Matching and Routing: Structures and Algorithms András Sebő, CNRS, IMAG-Leibniz, Grenoble.
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Transcript of Matching and Routing: Structures and Algorithms András Sebő, CNRS, IMAG-Leibniz, Grenoble.
Reaching with directed paths
X0 V di-cut if x0 X0 and no leaving edge
Proposition: x X0 no (x0,x) path
V0:={x: x reachable from x0}
x
x
V0
x0
Proposition V0 X0 di-cut X0.
Thm : V0 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,bV, ,{-1,1} )
+if x=a=ba b (x) = if x=ab, if x=ba 0 if x{a,b}
bidirected graph: G=(V,E), eE a,b,,: e=ab
bidirected (a,bpath : P E, eP e = abThen b (b) is reachable from a (a).
{ - + + + a - - b
+ -
How do the Paths Go ?If P is an (a,bpath, then there exists anorder {e1,…,ep} of its edges so that for all
i=1,…p {e1,…,ei} is a path
Corollary: v P is reachable from a .
Skew Transitivity Lemma:
If (a,bpath and (b-,cpath, then:
either there is also an (acpath, or both the (a,b-), (b,cpaths exist.
--+ +
We proved in case x0 V+ \ V- , eg, if d(x0)=1
(i) x0 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 ‘’ ‘’ ‘’ ‘’
x0
V+ \ V-- - + - + - - -
x0-
x’0-
V- \ V+
x0’
+
It follows immediately for the general case that:
(i) x0 V+
(ii) V+ V- does not induce any useful edge.
(iii) For a component C of G - V+ V-
either C V+ V- ,
x0 C and 1 useful edge entering C,
x0 C and no “ “ “
or C V+ = C V- = and 0 useful ….
x0‘
x0
Remark: particular bidirected cuts
(i) x0 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 x0-, if
(i) x0 X+
(ii) X:= X+ X- does not induce any useful edge.
(iii) For a component C of G-X
either C X+ X- ,
x0 C and 1 useful edge entering C,
x0 C and no “ “ “
or C X+ = C X- =
What do bi-cuts exclude ?Proposition: If (X+ , X- ) is a bi-cut for x0 ,
then x X+ there is no (x0- , x+) path,
x X- there is no (x0- , x
-) path.
Proof : Choose x V, and P an (x0- , x) path, so
that P is a counterexample, and P min …
X+\X- X-\X+
- - + - + - - -
V+:={vV: (x0- ,v+) path}, V-:={vV: (x0
- ,v -) path}
Proposition V+ X+ , V- X- (X+ , X- ) bi-cut
+ +
- - + - + - - -
Proof : x0 V+ , since is an (x0-,x0
+) path.
If say e=a-b- is useful, a,bV+, let P (x0-,a+) path.
Both e P and e P lead to a contradiction.
Let e=p+r+ (pV- ) 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 +,- reachableC not reachable at all
+
trick: to switch the signs in a point does not
change the accessibility of vV .
Claim : V+r-(C)= V-
r-(C)
Enough: :
x Vr-+
(C), x V+ \ V - xV
–
If an (x0-,x-)-path enters last on e : DONE
If an (x0-,x-)-path enters last on p’r’: contradiction
in either case of the Skew Transitivity Lemma.
p p’
rr’
+ +
+ +
x -+
eV-
(l,u)-factors (l u) + parity constraints in some vertices
Find F E with minimum deficiency, where def :=
{dF(v) - u(v): dF(v) > u(v)} + { l(v) - dF(v): dF(v) < l(v)}
x0-
dF < l : - dF > u : +
F
Fu< dF < l ,
u dF < l ,
u< dF l ,
Lemma: If one of the following statements hold, then F does not have minimum deficiency:
a. x0 V+ V-
b. xV+: dF(x)<u(x), and not dF(x)+1=u(x)=l(x)
c. xV- : dF(x)> l(x), and not dF(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
x0
V+ \ V- = {x0 } TUTTE SET
max cardinality (l,u)-factor: min deficiency u-factor
V-
V- is a stable set (avoided with +,+ loops).
orientationsAgain: u, l, (+ parity) : l din u
x0-
dout < l : + dout > u : -u< dout < l ,
u dout < l ,
u< dout 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) (aA) demands t(h) (hB)
number of ressources in hour h. Feasible Y B
uU 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: {xV(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 x0
- to xA there is no red-red alternating path
- to xC, 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 AC
(where A is a Tutte-set and C the of even) ,
because there is no red-red path to AC , and
A(G) C(G) is the set of all such vertices.
Conclusion about bidirected graphsand 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
Theorem: G bipartite, w: E {-1,1}, cons.,x0V
Then | (u) – (v) | = 1 for all uvE, and for all
D D : (D) contains 1 negative edge if x0 D 0 0 negative edge if x0 DProof (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 x0 ), if (i) (x0)=0
(ii) | (u) – (v) | = 1 for all uvE,
(iii) for all D D() : (D) contains
1 negative edge if x0 D 0 0 negative edge if x0 D
Algorithm:
INPUT: G,w,x0
TASK: Find the distances from x0
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
ApplicationsMethod 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
Answers:The algorithm that finds distances from x0 is
polynomial: O(n4). 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 + b1/2 +...+ bp/2
Mader-Structure (with L. Szegő, ‘03)For edge-Mader (simpler):
Xi :={v V: opt T-path
packing+path only from ti}
Follows from matroid matching or the ellipsoid method, but NO ‘NORMAL’ ALG !
t2t1 tk
…X1 X2 Xk
…
…
t2t1 tk
…X1 X2 Xk
…
…
+ path from ij no +path
Jump systemsA jump system (Bouchet, Cunnigham ’93) is a set of integer vectors J so that for every u,v J and
step u+ei from u towards v, either u+ei J or
there exists a step u+ei+ej J from u+ei 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’.