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

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

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 Def conservative (cons) : no circuit with neg total weight x0 V, l(u)= min weight of an (x0,u) path

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

Various degree constraints and bidirected graphs + path - 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 a c b For bidirected graphs: a c

maxfix covers minimize vV(G) dF(v)2 - const(=|E(G)|) Input: H graph, kIN Task: Find S  V(H) |S|=k that S hits a max number of edges of H. Contains Vertex Cover. 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)F is not optimal better F’ with vV(G) | dF(v) – dF’(v) |  4 14 14 12 Cor : Pol solvable

50 24 4 number of years (edges of L(G) hit): : Many happy returns of the day Aki nem hiszi számoljon utána …

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 a set of chains covering V/X}

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 orthogonal and strong follows: 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

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. improving paths : Def (Graver ‘75, Scarf, Bárány, Lovász, …) A matrix; T is a test-set if for all b and c, Ax  b, x integer has a better solution than x0   also among x0 + t (tT). neighbours of the 0, Hilbert b., lattice-free bodies, empty simplices… Complexity of “Is a given integer simplex empty ?” .

Jump systems (js) JZn is a jump 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): Edmonds type alg for degree seq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 Subsets of T covered by T-path-packings(Schrijver’s proof of Mader) general factor (gf) Jump system intersection

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