Minmax Relations for Cyclically Ordered Graphs

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

-ConjofGallai (Bessy,Thomassé’64)-Cleaning the notions in it-Newresultsongraphs without cyc. ord. Algorithms, Polyhedra max 1Tx : x(S)  1, S stable,x 0, min 1Tx : x(C)  1, dir.cycle C,x 0 integer Solve them ? But first, put … Yes . … a cyclic order on the vertices

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 Pmin{ 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-chromminC C min{k,|V(C)|}ccovers (noloops:max k-chrom min|X|+k |c|:XV,ccovers 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: ind(C)=2 clockwise C Bessy,Thomassé: invariance of # through opening !

COMPATIBILITY A 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

clockwise Cyclic stability C Bessy,Thomassé: invariance of the index through interchanging nonadjacent consecutive points ! equivalent S cyclic stable, if stable and interval in equivalent order. 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 |SC|  1 cycle C with ind(C) =1 .The rest: Thm: |SC|  i(C)  cycle C <=> S cyclic stable

e backward arcs vin vout if f(e) > lower capacity 0 -1 vin -2 -1 0 1 vout Proof Algorithm: flow = dual, p(vin) - p(vout) =:xv primal cost = 1 lower capacity wv =1 If coherent & strong then 0-1 No neg cycle =>  potential =: S for which: From this: primal

II. x(S)  1,S cyclic stable,x 0 (antiBT) Cyclic q-coloring: G=(V,A) q=13.28 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 Thm (BT 2003) : min q = max |C| / ind(C)

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 setsThm: Antiblocking pair (with four proofs)

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

backward cost = -lv arcs vin vout vin vout upper=lower capacity=wv cost = k cost = uv Etc, Q.E.D Proof:x(C)  k i(C) l  x uhas integer primal, dual, k,l,u P 0-1 & IDP & « kP is box TDI »: 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. = min{C  Cmin{ k ind(C) , |C| }: C cover}

2 2 2 2 III. (blocking) x(C)  i(C)  cycle C, x 0 min 1Tx 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

2 2 2 2 III. (blocking) x(C)  i(C)  cycle C, x 0 cyclic feedback : min 1Tx 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: ‘’ = ‘’

Summarizing « Good characterization », and pol algsfor the following variants: choose btw • Antiblocking (containing max cycl stable) blocking (containing min cycl feedback), etc 2. One of the pairs • k=1 or k>1 • Vertex or arc version • 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 • 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 . • If C is an arbitrary circuit and B(ack arcs) Then CT 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.

≥- (r-1) -1 u v r-1 u v 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. |(v)| < |(u)| |(u)| < |(v)| uv arc: |p(u)-p(v)| ≤ 1 |(v)| =p(v)r + q(v) replace p by q ! Fact: {uv arc: |p(u)-p(v)| = 1} = reversed arcs, cut

Minmax Relations for Cyclically Ordered Graphs