1. Minmax Relations for Cyclically Ordered Graphs András Sebő, CNRS, Grenoble

2. Algorithms, Polyhedra max 1 T x : x(S)  1,  S stable, x  0, min 1 T x : 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.

3. 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.

4. 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

5. 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-chrom  min  C  C min{k,|V(C)|} c covers ( no loops: max k-chrom  min |X| + k | c |: X  V, c covers V / X not partitioned ! (Thm:S.04)

6. 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.

7. The winding of a cycle or of a set of cycles: C ind(C)=2 clockwise Bessy,Thomassé: invariance of # through opening !

8. COMPATIBILITY A cyclic order is called compatible, if every arc e in a cycle, is also in a cycle C e 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 C e of G-(B/e): ind(C e )=1

9. 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 }

10. 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.

11. 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 => x u +x v  x(C uv )  ind(C uv )=1 Q.E.D. We got back only part of what we have lost: primal is 0-1, and stable using only |S  C|  1  cycle C with ind(C) =1.The rest: Thm: |S  C|  i(C)  cycle C S cyclic stable

12. Proof Algorithm: flow = dual, p(v in ) - p(v out ) =:x v primal lower capacity w v =1 arcs backward e if f(e) > lower capacity v in v out =: S for which: If coherent & strong then 0-1 From this: primal cost = 1 No neg cycle =>  potential 0 -2 0 v out v in 1

13. 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:

14. 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)

15. Thm 1: x(C)  k ind(C)  cycle C, 1  x  0 TDI. max prim=min |X|+  C  C k ind(C) :X  V, 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

16. 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  u has integer primal, dual,  k,l,u P 0-1 & IDP & « kP is box TDI »: upper=lower capacity=w v v in v out arcs backward cost = k v in v out cost = -l v cost = u v Etc, Q.E.D = min{  C  C min{ k ind(C), |C| }: C cover}

17. x(C)  i(C)  cycle C, x  0 min 1 T x cyclic feedback sets: solutions not consec Thm : 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 22 2

18. x(C)  i(C)  cycle C, x  0 cyclic feedback : min 1 T x not cyclic Thm : 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 22 2

19. 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

20. 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 !

21. 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.

22. Application: cyclic colorations 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. |  (u)| < |  (v)| |  (v)| < |  (u)| ≥- (r-1) uv v u 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 !