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

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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 !

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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 + -

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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. -- + +

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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’ +

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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

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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 “ “ “

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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 - =

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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 +

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- - + - + - - - 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 +

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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-

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(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,

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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)__ 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
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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).

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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)

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The bipartite or network flow cases The odd components disappear if: l__
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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)

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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}

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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.

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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 11. 2.)minmax

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x0x0

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x0x0 x0x0

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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.

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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

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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, 582-603

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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.

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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

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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’.

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