Making Bipartite Graphs DM-irreducible

Transcription

1 Making Bipartite Graphs DM-irreducible Kristóf Bérczi 1, Satoru Iwata 2, Jun Kato 3, Yutaro Yamaguchi 4 1. Eötvös Lorand University, Hungary. 2. University of Tokyo, Japan. 3. TOYOTA Motor Corporation, Japan. 4. Osaka University, Japan. ISMP July 5, 2018

2 Dulmage Mendelshon Decomposition [Dulmage Mendelsohn 1958,59] G = V +, V ; E : Bipartite Graph V + V 2

3 Dulmage Mendelshon Decomposition [Dulmage Mendelsohn 1958,59] G = V +, V ; E : Bipartite Graph V + V V 0 + > V 0 or V 0 = V + i = V i (i 0, ) V + < V or V = Max. Matching in G is a union of Perfect Matchings in G V i V 0 V 1 V 2 V Unique Partition of Vertex Set reflecting Structure of Maximum Matchings (Definition will be given later) 3

4 Dulmage Mendelshon Decomposition [Dulmage Mendelsohn 1958,59] G = V +, V ; E : Bipartite Graph V + V V 0 + > V 0 or V 0 = V + i = V i (i 0, ) V + < V or V = Max. Matching in G is a union of Perfect Matchings in G V i Covering ALL vertices in one side V 0 V 1 V 2 V Unique Partition of Vertex Set reflecting Structure of Maximum Matchings (Definition will be given later) 4

5 Dulmage Mendelshon Decomposition [Dulmage Mendelsohn 1958,59] G = V +, V ; E : Bipartite Graph V + V Max. Matching in G is a union of Perfect Matchings in G V i Edges between V i and V j i j cannot be used. V 0 V 1 e: Edge in G V i, Perfect Matching in G V i using e V 2 V Unique Partition of Vertex Set reflecting Structure of Maximum Matchings (Definition will be given later) 5

6 Dulmage Mendelshon Decomposition [Dulmage Mendelsohn 1958,59] G = V +, V ; E : Bipartite Graph V + V Max. Matching in G is a union of Perfect Matchings in G V i Edges between V i and V j i j cannot be used. V 0 V 1 e: Edge in G V i, Perfect Matching in G V i using e V 2 V Unique Partition of Vertex Set reflecting Structure of Maximum Matchings (Definition will be given later) 6

7 Our Problem Input Goal G = V +, V ; E : Bipartite Graph Find a Minimum Number of Additional Edges to Make G DM-irreducible DM-decomposition consists of a Single Component = e, M: Perfect Matching s.t. e M + α (Some Connectivity) V 0 V 1 V 2 V 1 V V 1 V 2 V 3 V V 7

8 Our Results (Summary) Input Goal G = V +, V ; E : Bipartite Graph Find a Minimum Number of Additional Edges to Make G DM-irreducible Min-Max Duality via Supermodular Arc Covering [Frank Jordán 1995] Unbalanced V + V Matroid Intersection Balanced V + = V & Perfectly Matchable Strong Connectivity Augmentation [Eswaran Tarjan 1976] Balanced & NOT P.M. Direct O nm -time Algorithm (Moreover, General Case) 8

9 Overview Supermodular Arc Covering Matroid Intersection Our Problem k-connectivity Augmentation Unbalanced Balanced & Perfectly Matchable Strong Connectivity Augmentation [Eswaran Tarjan 1976] 9

10 Overview Supermodular Arc Covering Matroid Intersection Our Problem k-connectivity Augmentation Unbalanced Balanced & Perfectly Matchable Strong Connectivity Augmentation [Eswaran Tarjan 1976] 10

11 Min-Max Duality Polytime by Ellipsoid [Frank Jordán 1995] Matroid Intersection Overview Supermodular Arc Covering Min-Max Polytime Our Problem Combinatorial Pseudopolytime [Végh Benczúr 2008] Polytime k-connectivity Augmentation Unbalanced Balanced & Perfectly Matchable Strong Connectivity Augmentation [Eswaran Tarjan 1976] 11

12 Min-Max Duality Polytime by Ellipsoid [Frank Jordán 1995] Overview Supermodular Arc Covering Combinatorial Pseudopolytime [Végh Benczúr 2008] Matroid Intersection Min-Max Polytime Our Problem O nm -time Polytime k-connectivity Augmentation Unbalanced Faster Weighted ver. OK Balanced & Perfectly Matchable Weighted ver. NP-hard Strong Connectivity Augmentation [Eswaran Tarjan 1976] 12

