Minmax Relations for Cyclically Ordered Graphs

András Sebő,

CNRS, Grenoble

Algorithms, Polyhedra

max 1Tx : x(S) 1, S stable, x 0,

min 1Tx : x(C) 1, dir.cycle C, x 0 integer

Solve them ? Yes . But first, put …… a cyclic order on the vertices

-Conj of Gallai (Bessy,Thomassé’64)-Cleaning the notions in it-New results on graphs without cyc. ord.

G=(V,A) digraph, cover: family F, U F = V

Acyclic iff order so that every arc is forward

Dilworth : G acyclic, transitive

max stable = min cover by paths(cliques)

Green-Kleitman : G acyclic, transitive

max k-chrom = min P P min{ k,|V(P)| }

on covers by paths.

Rédei ‘34: tournament Hamiltonian path

Camion ‘59: strong tournament Ham cycle

Gallai-Roy ‘68: digraph (G)-vertex-path.

Bondy ‘76 : strong ‘’ (G)-vertex-cycle.

? methods for ‘big enough particular cases’ of

stable sets, path partitions, cycle covers,

feedback (arc-)sets, etc. by putting on …

… cyclic orders

Gallai-Milgram (1960): G graph. The vertices of G can be partitioned into at most (G) paths.

Ex-conjecture of Gallai (1962):G strong(ly conn) graph the vertices can be covered by (G) cycles

Thm: Bessy,Thomassé (2003)

Conjecture of Linial :

max k-chrom min P P min{ k, |V(P)|} on path partitions.

Whose ex-conjecture? In a strong graph with loops:

max k-chrommin C C min{k,|V(C)|}c covers

(no loops:max k-chrom min |X| + k | c|: XV,c covers V / X

not partitioned !

(Thm:S.04)

structural versions (complementary slackness):

Gallai-Milgram : For each optimal path partition there exists a stable set with one vertex on each path.

BT: G strong => There exists a circuit cover and a

stable set with one vertex in each circuit. Conjecture of Berge: For each path partition minimizing

P P min{ |V(P)|, k } a k-colored subgraph where each path meats ’’ ’’ colors.

S.’04 : G strong. There exists a circuit cover and a k-colored subgraph so that each circuit of the cover

meets C C min{ |V(C)|, k } colors.

The winding of a cycle or of a set of cycles:

C

ind(C)=2clockwise

Bessy,Thomassé: invariance of # through opening !

COMPATIBILITYA cyclic order is called compatible, if every arc e

in a cycle, is also in a cycle Ce of winding 1, and the other arcs are forward arcs.

generalizes acyclic: adjacent => forward path

Thm (Bessy, Thomassé 2002) for every digraph

Proof: F (incl-wise) min FAS s.t (|F C|:C cycle) min

G-F acyclic, compatible order

e B(ackward arcs) in some shift

F and B are min feedback arc-sets

cycle Ce of G-(B/e): ind(Ce)=1

Cyclic stability

C

clockwise

Bessy,Thomassé: invariance of the index through interchanging nonadjacent consecutive points !

S cyclic stable, if stable and interval in equivalent order.

equivalent

Thm (Bessy, Thomassé 2003) : max cyclic stable=

min { ind(C ) , C cycle cover }

I. x(C) ind(C) cycle C, x 0 (BT)

Thm : If the optimum is finite, then integer primal and dual optimum and in polytime.

Easy from Bessy-Thomassé through replication. Easy from mincost flows as well. But we lost something: the primal has no meaning !

With an additional combinatorial lemma get BT.

Get back the lost properties !

Corollary (>~Gallai’s conj): G strong, compatible

=>S stable & C cover such that |S| = ind(C) (|C|)

Proof: uv E => xu+xv x(Cuv) ind(Cuv)=1 Q.E.D.

