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The Triangle-free 2-matching Polytope Of Subcubic Graphs Kristóf Bérczi Egerváry Research Group (EGRES) Eötvös Loránd University Budapest ISMP 2012 Motivation 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 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 k=n-3: Deleting n-4 nodes G remains connected. n-4 n-4 G G n-4 n-4 Degrees at most 2 in G. No cycle of length 4. Definitions G=(V,E) undirected, simple, b:V→Z+ Def.: A b-matching is a subset F⊆E s.t. dF(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 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.: C4-free 2-matching Notation: C3=∆, C4=◊ Example: k=3 Previous work 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: • Kt,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: • Kt,t- and Kt+1-free t-matchings in degree-bounded graphs Polyhedral descriptions 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. K F matching The b-factor polytope matching Def.: The b-factor polytope is the convex hull of incedence vectors of b-factors. matchings Thm.: matching The b-factor polytope is determined by 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! matchings Triangle-free 2-factors Thm.: (Hartvigsen and Li ’07) matching Conjecture: For subcubic G, the ∆-free 2-factor polytope is determined by NOT TRUE !!! Subcubic graphs Problem with degrees „Usual” way of proof: G 3 ∆ -free 2-factors G’ 3 3 ∆ -free 2-matchings 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. Triangle-free 2-matchings Thm.: (Hartvigsen and Li ’12) For subcubic G, the ∆-free 2-matching polytope is determined by New proof Perfect matchings Thm.: (Edmonds ‘65) The p.m. polytope is determined by Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver) Another proof Thm.: (Edmonds ‘65) The p.m. polytope is determined by Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver) Another proof Thm.: (Edmonds ‘65) The p.m. polytope is determined by Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, and Schrijver) Plan Tricky ! Technical … Are inequalities true for x’? Define shrinking Shrink the complement, Extend put convex OR combinations combination together to the original problem Define tightness Hartvigsen and Li Yipp ! Shrinking Shrinking a tight ∆ Shrinking a tight tri-comb Conclusions Now: • New proof for the description of the ∆-free 2matching 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 Thank you for your attention!