Prsentation PowerPoint

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CONCEPTION DE RESEAUX AVEC CONTRAINTES DE BORNES
A. Ridha Mahjoub LIMOS, Université Blaise Pascal,
Clermont-Ferrand, France
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Introduction
Survivability The ability to restore network
service in the event of a catastrophic failure.
Goal Satisfy some connectivity requirements in
the network.
Motivation Design of optical communication
networks.
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Introduction
Bounded lengths
Motivation
To have effective rerouting.
Two rerouting strategies
Local rerouting
Each edge must belong to a bounded cycle
(ring). SONET/SDH networks
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Introduction
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Introduction
End-to-end rerouting
the paths between the terminals should not exceed
a certain length (a certain number of hops)
(hop-constrained paths). ATM networks, INTERNET
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Contents 1. Design of survivable networks with
bounded rings 1.1) Node case 1.2) Edge case
2. Design of survivable networks with bounded
paths 2.1) A general model 2.2) Special
cases and related problems 3. Formulation for L
3 4. The 2-edge connected hop-constrained
network design problem 4.1.
Complexity 4.2. Polyhedral results and
BranchCut algorithm 4.4. Formulation for
L4 5. Open questions and concluding remarks
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1.Bounded rings 1.1. Node case
1. Bounded rings
1.1. 2-node connected graphs Fortz, Labbé,
Maffioli (1999) Fortz, Labbé (2002) The
problem Given a graph G(V,E) with weights
and lenghts associated with the edges, and a
constant B, determine a minimum 2- node
connected spanning subgraph such that each edge
belongs to a cycle of length no more than B.
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1.Bounded rings 1.1. Node case
Extended formulations Valid inequalities
Separation algorithms Lower bounds on the
optimal value Cutting plane algorithms
Cyclomatic inequalities (unitary lengths case)
Let (V1,,Vp) be a partition of V. Then the
inequality
is valid for the problem. ?(V1,,Vp) is the set
of edges between the elements of the partition.
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1.Bounded rings 1.1. Node case
To separate the cyclomatic inequalities
approximatly Fortz, Labbé and Maffioli
considered the inequalities x(?(V1,...,Vp))
a(p-1), with
where n is the number of nodes in the garph.
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1.Bounded rings 1.1. Node case
Consider the constraints x(?(V1,...,Vp))
p-1. These are called partition inequalities.
They arise as valid inequalities in many
connectivity problems. They are valid for the
problem, when considered on G\v, v?V. The
separation problem for these inequalities reduce
to E min cut problems Cunningham (1985) .
It can also be reduced toV min cut problems
Barahona (1992).
Both algorithms provide the most violated
inequality if there is any.
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1.Bounded rings 1.2. Edge case
1.2. 2-edge connected graphs Fortz,
M., McCormick, Pesneau (2003)
The problem Given a graph G(V,E) with
weights on the edges, and an integer B,
determine a minimum 2-edge connected spanning
subgraph such that each edge belongs to a cycle
of length no more than B.
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1.Bounded rings 1.2. Edge case
Formulation
Valid inequalities
d(W) is called a cut
cut inequalities
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1.Bounded rings 1.2. Edge case
Let p(V0,V1,...,Vp) be a partition of V such
that pB.

