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Exploring Advances in Matching Theory: Celebrating András Sebő's Contributions

This paper commemorates András Sebő's contributions to matching theory, reflecting on key concepts such as matroid structures, k-chromatic polynomials, and the Edmonds-Gallai theorem. It explores advanced topics including maximum fix covers, parity flows, stable sets, and the complexity of matching in bipartite graphs. By connecting historical developments and modern theories, the work highlights the significance of matchings and alternating paths in combinatorial optimization. This celebration marks over fifty years of progress and innovation inspired by Sebő's impactful research.

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Exploring Advances in Matching Theory: Celebrating András Sebő's Contributions

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  1. jump systems maxfix cover structure test-sets matroids k-chrom polyhedra b-matchings (multi)flows parity stable sets factors hypergraph matching, coloring ays to matching generalizations András Sebő, CNRS, Grenoble (France) For the 50th birthday of the Hungarian Method MATCHINGS,ALTERNATING PATHS

  2. G 1 -1 The Fifty Year Old : Many happy returns of the day x0

  3. Parity of Degrees and Negative Circuits 1 : if not in x0 -1 : if in Edmonds (65): Chinese Postman through matchings odd degree subgraphs:Edmonds,Johnson(73)minmax, alg: EJ, Barahona, Korach;sequence of sharper thms: Lovász (76), Seymour (81), Frank, Tardos (84), … , S Idea:1. minimum  no negative circuit (Guan 62) 2. identify vertices that are at distance  0, induction Defconservative (cons) : no circuit with neg total weight l(u)= min weight of an (x0,u) path x0V,

  4. 1 0 -1 -2 x0 x0 D: Thm: cons, bipartite, all distances <0  negative forest Thm:(S 84) G bipartite,w:E {-1,1}, conservativeThen | l (u) – l (v) | = 1 for all uvE, andfor allD D : d(D) contains 1 negative edge if x0 D0 0 negative edge if x0D Applications: matching structure; Integer packings of cuts, paths (Frank Szigeti, Ageev Kostochka Szigeti, …)

  5. + - path + + + + - - + - - - + + Various degree constraints and bidirected graphs b a c Def: Edmonds, Johnson (‘70) bidirected graph : ~alt path: edges are used at most once; was defined to handle a ‘general class of integer programs’ containing b-matchings. One of the reasons ‘labelling’ works for bipartite graphs: Transitivity : (a,b) & (b,c) alt paths  (a,c) Broken Transitivity:(S ’86) If  (a,bb)&(b-b,cg) path, then: either(a,cg) path,or both (a,b-b) & (bb,cg) paths. Tutte&Edmonds-Gallai type thms+‘structure algorithms’ for lower,upper bounds and parity, including digraphs. b For bidirected graphs: a c

  6. 14 14 maxfix covers Input: H graph, kIN Task: Find S  V(H) |S|=k that S hits a max number of edges of H. ContainsVertexCover. Let H=L(G) be a line graph! How many edges remain in F = L(G) – S ? minimize vV(G)dF(v)2 - const(=|E(G)|) Thm:(Apollonio, S.’04)Fisnot optimal better F’ withvV(G) | dF(v) – dF’(v) |  4 12 Cor : Pol solvable

  7. 4 0 24 50 number of years (edges of L(G) hit): : Many happyreturns of the day Aki nem hiszi számoljon utána …

  8. Independent sets in graphs (stable set) in matroids in posets(antichains) Extensions by Dilworth, Greene-Kleitman (further by Frank, K. Cameron, I. Hartman) : max union of k antichains = min{ |X| + k |c| : XV, c is aset of chains covering V/X}

  9. Conjecture of Linial : max k-chrom  min { |X| + k |P|: XV,P path partition of V / X } k=1 : Gallai-Milgram (1960)   min|P| orthogonal version :  paths and stable, 1 on each strong version:Gallai’s conj 62,Bessy,Thomassé 03 strongly conn, pathcycle, partitioncover orthogonalandstrongfollows:BT is a minmax k arbitrary, orthogonal conjecture (Berge): open ‘’strong’’ conjecture (who ?) : Thm S ’04 minmax orthog and strong conjecture : - ‘’ - compl slack no partition

  10. Test-sets, neighbors improving paths : switching: neighbors on the matching polytope If there exists a larger (b, T, …)- ‘matching’, then there is also one that covers 2 more vertices. Def (Graver ‘75, Scarf, Bárány, Lovász, …)A matrix; T is a test-setif for all b and c, Ax  b, x integer has a better solution than x0 also among x0 + t (tT). neighboursofthe0,Hilbert b.,lattice-free bodies,emptysimplices… Complexity of “Is a given integer simplex empty ?” .

  11. general factor (gf) Jump systems (js) JZnis ajump system (Bouchet, Cunnigham ’93), if u,vJ and step u+ei from u towards v, either u+ei J, or  step u+ei+ej J from u+ei towards J. Examples: matroid independent sets,bases;{0,ei+ej} Degree sequences of graphs (B.,C.: J1,J2 js  J1+J2 js) Cornuéjols(86):Edmondstypealg fordegreeseqJgen box Lovász(72): Tutte-type, Edmonds-Gallai-type thms for gf Then gf can be pol. reduced to bounds+ parity (S 86) Lovász (95): gen minmax result including J1Jbox Pol red of J1Jgen box to J1Jbox+paritylike for graphs (S 96) gen box :  of 1 dim js SubsetsofTcovered by T-path-packings(Schrijver’s proof of Mader) Jump system intersection

  12. jump systems maxfix cover structure test-sets matroids k-chrom polyhedra b-matchings (multi)flows parity stable sets factors hypergraph matching, coloring Many happy returns of this day MATCHINGS,ALTERNATING PATHS

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