MATCHINGS,ALTERNATING b - m a t c h i n g s factors s t a b l e s e t s maxfix cover p a r i t y struct ure (m u l t i) f lo ws jump systems m at r oi ds hypergra ph matc hing, coloring polyhedra k-chrom test-se ays to matching generalizations András Sebő, CNRS, Grenoble (France) For the 50th birthday of the Hungarian Method
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Thm: cons, bipartite, all distances <0 negative forest
x0
1
0
-1
-2
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,calt paths (a,c)
Broken Transitivity:(S ’86) If (a,b&(b-,cpath, then: either (acpath, or both (a,b-) & (b,cpaths.
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 coversInput: H graph, kIN
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 vV(G) dF(v)2 - const(=|E(G)|)
Thm:(Apollonio, S.’04)F is not optimal better F’
with vV(G) | dF(v) – dF’(v) | 4
Cor : Pol solvable
14
1214
number of years (edges of L(G) hit): : 042450
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| : XV, c is a set of chains covering V/X}
Conjecture of Linial : max k-chrom
min { |X| + k |P|: XV,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, pathcycle, partitioncover
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 x0 also among x0 + t (tT).
neighbours of the 0, Hilbert b., lattice-free bodies, empty simplices…
Complexity of “Is a given integer simplex empty ?” .
improving paths :
Jump systems (js) JZn 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.