Let e ? ? (V0,Vp)
cycle inequalities
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1.Bounded rings 1.2. Edge case
Le problème est équivalent au programme
min
Subject to
x(d (W)) 2 for all ? (W)
for all partition
and e?
0 x(e) 1 for all e ? E,
x(e)?0,1 for all e ? E.
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1.Bounded rings 1.2. Edge case
If we add the constraints
x(dG-v(W)) 1, for all W?
V\v, v?V
we obtain a formulation for the 2-node case.
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1.Bounded rings 1.2. Edge case
Separation of cycle inequalities
- If the solution is in 0-1, the separation can
be done easily
The minimum L-st-path cut problem Let G(V,E) be
a graph. Let s,t ?V and L a fixed integer. We
call L-st-path cut any edge set C that intersects
every st-path of length L.
Given weigts on the edges,the minimum st-L-path
cut problem is to find an L-st-path cut of
minimum weight.
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1.Bounded rings 1.2. Edge case
Lemma The separation problem for cycle
inequalities reduces to the minimum
(B-1)-st-path cut problem.
Let est an edge of G. Let C be a (B-1)-st-path
cut. We have that G\C does not contain a cycle
of lenght B. If x is a solution and C is
minimum (B-1)-st-path cut, then - if x(C) lt
x(e), then there is a violated cycle
inequality. - if not, then there is no violated
cycle inequalities.
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1.Bounded rings 1.2. Edge case
Solving the minimum st-L-path cut problem when L
3
L 2
It suffices to calculate a min cut separating s
and t in the graph Induced by the st-paths of
length 2.
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1.Bounded rings 1.2. Edge case
L3
We may suppose each node is adjacent to s or t,
and no edges between s and t.
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1.Bounded rings 1.2. Edge case
L3
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1.Bounded rings 1.2. Edge case
Theorem The minimum st-L-path cut problem for
L3 can be solved in polynomial time.
Corollary The separation problem for the cycle
inequalities for B4 can be solved in polynomial
time.
Theorem (Baier, Erlobach, Hall, Schilling,
Skutella (2006)) The minimum st-L-path cut
problem is NP-hard for L4. (Reduction from
the vertex cover problem)
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2. Bounded paths 2.1. General model
2. Bounded paths
2.1. A general model
Given a graph with weights on the
edges, a set D of terminal- pairs
(origine-destinations), two intgers K, L, find a
minimum weight subgraph such that
between each pair of terminals in D
there are at least K edge-disjoint paths of
length (in number of edges (hops)) no
more than L.
The hop-constrained network design problem (HCNDP)
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2. Bounded paths 2.1. General model
The HCNDP is NP-hard in general NP-hard even
for L2 (Dahl (1998))
Polynomially solvable when D1
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2. Bounded paths 2.2. Special cases
2.2. Special cases and related problems
2.2.1. D1, K1, L fixed
The minimum hop-constrained path problem
Description of the associated polyhedron for L 3.
Dahl (1999)
Formulation in the natural space of variables
Valid inequalities Description of the associated
polytope when L2,3.
Dahl Gouveia (2004)
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2. Bounded paths 2.2. Special cases
Description of the assiciated polyhedron for all
L. Nguyen (2003)
Description of the polyhedron of the directed
st-walks having exactly L arcs. Coullard,
Gamble, Liu (1994)
Description of the polytope of the directed
st-walks having no more than L4 arcs. Extended
formulation for the underlaying problem
Dahl, Foldnes, Gouveia (2004)
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2. Bounded paths 2.2. Special cases
2.2.2. K1, L fixed, D is rooted
The minimum hop constrained spanning tree
problem Determine a minimum spanning tree such
that the number of links between a root node and
any node in the tree does not exceed a bound L.
(NP-hard (even for L2))
Multicommodity flow formulations Hop-indexed
formulation Lagrangean relaxations
Gouveia (1996,1998)
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2. Bounded paths 2.2. Special cases
Other Lagrangean relaxations
Gouveia Requejo (2001)
Description of the associated polytope on a wheel
when L2 Dahl (1998)
Minimum spanning trees with bounded
diameter Integer programming formulation.
Gouveia Magnanti (2000) other modeling
approach when the diameter is odd.
Gouveia, Magnanti Requejo (2004)
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2. Bounded paths 2.2. Special cases
2.2.3. K1 (and L arbitrary)
Extended formulation, Lagrangean relaxation
Balakrishnan, Altinkemer (1992)
Multicommodity flow formulation and
heuristics Pirkul, Sony (2003)
2.2.4. K1, L2
Formulation of the problem in the natural space
of variables Valid inequalities Greedy
approximation algorithms Cutting plane algorithm
Dahl, Johannessen (2004)
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2. Bounded paths 2.2. Special cases
Length constrained 2-connected graphs
Ben Ameur (1998, 2000)
Classes of length constrained 2-connected
graphs Lower bounds on the number of edges
Valid inequalities for the 2-connected polytope
with length constraints
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3. Formulation, L3 3.1. Valid
inequalities
3. Formulation for L 3
3.1. Valid inequalities
s
t
st-cut inequalities
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3. Formulation, L3 3.1. Valid
inequalities
s
t
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3. Formulation, L3 3.1. Valid
inequalities

