Excluded t-factors in Bipartite Graphs:

Transcription

1 Excluded t-factors in Bipartite Graphs: A nified Framework for Nonbipartite Matchings and Restricted -matchings Blossom and Subtour Elimination Constraints Kenjiro Takazawa Hosei niversity, Japan IPCO017 niversity of Waterloo June 5-7, 017

2 Matching, -matching, and t-matching G = (V,E) : Simple, ndirected v δv Definition F E is a matching F δv 1 v V F E is a -matching F δv v V F F E is a t-matching F δv t v V F F 1 F 3 Just keep t=1, in mind No theoretical difference in t Z >0 t = 3

3 Our Framework : What are contained? Our Framework 3 Matching Restriction Triangle-free -matching with edge-multiplicity Square-free -matching in bipartite graph Hamilton cycle

4 Our Result : What did we solve? Our Framework Matching Our Result Min-max theorem LP with dual integrality Combinatorial algorithm 4 Triangle-free -matching with edge-multiplicity Square-free -matching in bipartite graph P NP-hard Hamilton cycle Even factor [Cunningham, Geelen 01] K t,t -free t-matching [Frank 03] -matching covering 3,4-edge cuts [Kaiser, Škrekovski 04,08] [Boyd, Iwata, T. 13]

5 Contents 5 1. Introduction. Previous work Triangle-free -matching with multiplicity Square-free -matching 3. Our framework: -feasible t-matching Min-max theorem Combinatorial algorithm 4. Weighted -feasible t-matching LP with dual integrality Combinatorial algorithm 5. Summary

6 Triangle-free -matching Definition (Triangle-free -matching) -matching x {0,1,} E is Triangle-free Excluding cycles of length 3 Allowing multiplicity : 6 F 1 F Theorem [Cornuéjols & Pulleyblank 80] Max. x(e) : P Max. w(e)x(e) : P Min-max theorem LP with dual integrality Combinatorial algorithm No multiplicity allowed: Max. F : Algorithm [Hartvigsen 84] Max. w(f): Open Discrete convexity [Kobayashi 14] F 3 F 4

7 Square-free -matching in bipartite graph 7 Definition (Square-free -matching) -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. 1]

8 Contents 8 1. Introduction. Previous work Triangle-free -matching with multiplicity Square-free -matching 3. Our framework: -feasible t-matching Min-max theorem Combinatorial algorithm 4. Weighted -feasible t-matching LP with dual integrality Combinatorial algorithm 5. Summary

9 Our Framework: -feasible t-matching 9 V : Vertex subset family Definition t-matching F E is -feasible F t 1 Excluding t-factors in G[] 1 t=1: F[] t=: F 1 1 = 1 = 1 1 ( : even) [T. 16] ( : odd) 1 = V {,V} -feasible -factor=hamilton cycle 1

10 Our Result 10 Our assumption G: Bipartite is factor-critical ( p. 13) Our result Min-max theorem Combinatorial algorithm Weighted (Assumption on w) LP with dual integrality Combinatorial algorithm Problems accepting this assumption: Square-free -matching = { : V, =4} t= Nonbipartite matching Triangle-free -matching Even factor K t,t -free t-matching Nonbipartite ( Next slides)

11 Special Case: Triangle-free -matching G=(V,E): Nonbipartite graph u v x 11 G =(V,E ): Bipartite graph V = V 1 V E = {u 1 v, v 1 u : uv E} t = 1 = { 1 : V, =3} z u 1 y u Proposition v 1 v max. triangle-free -matching in G = max. -feasible 1-matching in G x 1 x y 1 y z 1 z

12 Special Case: Nonbipartite Matching G=(V,E): Nonbipartite graph u x v 1 G =(V,E ): Bipartite graph V = V 1 V E = {u 1 v, v 1 u : uv E} t = 1 = { 1 : V, is odd} z u 1 y u Proposition v 1 v max matching in G = max -feasible 1-matching in G x 1 x u Dipaths and even dicycles = Even factor [Cunningham, Geelen 01] z x v y y 1 z 1 y z

13 Algorithm + Factor-criticality of 13 Nonbipartite matching: Shrink odd cycles [Edmonds 65] v Shrink Augment Expand -feasible t-matching: Shrink v factor-critical Perfect matching covering -v u v u v Assumption on (G,,t) is factor-critical Shrink Augment u, v : Degree 1 (=t-1) -{u,v} : Degree (=t) Expand Feasible for crossing

14 Min-max Theorem Theorem G: Bipartite is factor-critical max{ F : F is a -feasible t-matching} t 1 =min{ t X + E C V X + σ (V X) } max{ M : M is a matching} = 1 Nonbipartite matching Triangle-free -matching Square-free -matching Even factor K t,t -free t-matching Theorem [Tutte 47, Berge 58] 14 min{ V + X odd(x): X V} X X

