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The Triangle-free 2-matching Polytope Of Subcubic Graphs

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

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The Triangle-free 2-matching Polytope Of Subcubic Graphs

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

  2. Motivation

  3. Hamiltonian cycleproblem Relaxation: Find a subgraph • withdegrees = 2 • containingno „short” cycles (lengthat most k) Fisher, Nemhauser, Wolsey ‘79: howsolutionsfortheweighted version approximatetheoptimalTSP Remark: for k > n/2therelax. and the HCP areequivalent

  4. Connectivityaugmentation Problem:Make G k-node-connectedbyadding a minimum number of newedges. k = n-1: trivial (completegraph) k = n-2: maximalmatchinginG

  5. n-4 n-4 k=n-3: Deleting n-4 nodes G remainsconnected. G G Degreesat most 2 in G. No cycleof length 4. n-4 n-4

  6. Definitions

  7. G=(V,E) undirected, simple, b:V→Z+ Def.: A b-matchingis a subset F⊆E s.t. dF(v) ≤ b(v)foreachnodev. If=holdseverywhere, then F is a b-factor. Ifb=tforeachnode: t-matching. Examples: b=1 b=2

  8. LetK be a list of forbiddensubgraphs. Def.: A K-free b-matchingcontains 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

  9. Previouswork Papadimitriu ‘80: • NP-hardfor k ≥ 5 Vornberger ‘80: • NP-hardincubicgraphsfor k ≥ 5 • NP-hardincubicgraphsfor k = 4 withweights Hartvigsen ’84: • Polynomialalgorithmfor k=3 Hartvigsen and Li ‘07, Kobayashi ‘09: • Polynomialalgorithmfor k=3 insubcubicgraphswithgeneralweigths Nam ‘94: • Polynomialalgorithmfor k=4 if ◊’s arenode-disjoint Hartvigsen ‘99, Király ’01, Pap ’05, Takazawa ‘09: • Resultsforbipartitegraphs and k=4 Frank ‘03, Makai ‘07: • Kt,t-free t-matchingsinbipartitegraphs B. and Kobayashi ’09, Hartvigsen and Li ‘11: • Polynomialalgorithmfor k=4 insubcubicgraphs B. and Végh ’09, Kobayashi and Yin ‘11: • Kt,t- and Kt+1-free t-matchingsindegree-boundedgraphs

  10. Polyhedraldescriptions

  11. The b-factorpolytope Def.: The b-factorpolytopeis theconvex hull of incedencevectorsofb-factors. Def.: (K,F) is a blossomif K⊆V, F⊆δ(K) and b(K)+|F| is odd. F K

  12. matching The b-factorpolytope matching Def.: The b-factorpolytopeis theconvex hull of incedencevectorsofb-factors. Thm.: The b-factorpolytopeis determinedby matchings matching

  13. The C(≤)k-freecase The weighted C(≤)k-free 2-matching (factor) problem is NP-hardfor k ≥ 4 Whataboutk = 3 ??? Problem:Give a description of the∆-free 2-matching (factor) polytope. UNSOLVED!

  14. matchings Triangle-free 2-factors Conjecture: Thm.: (Hartvigsen and Li ’07) Forsubcubic G, the ∆-free 2-factor polytope is determinedby NOT TRUE !!! matching

  15. Subcubicgraphs Problemwithdegrees „Usual” way of proof: G G’ ∆ -free 2-factors ∆ -free 2-matchings 3 3 3

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

  17. Triangle-free 2-matchings Thm.: (Hartvigsen and Li ’12) Forsubcubic G, the ∆-free 2-matching polytope is determinedby

  18. New proof

  19. Perfectmatchings Thm.: (Edmonds ‘65) The p.m. polytopeis determinedby Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, andSchrijver)

  20. Anotherproof Thm.: (Edmonds ‘65) The p.m. polytopeis determinedby Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, andSchrijver)

  21. Anotherproof Thm.: (Edmonds ‘65) The p.m. polytopeis determinedby Proof: (Aráoz, Cunningham, Edmonds and Green-Krótki, andSchrijver)

  22. Technical… Extendconvexcombinationtotheoriginalproblem Plan Shrinkthecomplement, putcombinationstogether Areinequalitiestruefor x’? Tricky ! OR Defineshrinking Definetightness Yipp ! Hartvigsen and Li

  23. Shrinking

  24. Shrinking a tight ∆

  25. Shrinking a tighttri-comb

  26. Conclusions

  27. Now: • New proofforthedescription of the ∆-free 2-matching polytope of subcubicgraphs • Slightgeneralization • list of triangles • b-matching; onnodes of triangles b = 2 • notsubcubic; degrees of trianglenodes ≤ 3 Open problems: • Algorithmfor maximum ◊-free 2-matching • Description of the∆-free 2-matchingpolytopeingeneralgraphs

  28. Thankyouforyourattention!

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