Matching Forests in Mixed Graphs and b -branchings in Digraphs - PowerPoint PPT Presentation

Notes on Equitable Partitions into Matching Forests in Mixed Graphs and b -branchings in Digraphs Kenjiro Takazawa Hosei Univ, JPN 6th I nternational S ymposium on C ombinatorial O ptimization @ Zoom May 4-6, 2020

Equitable Coloring 2 [Hajnal, Szemerédi ’69] (Conjecture by P . Erdős ’64 ) Graph ( V , E ) with maximum degree Δ ➔ V can be partitioned into Δ+1 stable sets S 1 , S 2 ,..., S Δ +1 such that (3, 3, 1) || S i | - | S j || ≤ 1 ∀ i , j ∊ {1,2,...,Δ+1} ➔ Not equitable |𝑊| |𝑊| ➢ Namely, | S i | = ∆+1 or ∆+1 [Folkman, Fulkerson ’67] etc. Bipartite graph ( V , E ) with maximum degree Δ (3, 2, 2) ➔ For any k ≥Δ , E can be partitioned into ➔ Equitable k matchings M 1 , M 2 ,..., M k such that || M i | - | M j || ≤ 1 ∀ i , j ∊ {1,2,..., k } (3, 3, 3, 2) (4, 3, 2, 2) ➔ Equitable ➔ Not equitable

Equitable Partition into Branchings 3 ◼ Digraph ( V , A ) Definition B ⊆ A is a branching if (i) indeg( v ) ≤ 1 ∀ v ∈ V (ii) No (undirected) cycle (6, 5, 1) ➔ Not equitable Theorem [Schrijver 03]? If A can be partitioned into k branchings ➔ A can be partitioned into k branchings B 1 , B 2 ,..., B k such that || B i | - | B j || ≤ 1 ∀ i , j ∊ [ k ] (4, 5, 4) ➔ Equitable

Overview of Our Results 4 Theorem For a mixed graph ( V , E ∪ A ), if E ∪ A can be partitioned into k matching forests ➔ E ∪ A can be partitioned into k matching forests F 1 , F 2 ,..., F k such that ||∂ F i | - |∂ F j || ≤ 2 ∀ i , j ∊ {1,2,..., k } Theorem Vector b ∊ Z V If A can be partitioned into k b -branchings 3 2 ➔ A can be partitioned into 2 １ k b -branchings B 1 , B 2 ,..., B k such that ➢ || B i | - | B j || ≤ 1 ∀ i , j ∊[ k ] ➢ |indeg i ( v ) - indeg j ( v )| ≤ 1 ∀ v ∊ V , ∀ i , j ∊[ k ] 2 2

Contents 5 ⚫ Introduction ◼ Equitable partition in graphs: ➢ Matching, Branching ⚫ Matching Forest ◼ Common generalization of matching and branching ◼ [Király, Yokoi ’18] Tri-criteria equitability 3 2 ➢ Cannot be optimized simultaneously 2 １ ◼ [This talk] Single-criterion equitability ➢ Can always be optimized ⚫ b -branching 2 2 ◼ Generalization of branchings allowing indegree ≥ 2 ◼ [This talk] ( n +1)-criteria equitability ➢ Can always be optimized simultaneously ⚫ Proof Sketch (for Matching Forests)

Matching Forest [Giles ’82] 6 ◼ Mixed graph G = ( V , E ∪ A ) u v u v Definition Undirected edge { u , v }∊ E covers both u and v Directed edge ( u , v )∊ A covers only v Definition F ⊆ E ∪ A is a matching forest if ➢ its underlying edge set is a forest ➢ every vertex is covered by ≤ 1 edge ➢ Namely, B = F ∩ A : Branching • M = F ∩ E : Matching s.t. ∂ M ⊆ V ∖∂ - B • ➢ Set of covered vertices ∂ F := ∂ - B ∪∂ M