15 Contents Introduction. Previous work Triangle-free -matching with multiplicity Square-free -matching 3. Our framework: -feasible t-matching Min-max theorem Combinatorial algorithm 4. Weighted -feasible t-matching LP with dual integrality Combinatorial algorithm 5. Summary

16 LP for Square-free -matching Max weight square-free -matching Maximize e E w(e) x(e) subject to e δv x(e) (v V) e E[] x(e) 3 ( V, =4) 0 x(e) 1 (e E) t 1 16 Theorem [Makai 07, T. 09] Assumption on w G: Bipartite w is vertex-induced on square i.e., w(u 1 v 1 )+w(u v ) = w(u 1 v )+w(u v 1 ) This LP has an integral opt solution The dual LP has an integral opt solution u 1 u v 1 v

17 Our Result: LP for -feasible t-matching 17 Max weight -feasible t-matching Maximize e E w(e) x(e) subject to e δv x(e) t (v V) e E[] x(e) x(e) 0 t 1 ( ) (e E) Theorem Proved by our G: Bipartite primal-dual algorithm is factor-critical w is vertex-induced on i.e., in G[], the weights of perfect matchings are identical This LP has an integral opt solution The dual LP has an integral opt solution u 1 u v 1 u 3 v v 3

18 Our Result: LP for -feasible t-matching 18 Max weight -feasible t-matching Maximize e E w(e) x(e) subject to e δv x(e) t (v V) e E[] x(e) x(e) 0 t 1 ( ) (e E) Special cases Subtour Elimination Const. for TSP t 1 t= = - 1 Blossom Const. for matching t=1, = (odd) t 1 = 1 u 1 u v 1 u 3 v v 3

19 Subtour Elimination for TSP 19 IP for TSP [Dantzig, Fulkerson, Johnson 54] Minimize e E w(e) x(e) subject to e δv x(e) = (v V) e E[] x(e) - 1 x(e) {0,1} Conjecture [Goemans 95] ( V) (e E) 1 w is metric Integrality gap 4 3 i.e., OPT(IP) 4 3 OPT(LP)

20 Blossom Const for Nonbipartite Matching 0 Max. weight matching Maximize e E w(e) x(e) subject to e δv x(e) 1 (v V) 1 e E[] x(e) ( V, is odd) x(e) 0 (e E) 1 Theorem [Cunningham, Marsh 78] This LP has an integral optimal solution The dual LP has an integral optimal solution

21 LP for Triangle-free -matching Max weight triangle-free -matching Maximimize e E w(e) x(e) subject to e δv x(e) (v V) e E[] x(e) ( V, =3) x(e) 0 (e E) 1 Theorem [Cornuéjols & Pulleyblank 80] This LP has an integer optimal solution 1 1

22 Contents 1. Introduction. Previous work Triangle-free -matching with multiplicity Square-free -matching 3. Our framework: -feasible t-matching Min-max theorem Combinatorial algorithm 4. Weighted -feasible t-matching LP with dual integrality Combinatorial algorithm 5. Summary

23 Summary Our Framework -feasible t-matching: F t 1 3 Special Cases Nonbipartite matching Triangle-free -matching with edge multiplicity Even factor Square-free -matching K t,t -free t-matching -matchings covering edge cuts Hamilton cycles Solved: G: Bipartite is factor-critical w is vertex-induced on Min-max theorem LP with dual integrality Combinatorial algorithm

24 Further Research 4 Application to TSP Subtour elimination Blossom constraint So what?? New class of ***-free -factors?? C 4 [Hartvigsen 06, Pap 07] C 6 with chords [T. 16] Matroids as Special Cases Matroids (t=1, ={Circuit}) Arborescences And more?? So what?? : Circuit

25 References 5 G. Cornuéjols, W. Pulleyblank: A matching problem with side conditions, Discrete Math. 9 (1980), W.H. Cunningham, J.F. Geelen, Vertex-disjoint dipaths and even dicircuits, manuscript, 001. D. Hartvigsen: Finding maximum square-free -matchings in bipartite graphs, J. Combin. Theory B, 96 (006), G. Pap: Combinatorial algorithms for matchings, even factors and square-free -factors, Math. Program. 110 (007), K. Takazawa: A weighted even factor algorithm, Math. Program. 115 (008), K. Takazawa: A weighted K t,t -free t-factor algorithm for bipartite graphs, Math. Oper. Res. 34 (009), K. Takazawa: Finding a maximum -matching excluding prescribed cycles in bipartite graphs, Discrete Optimization, to appear.

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