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

The Fifty Year Old : x0x0 Many happy returns of the day 1 G

-1 : if in Parity of Degrees and Negative Circuits Idea:1. minimum  no negative circuit (Guan 62) 1 : if not in x0x0 odd degree subgraphs: Edmonds, Johnson (73) minmax, alg: EJ, Barahona, Korach; sequence of sharper thms: Lovász (76), Seymour (81), Frank, Tardos (84), … Edmonds (65): Chinese Postman through matchings Def conservative (cons) : no circuit with neg total weight x 0  V, (u)= min weight of an (x 0,u) path 2. identify vertices that are at distance  0, induction, S

Thm:(S 84) G bipartite, w: E  {-1,1}, conservative Then |  (u) –  (v) | = 1 for all uv  E, and for all D  D :  (D) contains 1 negative edge if x 0  D 0 x0x0 0 negative edge if x 0  D Applications: matching structure; Integer packings of cuts, paths (Frank Szigeti, Ageev Kostochka Szigeti, …) Thm: cons, bipartite, all distances <0  negative forest x0x D:

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 , b   & ( b - , c   path, then: either  ( a   c   path, or  both (a , b -  ) & ( b , c   paths.  Tutte & Edmonds-Gallai type thms+‘structure algorithms’ for lower,upper bounds and parity, including digraphs. Various degree constraints and bidirected graphs path a b c For bidirected graphs: a c b

maxfix covers 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) d F (v) 2 - const(=|E(G)|) Thm:(Apollonio, S.’04)F is not optimal  better F’ with  v  V(G) | d F (v) – d F’ (v) |  4 Cor : Pol solvable

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

Independent sets in matroids in graphs (stable set) 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 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-set if for all b and c, Ax  b, x integer has a better solution than x 0   also among x 0 + t (t  T). neighbours of the 0, Hilbert b., lattice-free bodies, empty simplices… Complexity of “Is a given integer simplex empty ?”. improving paths :

Jump systems (js) J  Z n is a jump system (Bouchet, Cunnigham ’93), if  u,v  J and step u+e i from u towards v, either u+e i  J, or  step u+e i +e j  J from u+e i towards J. Examples: matroid independent sets, bases; {0,e i +e j } Degree sequences of graphs (B.,C.: J 1,J 2 js  J 1 +J 2 js) Cornuéjols(86): Edmonds type alg for degree seq  J gen 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 J 1  J box Pol red of J 1  J gen box to J 1  J box+parity like for graphs (S 96) general factor (gf) gen box :  of 1 dim js Subsets of T covered by T-path-packings(Schrijver’s proof of Mader) Jump system intersection

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