We got back only part of what we have lost: primal

is 0-1, and stable using only

|SC| 1 cycle C with ind(C) =1 .The rest:

Thm: |SC| i(C) cycle C <=> S cyclic stable

Proof

Algorithm: flow = dual, p(vin) - p(vout) =:xv primal

lower capacity wv =1

arcsbackward

e

if f(e) > lower capacityvin vout

=: S for which:

If coherent & strong then 0-1

From this: primal

cost = 1

No neg cycle

=> potential

0

-1

-2-1 0

vout

vin

1

II. x(S) 1, S cyclic stable, x 0 (antiBT)

Place the vertices on a cycle of length q, following an equivalent cyclic order, so that the endpoints of arcs are at ‘distance’ 1

min q ?

|C| ind(C)q

G=(V,A) q=13.28

Thm (BT 2003) : min q = max |C| / ind(C)

Cyclic q-coloring:

Proof : r := max |C| / ind(C) Define arc-weights: -1 on forward arcs r-1 on backward arcs –

-|C|+r ind(C) 0 C :No negative cycles potentials … form a coloration + … Q.E.D.

x(C) ind(C) cycle C, x 0 (BT)

x(S) 1 cyclic stable S, x 0 (antiBT)

dual: colorations with cyclic stable sets

Thm: Antiblocking pair (with four proofs)

Thm 1: x(C) k ind(C) cycle C, 1 x 0 TDI.

max prim=min |X|+C C k ind(C) :XV, C covers V\X

=max union of k cyclic stable sets

Thm 2: (BT) has the Integer Decomp Property, i.e.

w k(BT) int =>w= sum of k integer points in (BT)

Proof:*circ = max |C|/i(C) *, so = everywhere!

=> * = circ = .

w (kBT)=>w/k(BT), that is, max w(C)/ind(C) k.

By the coloring theorem (after replication) :

w is the sum of k cyclic stable sets. Q.E.D.

I

Thm: max cyc k-col = min k i(C ) + |not covered|

Proof: kP {x: 0 x 1} = conv {cycl k-col} (IDP) r Formula because of box TDI.

Proof: x(C) k i(C) l x uhas integer primal, dual, k,l,u

P 0-1 & IDP & « kP is box TDI »:

upper=lower capacity=wv

vin vout arcsbackward

cost = k

vin vout

cost = -lv

cost = uv

Etc, Q.E.D

= min{ C C min{ k ind(C) , |C| }: C cover}

x(C) i(C) cycle C, x 0 min 1Txcyclic feedback sets: solutions

not consecThm : integer primal and dual, and in polytime.

upper capacity w(v) , costs = -1 , …

feedback cyclic feedback feedback arc cyclic FAS backward arcs

III. (blocking)

2 2 2 2

x(C) i(C) cycle C, x 0

cyclic feedback : min 1Tx

not cyclicThm : integer primal and dual, and in polytime.

upper capacity w(v) , costs = -1 , …

feedback cyclic feedback feedback arc cyclic FAS backward arcs

Attila Bernáth: ‘’ = ‘’

III. (blocking)

2 2 2 2

Summarizing « Good characterization », and pol algs for

the following variants: choose btw

1. Antiblocking (containing max cycl stable) blocking (containing min cycl feedback), etc

2. One of the pairs

3. k=1 or k>1

4. Vertex or arc version

5. Arbitrary or transitive

The poset of orders (Charbit, S.)

cyclic order 1 ≤ cyclic order 2 (def)

ind 1 (C) ≤ ind 2 (C) for every circuit C.

Exercises: 1. po well-defined on equiv classes

2. Minimal elements: compatible classes

3. The winding is invariant on any undirected cycle as well – through the operations !

Characterizing Equivalence (Charbit, S.)

Problems:

1.* If ind 1 (C) = ind 2 (C) for every undirected cycle, then order 1 ~ order 2 .

2. If C is an arbitrary circuit and B(ack arcs) Then C

T B= |C| - 2 ind.

3. Every C is a linear combination of incidence vectors of directed circuits .

Thm: If G strongly connected, then order 1 ~ order 2 iff ind 1 (C) = ind 2 (C) for every cycle C.

Application: cyclic colorationsr := max |C| / ind(C) Define arc-weights:

-1 on forward arcs, r-1 on backward arcs – -|C|+r ind(C) 0 C : no negative cycles potentials … form a coloration + … Q.E.D.

|(u)| < |(v)| |(v)| < |(u)|

≥- (r-1)

u v v u-1 r-1

|(v)| =p(v) r + q(v) uv arc: |p(u)-p(v)| ≤ 1

Fact: {uv arc: |p(u)-p(v)| = 1} = reversed arcs, cut

replace p by q !