1. 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

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

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

4. 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 negative edge if x0 D
Applications: matching structure; Integer packings of cuts, paths (Frank Szigeti, Ageev Kostochka Szigeti, …)

5. 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

6. 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

7. 50 24 4 number of years (edges of L(G) hit): :
Many happy returns 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 a set 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
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

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.
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 ?” .

11. 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

12. 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