Excluded t -factors in Bipartite Graphs: A Unified Framework for ➢ Nonbipartite Matchings and Restricted 2-matchings ➢ Blossom and Subtour Elimination Constraints Kenjiro Takazawa Hosei University, Japan IPCO2017 University of Waterloo June 25-27, 2017
Matching, 2-matching, and t -matching 2 v G = ( V , E ) : Simple, Undirected δ v Definition F F 1 F ⊆ E is a matching ⟺ | F ∩δ v | ≤ 1 ∀ v ∈ V F ⊆ E is a 2-matching F 2 F 3 ⟺ | F ∩δ v | ≤ 2 ∀ v ∈ V F ⊆ E is a t -matching ⟺ | F ∩δ v | ≤ t ∀ v ∈ V ➢ Just keep t =1,2 in mind t = 3 ➢ No theoretical difference in ∀ t ∈ Z >0
Our Framework : What are contained ? 3 Our Framework Matching Restriction Triangle-free 2-matching with edge-multiplicity Square-free 2-matching in bipartite graph Hamilton cycle
Our Result : What did we solve ? 4 Our Framework Our Result ➢ Min-max theorem Matching ➢ LP with dual integrality ➢ Combinatorial algorithm Even factor Triangle-free 2-matching [Cunningham, Geelen ’01] with edge-multiplicity K t,t -free t -matching Square-free 2-matching [Frank ’03] in bipartite graph ↑ P ↓ NP-hard 2-matching covering Hamilton cycle 3,4-edge cuts [Kaiser, Škrekovski ’04,08] [Boyd, Iwata, T . ’13]
Contents 5 1. Introduction 2. Previous work Triangle-free 2-matching with multiplicity • Square-free 2-matching • 3. Our framework : 4. Weighted U -feasible t -matching U -feasible t -matching ➢ Min-max theorem ➢ LP with dual integrality ➢ Combinatorial algorithm ➢ Combinatorial algorithm 5. Summary
Triangle-free 2-matching 6 Definition (Triangle-free 2-matching) F 1 ○ 2-matching x ∊{0,1,2} E is Triangle-free ⟺ Excluding cycles of length 3 × F 2 ➢ Allowing multiplicity 2: Theorem [Cornuéjols & Pulleyblank ’80] Max. ∑x ( e ) : P F 3 ◎ Max. ∑w ( e ) x ( e ) : P ➢ Min-max theorem ➢ LP with dual integrality ➢ Combinatorial algorithm ◎ F 4 No multiplicity allowed: ➢ Max. | F | : Algorithm [Hartvigsen ’84] ➢ Max. w ( F ): Open ➢ Discrete convexity [Kobayashi ’14]
Square-free 2-matching in bipartite graph 7 Definition (Square-free 2-matching) × 2-matching F ⊆ E is Square-free ⟺ Excluding cycles of length 4 Previous work for bipartite graphs Max. |F | : P ➢ Min-max theorem [Z. Király ’99, Frank ’03] ➢ Combinatorial algorithm [Hartvigsen ’06; Pap ’07] ◎ ➢ Canonical decomposition [ T . ’15] Max. w ( F ): NP-hard [Z. Király ’99] ➢ P under a certain assumption on w ( ☞ p.16) ✓ LP with dual integrality [Makai ’07] ✓ Combinatorial algorithm [ T . ’09] Max. | F | in nonbipartite graphs: Open ➢ Discrete convexity [Kobayashi, Szabó, T . ’12]
Contents 8 1. Introduction 2. Previous work Triangle-free 2-matching with multiplicity • Square-free 2-matching • 3. Our framework : 4. Weighted U -feasible t -matching U -feasible t -matching ➢ Min-max theorem ➢ LP with dual integrality ➢ Combinatorial algorithm ➢ Combinatorial algorithm 5. Summary
