Matching and Routing: Structures and Algorithms András Sebő, CNRS, IMAG-Leibniz, Grenoble

Reaching with directed paths X 0  V di-cut if x 0  X 0 and no leaving edge Proposition: x  X 0  no (x 0,x) path V 0 :={x: x reachable from x 0 } x x V0V0 x0x0 Proposition  V 0  X 0  di-cut X 0. Thm : V 0 is a di-cut. The Most Exclusive !

Bidirected Graphs Defined by Edmonds, Johnson (‘70) to handle a general class of integer linear programs containing matchings and their generalizations and tractable with the same methods. a  b   {0,+1, -1, +2, -2} V, (a,b  V, ,  {-1,1} )  +  if x=a=b a  b  (x) =  if x=a  b,  if x=b  a 0 if x  {a,b} bidirected graph: G=(V,E), e  E    a,b, ,  : e=a  b  bidirected ( a , b   path : P  E,  e  P e = a  b   Then b  (b) is reachable from a  (a). { a - - b + -

How do the Paths Go ? If P is an ( a , b   path, then there exists an order {e 1,…,e p } of its edges so that for all i=1,…p {e 1,…,e i } is a path Corollary:  v  P is reachable from a . Skew Transitivity Lemma: If  ( a , b   path and ( b - , c   path, then: either there is also an ( a   c   path, or both the (a , b -  ), ( b , c   paths exist

We proved in case x 0  V + \ V -, eg, if d(x 0 )=1 (i)x 0  V + \ V - (ii) V +  V - does not induce any useful edge. (iii) For a component C of G - V +  V - either C  V +  V - & 1 useful edge entering C. or C  V + = C  V - =  & 0 ‘’ ‘’ ‘’ ‘’ x0x0 V + \ V x0-x0- x’ 0 - V - \ V + x0’x0’ +

It follows immediately for the general case that: (i)x 0  V + (ii) V +  V - does not induce any useful edge. (iii) For a component C of G - V +  V - either C  V +  V -, x 0  C and  1 useful edge entering C, x 0  C and no “ “ “ or C  V + = C  V - =  and 0 useful …. x0‘x0‘ x0x0

Remark: particular bidirected cuts (i) x 0  X + \ X - (ii) X:= X +  X - does not induce any useful edge. (iii) For a component C of G-X either C  X +  X - &  1 useful edge entering C. or C  X + = C  X - =  and no “ “ “

Bidirected Cuts in General (X +, X - ), X +, X -  V is a bi-cut for x 0 -, if (i) x 0  X + (ii) X:= X +  X - does not induce any useful edge. (iii) For a component C of G-X either C  X +  X -, x 0  C and  1 useful edge entering C, x 0  C and no “ “ “ or C  X + = C  X - = 

What do bi-cuts exclude ? Proposition: If (X +, X - ) is a bi-cut for x 0, then x  X +  there is no (x 0 -, x + ) path, x  X -  there is no (x 0 -, x - ) path. Proof : Choose x  V, and P an (x 0 -, x) path, so that P is a counterexample, and P min … X + \X - X - \X V + :={v  V:  (x 0 -,v + ) path}, V - :={v  V:  (x 0 -,v - ) path} Proposition  V +  X +, V -  X -  ( X +, X - ) bi-cut +

Proof : x 0  V +, since  is an (x 0 -,x 0 + ) path. If say e=a - b - is useful, a,b  V +, let P (x 0 -,a + ) path. Both e  P and e  P lead to a contradiction. Let e=p + r + (p  V - ) be a useful edge entering C. We show e is the only one and C  V +  V -. Theorem(bidirected structure) (V +, V - ) is a bi-cut. + What does a component of G - V +  V - look like ? V + \ V - V - \ V + entirely +,- reachable C not reachable at all +

trick: to switch the signs in a point does not change the accessibility of v  V. Claim : V + r - (C)= V - r - (C) Enough:  : x  V r - + (C), x  V + \ V -  x  V – If an (x 0 -,x - )-path enters last on e : DONE If an (x 0 -,x - )-path enters last on p’r’: contradiction in either case of the Skew Transitivity Lemma. p p’ r r’ x - + e V-V-

(l,u)-factors (l  u) + parity constraints in some vertices Find F  E with minimum deficiency, where def :=  {d F (v) - u(v): d F (v) > u(v)} +  { l(v) - d F (v): d F (v) < l(v)} x0-x0- d F < l : - d F > u : +  F  F F u< d F < l,  u  d F < l,  u< d F  l, 

Lemma: If one of the following statements hold, then F does not have minimum deficiency: a. x 0  V +  V - b.  x  V + : d F (x)<u(x), and not d F (x)+1=u(x)=l(x) c.  x  V - : d F (x)> l(x), and not d F (x) -1=u(x)=l(x) Lovász’s structure theorem (containing Tutte’s existence theorem, Lovász’s minmax theorem for the deficiency): universal U, L is a ‘Tutte-set’ V + \ V - V - \ V + OPT: U: oversat L: undersat u=l and+-1 (+ odd) feasible

Max Cardinality Factors, Matchings x0x0 V + \ V - = {x 0 }  TUTTE SET max cardinality (l,u)-factor: min deficiency u-factor  V- V- V - is a stable set (avoided with +,+ loops).

orientations Again: u, l, (+ parity) : l  d in  u x0-x0- d out < l : + d out > u : - u< d out < l,  u  d out < l,  u< d out  l,  Minmax theorems (Frank, S. Tardos) + structure theorem (best certificate)