13 Min-Max Duality Polytime by Ellipsoid [Frank Jordán 1995] Overview Supermodular Arc Covering Combinatorial Pseudopolytime [Végh Benczúr 2008] Matroid Intersection Min-Max Polytime Our Problem O nm -time Polytime k-connectivity Augmentation Unbalanced Faster Weighted ver. OK Balanced & Perfectly Matchable Weighted ver. NP-hard Strong Connectivity Augmentation [Eswaran Tarjan 1976] 13

14 How to Compute DM-decomposition G = V +, V ; E : Bipartite Graph V + V Find a Maximum Matching M in G Orient Edges so that M Both Directions E M Left to Right V 0 : Reachable from V + + M V : Reachable to V M V i : Strongly Connected Component of G V 0 V 14

15 How to Compute DM-decomposition G = V +, V ; E : Bipartite Graph V + V Find a Maximum Matching M in G Orient Edges so that M Both Directions E M Left to Right V 0 : Reachable from V + + M V : Reachable to V M V i : Strongly Connected Component of G V 0 V 15

16 How to Compute DM-decomposition G = V +, V ; E : Bipartite Graph V + V Find a Maximum Matching M in G Orient Edges so that M Both Directions E M Left to Right V 0 : Reachable from V + + M V : Reachable to V M V i : Strongly Connected Component of G V 0 V 16

17 How to Compute DM-decomposition G = V +, V ; E : Bipartite Graph V + V Find a Maximum Matching M in G Orient Edges so that M Both Directions E M Left to Right V 0 : Reachable from V + + M V : Reachable to V M V 0 V V i : Strongly Connected Component of G V 0 V 17

18 How to Compute DM-decomposition G = V +, V ; E : Bipartite Graph V + V Find a Maximum Matching M in G Orient Edges so that M Both Directions E M Left to Right V 0 : Reachable from V + + M V 1 V 2 V : Reachable to V M V i : Strongly Connected Component of G V 0 V 18

19 When Balanced & Perfectly Matchable V 1 V 1 V 2 V 3 V 2 V 3 V 1 V 4 DM-decomposition = V 4 Decompoisition into Strg. Conn. Comps. Make it Strg. Conn. by Adding Edges Obs. DM-irreducibility is Equivalent to Strong Connectivity of the Oriented Graph 19

20 Strong Connectivity Augmentation Input Goal G = V, E : Directed Graph (NOT Strg. Conn.) Find a Minimum Number of Additional Edges to Make G Strongly Connected : Strg. Conn. Comp. 20

21 Strong Connectivity Augmentation Input Goal G = V, E : Directed Graph (NOT Strg. Conn.) Find a Minimum Number of Additional Edges to Make G Strongly Connected Each Source needs an Entering Edge : Strg. Conn. Comp. 21

22 Strong Connectivity Augmentation Input Goal G = V, E : Directed Graph (NOT Strg. Conn.) Find a Minimum Number of Additional Edges to Make G Strongly Connected Each Source needs an Entering Edge : Strg. Conn. Comp. Each Sink needs a Leaving Edge 22

23 Strong Connectivity Augmentation Input Goal G = V, E : Directed Graph (NOT Strg. Conn.) Find a Minimum Number of Additional Edges to Make G Strongly Connected max # of Sources, # of Sinks edges are Necessary. 23

24 Strong Connectivity Augmentation Input Goal G = V, E : Directed Graph (NOT Strg. Conn.) Find a Minimum Number of Additional Edges to Make G Strongly Connected max # of Sources, # of Sinks edges are Necessary. Thm. Cor. It is also Sufficient. One can find such an edge set in Linear Time. [Eswaran Tarjan 1976] If the input is Balanced with Perfect Matching, Our Problem can be solved in Linear Time. 24

25 When Balanced & NOT Perfectly Matchable Idea Reduce to P.M. Case by Connecting Exposed Vertices Each V i i 0, remains as it was V 0 V 1 V 2 V 25

26 When Balanced & NOT Perfectly Matchable Idea Reduce to P.M. Case by Connecting Exposed Vertices Each V i i 0, remains as it was V V i V 0 V + M + max # of Sources, # of Sinks # of Additional Edges 26