Our Framework: U -feasible t -matching 9 U ⊆ 2 V : Vertex subset family Definition U 2 t- matching F⊆E is U -feasible U 1 × 𝒖 𝑽 −𝟐 ∀ U ∈ U ⟺ |𝑮 𝑽 | ≤ 𝟑 ⟺ Excluding t -factors in G [ U ] ∀ U ∈ U U 1 𝑉 −1 |𝑉| t =1: |𝐺[𝑉]| ≤ = (| U |: even) 2 − 1 2 𝑉 −1 (| U |: odd) 2 × × 2 𝑉 −1 t =2: |𝐺 𝑉 | ≤ = 𝑉 − 1 2 [ T . ’16] ➢ U =2 V ∖{ ∅ , V } U 2 U -feasible 2-factor= Hamilton cycle U 1
Our Result 10 Our assumption G : Bipartite ∀ U ∊ U is “factor - critical” ( ☞ p. 13) Our result Weighted (Assumption on w ) ➢ Min-max theorem ➢ LP with dual integrality ➢ Combinatorial algorithm ➢ Combinatorial algorithm × ◎ Problems accepting this assumption: ➢ Square-free 2-matching U = { U : U ⊆ V , | U |=4} • U U t =2 • ➢ Nonbipartite matching Nonbipartite ➢ Triangle-free 2-matching ( ☞ Next slides) ➢ Even factor ➢ K t,t -free t -matching
Special Case: Triangle-free 2-matching 11 u v G =( V , E ): Nonbipartite graph x G ’=( V ’, E ’): Bipartite graph ➢ V ’ = V 1 ∪ V 2 y z ➢ E ’ = { u 1 v 2 , v 1 u 2 : uv ∊ E } t = 1 u 1 u 2 U = { U 1 ∪ U 2 : U ⊆ V , | U |=3} v 1 v 2 Proposition |max. triangle-free 2-matching in G | x 1 x 2 ＝ |max. U -feasible 1-matching in G ’| y 1 y 2 z 1 z 2
Special Case: Nonbipartite Matching 12 u v G =( V , E ): Nonbipartite graph x G ’=( V ’, E ’): Bipartite graph ➢ V ’ = V 1 ∪ V 2 y z ➢ E ’ = { u 1 v 2 , v 1 u 2 : uv ∊ E } t = 1 u 1 u 2 U = { U 1 ∪ U 2 : U ⊆ V , | U | is odd} v 1 v 2 Proposition 2 ∙ |max matching in G | x 1 x 2 ＝ |max U -feasible 1-matching in G ’| v u y 1 y 2 Dipaths and even dicycles x = Even factor [Cunningham, Geelen ’01] z 1 z 2 y z
Algorithm + Factor-criticality of U ∈ U 13 ➢ Nonbipartite matching: Shrink odd cycles [Edmonds ’65] U v v factor-critical Augment Shrink Expand Perfect matching covering U - v ➢ U -feasible t -matching: Shrink U ∊ U U U ’ U u v u v Assumption on ( G , U , t ) ➢ u, v : Degree 1 (= t -1) ➢ U- { u,v } : Degree 2 (= t ) U ∊ U is “ factor-critical ” Augment Shrink Expand ➢ Feasible for U ’ ∊ U crossing U
Min-max Theorem 14 Nonbipartite matching Theorem Triangle-free 2-matching ➢ G : Bipartite Square-free 2-matching Even factor ➢ ∀ U∈ U is “factor - critical” K t,t -free t -matching max{| F | : F is a U -feasible t -matching} 𝑢 𝑉 −1 =min{ 𝑢 𝑌 + 𝐹 𝐷 𝑊−𝑌 } + σ 𝑉∈𝒱(𝑊−𝑌) 2 Theorem [Tutte ’47, Berge ’58] max{| M | : M is a matching} 1 = 2 min{ 𝑊 + 𝑌 − odd(𝑌): 𝑌 ⊆ 𝑊} 𝑌 𝑌
Contents 15 1. Introduction 2. Previous work Triangle-free 2-matching with multiplicity • Square-free 2-matching • 3. Our framework : 4. Weighted U -feasible t -matching U -feasible t -matching ➢ Min-max theorem ➢ LP with dual integrality ➢ Combinatorial algorithm ➢ Combinatorial algorithm 5. Summary
