ones ➢ ∀ s ∈ S reaches T ➢

Results on Bibranchings 18 Results on Bibranchings (A) TDI linear system [Schrijver 82] ➢ NOT follows from TDIness of matroid intersection (B) Algo. for min-weight bibranching [Keijsper, Pendavingh 98] ➢ NOT greedy , but as fast as bipartite edge cover algorithm (C) Packing Theorem [Schrijver 82] ➢ Used in a proof for Woodall’s conjecture for source-sink connected digraphs [Schrijver 82] (D) M ♮ -convex submodular flow formulation [ T . 12] (E) (D) is obtained from (A) by Benders decomposition [Murota, T . 17]

(A) TDI System for Bibranching 19 S T Definition [Schrijver 82] ⚫ C ⊆ A is a bicut ⇔ C = δ in ( U ), where ∅≠ U ⊆ T or T ⊆ U ⊊ V C 1 C 2 Linear Program (P) in variable x ∈ R A U 2 U 1 min. ∑ w ( a ) x ( a ) s.t. x ( C ) ≥ 1 ( C : bicut) x ( a ) ≥ 0 ( a ∈ A ) TDI theorem [Schrijver 82] This linear system is TDI B ⊆ A is a bibranching ⟺ In ( V , B ), ∀ v ∈ T is reachable from S ➢ ∀ u ∈ S reaches T ➢

(C) Packing Disjoint Bibranchings 20 S T Disjoint Bibranchings Theorem [Schrijver 82] D has k arc-disjoint bibranchings C 1 C 2 ⇔ | C | ≥ k ( C : bicut) U 2 U 1 ⚫ Can be proved by supermodular colouring [Schrijver 85][Tardos 85] ⚫ Applied to a proof for Woodall’s conjecture for source-sink connected digraphs [Schrijver 82] Woodall’s conjecture [1978] Every directed cut has ≥ k arcs ⇒ ∃ k disjoint directed-cut covers (dijoins)

Outline 21 Our result (3) b -bibranching ➢ TDI system ➢ Packing ➢ M-convex submodular flow formulation (2-2) Bibranching ➢ TDI system [Schrijver 82] ➢ Packing [Schrijver 82] ➢ M-convex submodular flow formulation [ T . 12] (2-1) b -branching ➢ TDI system [Kakimura, Kamiyama, T . 18] ➢ Packing [Kakimura, Kamiyama, T . 18] (1) Branching ➢ TDI system [Edmonds 70] ➢ Packing [Edmonds 73]