1. The Triangle-free 2-matching Polytope Of Subcubic Graphs Kristóf Bérczi Egerváry Research Group (EGRES) Eötvös Loránd University Budapest ISMP 2012

2. Motivation

3. Hamiltonian cycle problem Relaxation: Find a subgraph with degrees = 2 containing no „short” cycles (length at most k) Fisher, Nemhauser, Wolsey ‘79: how solutions for the weighted version approximate the optimal TSP Remark: for k > n/2 the relax. and the HCP are equivalent

4. Connectivity augmentation Problem: Make G k-node-connected by adding a minimum number of new edges. k = n-1: trivial k = n-2: G G G is a matching. n-3

5. Connectivity augmentation Problem: Make G k-node-connected by adding a minimum number of new edges. k = n-1: trivial (complete graph) k = n-2: maximal matching in G

6. k=n-3: Deleting n-4 nodes G remains connected. G Degrees at most 2 in G. No cycle of length 4. n-4

7. Definitions

8. G=(V,E) undirected, simple, b:V → Z + Def.: A b-matching is a subset F ⊆ E s.t. d F (v) ≤ b(v) for each node v. If = holds everywhere, then F is a b-factor. If b=t for each node: t-matching. Examples: b=1 b=2

9. Let K be a list of forbidden subgraphs. Def.: A K-free b-matching contains no member of K. Def.: A C (≤)k -free 2-matching contains no cycle of length (at most) k. Hamiltonian relax.: C ≤k -free 2-factor Node-conn. aug.: C 4 -free 2-matching Notation: C 3 =∆, C 4 =◊ Example: k=3

10. Papadimitriu ‘80: NP-hard for k ≥ 5 Vornberger ‘80: NP-hard in cubic graphs for k ≥ 5 NP-hard in cubic graphs for k = 4 with weights Hartvigsen ’84: Polynomial algorithm for k=3 Hartvigsen and Li ‘07, Kobayashi ‘09: Polynomial algorithm for k=3 in subcubic graphs with general weigths Nam ‘94: Polynomial algorithm for k=4 if ◊’s are node-disjoint Hartvigsen ‘99, Király ’01, Pap ’05, Takazawa ‘09: Results for bipartite graphs and k=4 Frank ‘03, Makai ‘07: K t,t -free t-matchings in bipartite graphs B. and Kobayashi ’09, Hartvigsen and Li ‘11: Polynomial algorithm for k=4 in subcubic graphs B. and Végh ’09, Kobayashi and Yin ‘11: K t,t - and K t+1 -free t-matchings in degree-bounded graphs Previous work

11. Polyhedral descriptions

12. The perfect matching polytope Def.: The perfect matching polytope is the convex hull of incedence vectors of perfect matchings. Thm.: (Edmonds ‘65) The p.m. polytope is determined by

13. The b-factor polytope Def.: The b-factor polytope is the convex hull of incedence vectors of b-factors. Def.: (K,F) is a blossom if K ⊆ V, F ⊆ δ(K) and b(K)+|F| is odd. F K

14. The b-factor polytope Def.: The b-factor polytope is the convex hull of incedence vectors of b-factors. Thm.: The b-factor polytope is determined by matching matchings matching

15. The C (≤)k -free case The weighted C (≤)k -free 2-matching (factor) problem is NP-hard for k ≥ 4 What about k = 3 ??? Problem: Give a description of the ∆-free 2-matching (factor) polytope. UNSOLVED!

16. Triangle-free 2-factors Thm.: (Hartvigsen and Li ’07) For subcubic G, the ∆-free 2-factor polytope is determined by NOT TRUE !!! Conjecture: matchings matching

17. Subcubic graphs Problem with ∆’s: „Usual” way of proof: GG’ Perfect matchings ∆-free 2-factors 2 3 1 1 1 1 1 11

18. Subcubic graphs Problem with degrees „Usual” way of proof: GG’ ∆ -free 2-factors ∆ -free 2-matchings 3 33

19. Tri-combs Def.: (K,F, T ) is a tri-comb if K ⊆ V, T is a set of ∆’s „fitting” K, F ⊆ δ(K) and |T|+|F| is odd.

20. Triangle-free 2-matchings Thm.: (Hartvigsen and Li ’12) For subcubic G, the ∆-free 2-matching polytope is determined by

21. New proof

22. Perfect matchings Thm.: (Edmonds ‘65) The p.m. polytope is determined by Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)

23. Another proof Thm.: (Edmonds ‘65) The p.m. polytope is determined by Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)

24. Another proof Thm.: (Edmonds ‘65) The p.m. polytope is determined by Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver)

25. Hartvigsen and Li Define tightness Plan Define shrinking Are inequalities true for x’? Tricky ! Technical … Yipp ! Extend convex combination to the original problem OR Shrink the complement, put combinations together

26. Shrinking

27. Shrinking a tight ∆

28. Shrinking a tight tri-comb

29. Conclusions

30. Now: New proof for the description of the ∆-free 2- matching polytope of subcubic graphs Slight generalization – list of triangles – b-matching; on nodes of triangles b = 2 – not subcubic; degrees of triangle nodes ≤ 3 Open problems: Algorithm for maximum ◊-free 2-matching Description of the ∆-free 2-matching polytope in general graphs

31. Thank you for your attention!