LP for Square-free 2-matching 16 U × Max weight square-free 2-matching Maximize ∑ e ∈ E w ( e ) x ( e ) subject to ∑ e ∈δ v x ( e ) ≤ 2 ( v ∈ V ) ∑ e ∈ E [ U ] x ( e ) ≤ 3 ( U ⊆ V, |U| =4) ◎ U 0 ≤ x ( e ) ≤ 1 ( e ∈ E ) 𝑢 𝑉 − 1 2 Theorem [Makai ’07, T . ’09] Assumption on w ➢ G: Bipartite ➢ w is vertex-induced on ∀ square U i.e., w ( u 1 v 1 )+ w ( u 2 v 2 ) = w ( u 1 v 2 )+ w ( u 2 v 1 ) u 1 v 1 u 2 v 2 This LP has an integral opt solution The dual LP has an integral opt solution
Our Result: LP for U -feasible t -matching 17 Max weight U -feasible t -matching U × Maximize ∑ e ∈ E w ( e ) x ( e ) subject to ∑ e ∈δ v x ( e ) ≤ t ( v ∈ V ) 𝒖 𝑽 −𝟐 ∑ e ∈ E [ U ] x ( e ) ≤ ( U ∊ U ) U ◎ 𝟑 x ( e ) ≥ 0 ( e ∈ E ) Theorem Proved by our primal-dual algorithm ➢ G : Bipartite ∀ U ∈ U is “factor - critical” ➢ ➢ w is vertex-induced on ∀ U ∊ U u 1 v 1 i.e. , in G [ U ], the weights of perfect matchings are identical u 2 v 2 u 3 v 3 This LP has an integral opt solution The dual LP has an integral opt solution
Our Result: LP for U -feasible t -matching 18 Max weight U -feasible t -matching U × Maximize ∑ e ∈ E w ( e ) x ( e ) subject to ∑ e ∈δ v x ( e ) ≤ t ( v ∈ V ) 𝒖 𝑽 −𝟐 ∑ e ∈ E [ U ] x ( e ) ≤ ( U ∊ U ) U ◎ 𝟑 x ( e ) ≥ 0 ( e ∈ E ) Special cases Subtour Elimination Const. for TSP 𝒖 𝑽 −𝟐 ➢ t =2 = | U | - 1 𝟑 u 1 v 1 Blossom Const. for matching u 2 v 2 𝒖 𝑽 −𝟐 𝑽 −𝟐 ➢ t =1, | U |=2∙(odd) = u 3 v 3 𝟑 𝟑
Subtour Elimination for TSP 19 IP for TSP [Dantzig , Fulkerson, Johnson ’54] Minimize ∑ e ∈ E w ( e ) x ( e ) subject to ∑ e ∈δ v x ( e ) = 2 ( v ∈ V ) ∑ e ∈ E [ U ] x ( e ) ≤ | U | - 1 ( U ⊆ V ) U x ( e ) ∈ {0,1} ( e ∈ E ) 2 𝑉 − 1 Conjecture [Goemans ’95] 2 𝟓 w is metric Integrality gap ≤ 𝟒 𝟓 i.e., OPT(IP) ≤ 𝟒 OPT(LP)
Blossom Const for Nonbipartite Matching 20 Max. weight matching Maximize ∑ e ∈ E w ( e ) x ( e ) subject to ∑ e ∈δ v x ( e ) ≤ 1 ( v ∈ V ) U 𝑽 −𝟐 ∑ e ∈ E [ U ] x ( e ) ≤ ( U ⊆ V , | U | is odd) 𝟑 x ( e ) ≥ 0 ( e ∈ E ) 𝑉 − 1 2 U Theorem [Cunningham, Marsh ’78] ➢ This LP has an integral optimal solution ➢ The dual LP has an integral optimal solution
LP for Triangle-free 2-matching 21 × Max weight triangle-free 2-matching Maximimize ∑ e ∈ E w ( e ) x ( e ) U 1 subject to ∑ e ∈δ v x ( e ) ≤ 2 ( v ∈ V ) U 2 ∑ e ∈ E [ U ] x ( e ) ≤ 2 ( U ⊆ V, |U| =3) x ( e )≥0 ( e ∈ E ) U ◎ 2 𝑉 − 1 2 Theorem [ Cornuéjols & Pulleyblank ’80] This LP has an integer optimal solution U ◎
Contents 22 1. Introduction 2. Previous work Triangle-free 2-matching with multiplicity • Square-free 2-matching • 3. Our framework : 4. Weighted U -feasible t -matching U -feasible t -matching ➢ Min-max theorem ➢ LP with dual integrality ➢ Combinatorial algorithm ➢ Combinatorial algorithm 5. Summary
Recommend
More recommend
