MATCHINGS,ALTERNATING PATHSb-m

atchings

factorsstable sets

maxfix c

over

parity

structure

(multi)flow

s

jump systems

mat

roid

s

hypergraph matching, coloring

polyhedra

k-ch

rom

test-sets

ays to matching generalizations

András Sebő, CNRS, Grenoble (France)

For the 50th birthday of the Hungarian Method

The Fifty Year Old :

x0

Many happy returns of the day

1

-1

G

-1 : if in

Parity of Degrees and Negative Circuits

Idea:1. minimum no negative circuit (Guan 62)

1 : if not inx0

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

x0 V, (u)= min weight of an (x0,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 uvE, and for all

D D : (D) contains 1 negative edge if x0 D 0

x0

0 negative edge if x0 D

Applications: matching structure; Integer packings of cuts, paths (Frank Szigeti, Ageev Kostochka Szigeti, …)

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.

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 seqJgen 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 J1Jbox

Pol red of J1Jgen box to J1Jbox+paritylike 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 PATHSb-m

atchings

factorsstable sets

maxfix c

over

parity

structure

(multi)flow

s

jump systems

mat

roid

s

hypergraph matching, coloring

polyhedra

k-ch

rom

test-sets

Many happy returns of this day