27 When Balanced & NOT Perfectly Matchable Idea Reduce to P.M. Case by Connecting Exposed Vertices Each V i i 0, remains as it was V V i V 0 V + M + max # of Sources, # of Sinks Const. Depending on M # of Additional Edges 27

28 When Balanced & NOT Perfectly Matchable Idea Reduce to P.M. Case by Connecting Exposed Vertices # of Sources or of Sinks depends on Max. Matching M Find Eligible Perfect Matchings in G V and in G V 0 Minimizing # of Sources in V and # of Sinks in V 0 Just by finding two edge-disjoint s t paths O n times Optimality is guaranteed by Min-Max Duality Thm. If the input is Balanced (in fact, NOT necessary), Our Problem can be solved in O nm time. [BIKY 2018] 28

29 When Balanced & NOT Perfectly Matchable Idea Reduce to P.M. Case by Connecting Exposed Vertices # of Sources or of Sinks depends on Max. Matching M Find Eligible Perfect Matchings in G V and in G V 0 Minimizing # of Sources in V and # of Sinks in V 0 Just by finding two edge-disjoint s t paths O n times Optimality is guaranteed by Min-Max Duality Thm. If the input is Balanced (in fact, NOT necessary), Our Problem can be solved in O nm time. [BIKY 2018] 29

30 When Balanced & NOT Perfectly Matchable Idea Reduce to P.M. Case by Connecting Exposed Vertices # of Sources or of Sinks depends on Max. Matching M Find Eligible Perfect Matchings in G V and in G V 0 Minimizing # of Sources in V and # of Sinks in V 0 Just by finding two edge-disjoint s t paths O n times Optimality is guaranteed by Min-Max Duality Thm. If the input is Balanced (in fact, NOT necessary), Our Problem can be solved in O nm time. [BIKY 2018] 30

31 Min-Max Duality Polytime by Ellipsoid [Frank Jordán 1995] Overview Supermodular Arc Covering Combinatorial Pseudopolytime [Végh Benczúr 2008] Matroid Intersection Min-Max Polytime Our Problem O nm -time Polytime k-connectivity Augmentation Unbalanced Faster Weighted ver. OK Balanced & Perfectly Matchable Weighted ver. NP-hard Strong Connectivity Augmentation [Eswaran Tarjan 1976] 31

32 Definition of DM-decomposition G = V +, V ; E : Bipartite Graph V + V f G X + Γ G X + X + X + V + f G is Submodular (Surplus for Hall s Condition) Minimizers form Distributive Lattice X 0 + X 1 + X k + : Maximal Chain V 0 + X 0 +, V 0 Γ G X 0 + V + i X + i X + i 1, V i Γ G X + + i Γ G X i 1 V + V + X + k, V V + Γ G X k 32

33 Definition of DM-decomposition G = V +, V ; E : Bipartite Graph V + V f G X + Γ G X + X + X + V + (Surplus for Hall s Condition) X + Γ G X + X + Γ G X + f G = 4 f G = 1 33

34 Definition of DM-decomposition G = V +, V ; E : Bipartite Graph V + V f G X + Γ G X + X + X + V + (Surplus for Hall s Condition) f G is Submodular Minimizers form Distributive Lattice X + 0 X + 1 X + k : Maximal Chain X 0 + X 1 + X

35 Definition of DM-decomposition G = V +, V ; E : Bipartite Graph V + V f G X + Γ G X + X + X + V + f G is Submodular (Surplus for Hall s Condition) Minimizers form Distributive Lattice X 0 + X 1 + X k + : Maximal Chain V 0 + X 0 +, V 0 Γ G X 0 + V + i X + i X + i 1, V i Γ G X + + i Γ G X i 1 V + V + X + k, V V + Γ G X k V 0 V 1 V 2 V 35

36 Rephrasing of DM-irreducibility G = V +, V ; E : Bipartite Graph V + V f G X + Γ G X + X + X + V + (Surplus for Hall s Condition) When V + < V (Unbalanced) V + is a unique minimizer Γ G X + > X + X + V + V When V + = V (Balanced) Only and V + are minimizers Γ G X + > X + X + V + 36

37 Rephrasing of DM-irreducibility G = V +, V ; E : Bipartite Graph V + V f G X + Γ G X + X + X + V + (Surplus for Hall s Condition) When V + < V (Unbalanced) V + is a unique minimizer Γ G X + > X + X + V + V 1 When V + = V (Balanced) Only and V + are minimizers Γ G X + > X + X + V + Symmetrically for V 37