Previous Work 7 ◼ Finding a max. weight matching forest (Not a topic of today) ➢ [Giles ’82] Polyhedral description, Primal -dual algorithm in O( n 2 m ) time ➢ [T . ’14] Improved to O( n 3 ) ➢ [Schrijver ’00] TDI -ness of the description ➢ [Schrijver ’03] Reduction to weighted linear matroid parity ◼ Partition into Matching Forests ➢ [Keijsper ’03] Partition into Δ +1 matching forests ➢ [Király, Yokoi ’18] Equitable partition into matching forests Theorem [Király, Yokoi ’18] If E ∪ A can be partitioned into k matching forests ➔ E ∪ A can be partitioned into k matching forests F 1 , F 2 ,..., F k such that || F i | - | F j || ≤ 1, || B i | - | B j || ≤ 2, || M i | - | M j || ≤ 2 ➔ E ∪ A can be partitioned into k matching forests F 1 , F 2 ,..., F k such that || F i | - | F j || ≤ 2, || B i | - | M j || ≤ 2, || M i | - | M j || ≤ 1 ➢ Tricriteria equitability ➢ These values are tight i.e., the three criteria cannot be optimized simultaneously

Delta-Matroid Structure 8 S 1 S 2 ◼ Subset family F ⊆ 2 V Definition s A set system ( V , F ) is a delta-matroid if ∀ S 1 , S 2 ∈ F , ∀ s ∈ S 1 Δ S 2 , t ➢ S 1 Δ { s } ∈ F or e e ∃ t ∈ ( S 1 Δ S 2 ) - { s }, S 1 Δ { s , t } ∈ F ➢ a d a d b c b c ◼ Undirected graph G =( V , E ) ➢ F M := { ∂ M ⊆ V | M is a matching in G } F M = { Ø, {a,b}, {a,d}, {b,c}, Theorem [ Chandrasekaran, Kabadi ’88; Bouchet ’89] {b,d}, {c,d}, {d,e}, ( V , F M ) is a delta-matroid {a,b,c,d}, {a,b,d,e}, {b,c,d,e} } ◼ Directed graph G =( V , A ) F B = { Ø, {a}, {b}, {c}, {d}, ➢ F B := { ∂ - B ⊆ V | B is a branching in G } {a,b}, {a,c},...,{c,d}, Theorem [ T . ’14] (Folklore?) {a,b,c}, {a,b,d},{a,c,d}, ( V , F B ) is a matroid (thus a delta-matroid ) {b,c,d}, {a,b,c,d} }

Equitability from the Structure 9 ◼ Mixed graph G =( V , E ∪ A ) a d ➢ F MF := { ∂ F ⊆ V | F is a matching forest in G } Theorem [ T . ’14] b c ( V , F MF ) is a delta-matroid F MF = { Ø, {a},{b},{c}, {a,b},{a,c},{a,d},{b,c}, Our Idea {c,d},{a,b,d},{a,c,d}, Define the equitability of matching forests {b,c,d}, {a,b,c,d} } by the size of ∂ F Theorem a b c d If E ∪ A can be partitioned into k matching forests ➔ E ∪ A can be partitioned into F 1 F 2 F 1 k matching forests F 1 , F 2 ,..., F k such that ||∂ F i | - |∂ F j || ≤ 2 ∀ i , j ∊ {1,2,..., k } ∂ F 1 = { a,b,c,d } ∂ F 2 = { b,c } ➢ Value 2 is optimal ➢ Optimality always attained

Contents 10 ⚫ Introduction ◼ Equitable partition in graphs: ➢ Matching, Branching ⚫ Matching Forest ◼ Common generalization of matching and branching ◼ [Király, Yokoi ’18] Tri-criteria equitability 3 2 ➢ Cannot be optimized simultaneously 2 １ ◼ [This talk] Single-criterion equitability ➢ Can always be optimized ⚫ b -branching 2 2 ◼ Generalization of branchings allowing indegree ≥ 2 ◼ [This talk] ( n +1)-criteria equitability ➢ Can always be optimized simultaneously ⚫ Proof Sketch (for Matching Forests)