The bipartite or network flow cases The odd components disappear if: l<u, or l=u=even, or the graph is bipartite. In this case bipartite matchings or network flows are sufficient. Direct algorithms based on alternating paths can mimic mfmc, eg: Thm:G=(A,B,E) bipartite, r(a) (a  A) demands t(h) (h  B) number of ressources in hour h. Feasible   Y  B  u  U r(u,Y)  t(Y), where r(u,Y):=max{r(u)-|N(u)\Y|,0}. The general case is ‘1 floor higher’: ellipsoids, or blossoms, or structure algorithms …

Solution of Exercises 9,10 C(G)={x  V(G): no red-red & no red-blue} A(G)={x  V(G): no red-red &  red-blue} Solution Exercise 9: V \ C(G) :  red-red or  red-blue  red-red: {x  V(G):  M x is uncovered} =: D(G) additional red-blue : neighbors of D (D  neighbors of D) = V \ C(G) =A(G)

Conclusion about bidirected graphs First, in terms of matchings: Exercise 8: If A is a Tutte set, then from x 0 - to x  A there is no red-red alternating path - to x  C, C even comp of G-A no red-red no red-blue C(G)={x  V(G): no red-red & no red-blue} A(G)={x  V(G): no red-red &  red-blue}

Confusing: A(G) itself is not maximal among Tutte-sets A. (The reason is that it corresponds to V + \ V - and not V +.) Gallai-Edmonds : A(G) is a Tutte-set. Proof : (V +, V - ) is a bidirected cut. S. of Exercise 10: C(G) is max among all C: triv A(G)  C(G) is max among all A  C (where A is a Tutte-set and C the  of even), because there is no red-red path to A  C, and A(G)  C(G) is the set of all such vertices.

Conclusion about bidirected graphs and the way they can be applied Thm : (V +, V - ) is a bidirected cut. (most excl) This certifies that other vertices are not reachable. Algorithmic proof: (ii) is not satisfied: useful edge induced by X +  X - (iii) ‘’ ‘’ ‘’ : two useful edges leaving a comp: new path by Skew Transitivity. Application: existence or max of subgraphs with constraints on degrees (bounds, parity, Orientation, etc.) Proof of optimality: 1.) Ex )minmax

x0x0

x0x0 x0x0

Theorem: G bipartite, w: E  {-1,1}, cons.,x 0  V Then |  (u) –  (v) | = 1 for all uv  E, and for all D  D :  (D) contains 1 negative edge if x 0  D 0 0 negative edge if x 0  D Proof (Sketch): Induction with resp. to |V|. If (b)=min { (x) : x  V} then Exercise 21 proves the statement for {b}  D. Contract  (b). The graph remains cons, since if not, there is a 0 circuit through b with b, Apply Exercise 19, contradicting Exercise 21. ~Similarly, the distances do not change. Induction.

 integer function is a potential (with center x 0 ), if (i)  (x 0 )=0 (ii) |  (u) –  (v) | = 1 for all uv  E, (iii) for all D  D (  ) :  (D) contains 1 negative edge if x 0  D 0 0 negative edge if x 0  D Algorithm: INPUT: G,w,x 0 TASK: Find the distances from x 0 While the distances do not form a potential find a better path or a negative circuit.(Ref: Finding …)  conservat. and upper bound for the distances

Applications Method for solving (in parallel + structure of ) : - Min (weighted) matchings - planar disjoint paths (+ planar max cut, via holes, air transport, Ising model) - the Chinese Postman problem - Minimizing T-joins, T-cuts (including min paths) - weighted algorithms for these, since they can mimic the +-1 bipartite case. ref: A.S., Potentials in Undirected Graphs and Planar multiflows, SIAM J. Computing, March 1997,

Answers: The algorithm that finds distances from x 0 is polynomial: O(n 4 ). It does not know about ‘efficient labelling’. Advantages: - it finds a potential, and The Best one, which helps in some applications to integer mflows. - the ‘classical’ algorithm reduces eg cardinality postman to weighted matchings. - helps in ‘reading’ matching th, and leads to generalizing it to conservative graphs (postman, planar disjoint paths, etc.) – used eg for integer multiflows.

G=(V,E) graph, T  V, k:=|T|. Ref:Schrijver Comb. Opt. T-path: path with different endpoints in T. edge-Mader : max = Min +  b 1 /2   b p /2  Mader-Structure (with L. Szegő, ‘03) For edge-Mader (simpler): X i :={v  V:  opt T-path packing+path only from t i } Follows from matroid matching or the ellipsoid method, but NO ‘NORMAL’ ALG ! t2t2 t1t1 tktk … X1X1 X2X2 XkXk … … t2t2 t1t1 tktk … X1X1 X2X2 XkXk … … + path from i  j no +path

Jump systems A jump system (Bouchet, Cunnigham ’93) is a set of integer vectors J so that for every u,v  J and step u+e i from u towards v, either u+e i  J or there exists a step u+e i +e j  J from u+e i towards J. Examples: matroid independent sets, bases. Degree sequences (B.C.): you have a proof according to Exercise 12 ! The covered vertices of T-path packings (with M. Sadli using Schrijver’s (cf The Book) simple proof. Insight and generalization of results on graph factors:jump system intersection theorems. There are ‘holes’.