38 Min-Max Duality (Unbalanced Case) Input Goal F G = V +, V ; E : Bipartite Graph V + < V Find a Smallest Set F of Additional Edges s.t. Γ G+F X + > X + X + V + X + X + 1 f G X + X + : Subpartition of V + Γ G X + X + f G = 1 f G = 0 f G+F = 1 f G+F = 1 38

39 Min-Max Duality (Unbalanced Case) Input Goal F Thm. G = V +, V ; E : Bipartite Graph V + < V Find a Smallest Set F of Additional Edges s.t. Γ G+F X + > X + X + V + X + X + Γ G X + X + 1 f G X + X + : Subpartition of V + min F G + F is DM irreducible max σ X + X + 1 f G X + X + : Subpartition of V + [BIKY 2018] 39

40 Min-Max Duality (Balanced Case) Input Goal G = V +, V ; E : Bipartite Graph V + = V Find a Smallest Set F of Additional Edges s.t. Γ G+F X ± > X ± X ± V ± Thm. min F G + F is DM irreducible X ± V ± max max τ G X + X + : Proper Subpartition of V +, max τ G X X : Proper Subpartition of V τ G X + σ X + X + 1 f G X + [BIKY 2018] G V, V + ; E : Interchanging V + and V 40

41 Supermodular Arc Covering Thm. V +, V : Finite Sets (possibly intersecting) F 2 V+ 2 V : Crossing Family (Constraint Set) g: F Z 0 Supermodular (Demand on F) The minimum cardinality of a multiset A: V + V Z 0 of directed edges in V + V that covers g is equal to max S F σ X +,X S g X+, X S: pairwise independent [Frank Jordán 1995] Packing (Max) vs. Covering (Min) type Strong Duality Polytime Solvability by Ellipsoid Method Including Directed k-connectivity Augmentation etc. 41

42 Supermodular Arc Covering Thm. V +, V : Finite Sets (possibly intersecting) F 2 V+ 2 V : Crossing Family (Constraint Set) g: F Z 0 Supermodular (Demand on F) The minimum cardinality of a multiset A: V + V Z 0 of directed edges in V + V that covers g is equal to max S F σ X +,X S g X+, X S: pairwise independent [Frank Jordán 1995] By defining F and g appropriately for Our Problem, we obtain Min-Max Duality Theorems for Our Problem (via some nontrivial Uncrossing arguments) 42

43 Overview Min-Max Duality Polytime by Ellipsoid [Frank Jordán 1995] Matroid Intersection Supermodular Arc Covering Min-Max Polytime Our Problem O nm -time Combinatorial Pseudopolytime [Végh Benczúr 2008] Polytime k-connectivity Augmentation Unbalanced Faster Weighted ver. OK Balanced & Perfectly Matchable Weighted ver. NP-hard Strong Connectivity Augmentation [Eswaran Tarjan 1976] 43

44 Weighted Problem (Unbalanced Case) Input Goal G = V +, V ; E : Bipartite Graph V + < V c: V + V E R >0 (Cost on Addition) Find a Minimum-Cost Set F of Additional Edges s.t. G + F is DM-irreducible Lem. G G + F: Forest s.t. Γ G v = 2 v V + [BIKY 2018] Γ G v = 2 G + F G Minimally DM-irreducible 44

45 Weighted Problem (Unbalanced Case) Input Goal G = V +, V ; E : Bipartite Graph V + < V c: V + V E R >0 (Cost on Addition) Find a Minimum-Cost Set F of Additional Edges s.t. G + F is DM-irreducible Lem. G G + F: Forest s.t. Γ G v = 2 v V + Reduce to find a Min-Weight Common Base in [BIKY 2018] M 1 : Graphic Matroid (with 2 V + -Truncation) M 2 : Partition Matroid (Degree Constraint on V + ) γ: V + V R 0 (Weight); γ e ቊ 0 e E c e e E 45

46 Our Results (Summary) Input Goal G = V +, V ; E : Bipartite Graph Find a Minimum Number of Additional Edges to Make G DM-irreducible Min-Max Duality via Supermodular Arc Covering [Frank Jordán 1995] Unbalanced V + V Matroid Intersection Balanced V + = V & Perfectly Matchable Strong Connectivity Augmentation [Eswaran Tarjan 1976] Balanced & NOT P.M. Direct O nm -time Algorithm (Moreover, General Case) 46