b -branching [Kakimura, Kamiyama, T 11 . ’20+] 3 2 ◼ Digraph ( V , A ) １ 2 ◼ Positive integer vector b ∈ Z V on V 2 Definition X 2 B ⊆ A is a b -branching if Not a b-branching: | B [ X ]|= 6 , b ( X )-1= ５ (i) indeg B ( u ) ≤ b ( u ) ( u ∈ V ) (ii) | B [ X ]| ≤ b ( X ) – 1 (∅≠ X ⊆ V ) ➢ Branching: b ( u ) ≡ 1 2 3 ➢ Special case of matroid intersection １ 2 (i): Direct sum of uniform matroids • (ii): Sparsity matroid (Count matroid) • 2 2 Sparsity matroid [cf. Frank ‘ s book 11] X Graph G =( V , E ), Vector b ∈ Z V , Integer k ≥0 A b-branching: ➔ { B ⊆ E : | B [ X ]| ≤ b ( X ) - k } is an | B [ X ]|= 7 , b ( X )-1= 8 independent set family of a matroid

Previous Work 12 ◼ Finding a max. weight b -branching (Not a topic of today) ➢ TDI description (  Matroid intersection polytope) ➢ Multi-phase greedy algorithm extending Arborescence Algorithm [Kakimura, Kamiyama, T . ’20+] ◼ Partition into b -branchings Theorem [Kakimura, Kamiyama, T. ’20+] A can be partitioned into k b -branchings iff ➢ indeg( u ) ≤ k ⋅ b ( u ) ∀ u ∊ V ➢ | A [ X ]| ≤ k ( b ( X ) - 1) ∅≠ ∀ X ⊆ V Definition (Recap) ➢ Obvious necessary condition works B ⊆ A is a b -branching if (i) indeg B ( u ) ≤ b ( u ) ( u ∈ V ) (ii) | B [ X ]| ≤ b ( X ) – 1 (∅≠ X ⊆ V )

Equitable Partition into b -branchings 13 Theorem B 1 If A can be partitioned into k b -branchings ➔ A can be partitioned into k B 3 b -branchings B 1 , B 2 ,..., B k such that ➢ || B i | - | B j || ≤ 1 ∀ i , j ∊[ k ] u ➢ |indeg i ( u ) - indeg j ( u )| ≤ 1 ∀ u ∊ V , ∀ i , j ∊[ k ] Namely, for each i ∊[ k ], B 2 |𝐵| |𝐵| ➢ | B i | = or 𝑙 𝑙 indeg 𝐵 (𝑣) indeg 𝐵 (𝑣) ➢ |indeg i ( u )| = or ∀ u ∊ V 𝑙 𝑙 ➔ These ( n +1)-criteria can be simultaneously optimized

Integer Decomposition Property (IDP) 14 2 P Definition x A polytope P has the integer decomposition property if x 1 ∀ k ∊ Z ++ , ∀ x ∊ kP ∩ Z A , x = x 1 +…+ x k ( x 1 ,..., x k ∊ P ∩ Z A ) ➢ True for x 2 Polymatroids [Giles ’75; Baum, Trotter ’81] • Intersection of two strongly base orderable matroids • [Davies, McDiarmid ’76; McDiarmid ’76] Branching polytope (below) • ➢ NOT True for every matroid intersection Branching polytope Theorem [Baum, Trotter ’81] ➢ x ( δ u ) ≤ 1 ( u ∈ V ) The branching polytope has IDP ➢ x ( A [ X ]) ≤ |X | – 1 (∅≠ X ⊆ V ) ➢ 0 ≤ x ( a ) ≤ 1 ( a ∈ A ) Theorem [McDiarmid ’83] The convex hull of the incidence vectors of the branchings of size ℓ has IDP

IDP for b -branchings 15 [Baum, Trotter ’81] [Kakimura, Kamiyama, T . ’20+] Branching The b -branching polytope has IDP [McDiarmid ’83] [Our Result] Branchings of size ℓ The convex hull of the incidence vectors of the b -branchings of size ℓ has IDP ➢ Further, the indegree can be fixed to be b ’( v ) (≤ b ( v )) [Our Result] The convex hull of the incidence vectors of the b -branchings of b -branching polytope ➢ size ℓ ; and ➢ x ( δ u ) ≤ b ( u ) ( u ∈ V ) ➢ indeg( v )= b ’( v ) ∀ v ∊ V ➢ x ( A [ X ]) ≤ b ( X ) – 1 (∅≠ X ⊆ V ) has IDP ➢ 0 ≤ x ( a ) ≤ 1 ( a ∈ A )