s
t
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3. Formulation, L3 3.1. Valid
inequalities
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3. Formulation, L3 3.1. Valid
inequalities
If at least K paths are required between s and
t, then
x(T) K
is valid for the corresponding polytope.
The separation problem for the L-st-path cut
inequalities can be solved in polynomial time,
if L 3.
Fortz, M., McCormick, Pesneau (2006)
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3. Formulation, L3 3.1. Valid
inequalities
Theorem (Huygens, M., Pesneau (2004)) For L3,
the HCNDP is equivalent to the following integer
program
x(T) K for all L-path-cut T,
for all (s,t) ? D
The linear relaxation of this program, when L3,
can be solved in polynomial time by the
ellipsoid method.
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3. Formulation, L3 3.1. Valid
inequalities
Remark
The formulation given above is not valid for
L4.
L4
s
t
Further inequalities are needed to formulate the
problem for L4
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4. The 2-edge case 4.1. Complexity
4. The Two edge connected hop-constrained
network design problem (THNDP)
That is the case when K 2
4.1. Complexity
The THNDP is NP-hard in general (the 2-edge
connected subgraph problem is a special case).
Even more
Theorem The THNDP is NP-hard when - D is
rooted - L2, and fixed - all edge weights are
1. Huygens, Labbé, M., Pesneau
(2005)
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4. The 2-edge case 4.1. Complexity
Proof (Outline)
Reduction from the dominating set problem
L3
G(V,E)
G(V,E)
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4. The 2-edge case 4.1. Complexity

• Lemma A minimum cardinality solution S to the
  rooted THNDP
• in G, w.r.t. s and the nodes of V2, can be
  chosen so that
• S contains all the paths between s and V2,
• S contains exactly V paths between V1 and V2
  that cover
• all the nodes of V2.

Thus the rooted THNDP in G reduces to finding a
minimum cardianlity subset of V1 that covers all
the nodes of V2. This subset corresponds to a
dominating set in G.
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4. The 2-edge case 4.1. Complexity
Proof (Outline)
Reduction from the dominating set problem
L3
G(V,E)
G(V,E)
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4. The 2-edge case 4.1. Complexity
However, If the graph is complete and all edge
weights are equal to 1, the rooted THNDP can be
solved in polynomial time for evey fixed
L2. (The algorithm is linear.)
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4. The 2-edge case 4.2. Polyhedral
results
4.2. Polyhedral results
4.2.1. THNDP polytope when L2, 3, D1.
Theorem (Huygens, M., Pesneau (2004)) If
D(s,t) and L2,3, then the THNDP polytope
is given by the inequalities
x(d (W )) 2 for all st-cut ?(W),
x(T) 2 for all
L-st-path-cut T,
0 x(e) 1 for all e ? E.
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4. The 2-edge case 4.2. Polyhedral
results
Theorem (Dahl, Huygens, M. Pesneau (2005)) If
D(s,t), L2, and K arbitrary, then the HCNDP
polytope is given by the inequalities
x (d (W)) K for all st-cut ? (W),
x (T) K for all
L-st-path-cut T,
0 x (e) 1 for all e ? E.
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4. The 2-edge case 4.2. Polyhedral
results
4.2.2. Valid inequalities (Huygens, Labbé, M.,
Pesneau (2005))
a) Double cut inequalities
L3
s1
t2
t1
e
FE\(V2,V3?V3,V4?e)
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4. The 2-edge case 4.2. Polyhedral
results
4.2.1. Valid inequalities (Huygens, Labbé, M.,
Pesneau (2005))
a) Double cut inequalities
L3
s1
t2
t1
e
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4. The 2-edge case 4.2. Polyhedral
results
4.2.1. Valid inequalities (Huygens, Labbé, M.,
Pesneau (2005))
a) Double cut inequalities
L3
s1
t2
t1
e
FE\(V2,V3?V3,V4?e)
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4. The 2-edge case 4.2. Polyhedral
results
b) Rooted-partition inequalities
Theorem Let Ut1,,tp be a subset of p
destination nodes relatively to node s. Let
(V0,V1,,Vp) be a partition of the node set V
such that s?V0, ti ?Vi, for all i1,,p. Then the
inequality
t1
V1
x(?(V0,V1,,Vp)) ?pp/L?
V0
t2
V2
is valid.
s
V3
rooted-partition inequality
t3
t4
Theorem If L2, a rooted-partition inequality
defines a facet only if p is odd and Viti for
i1,,p.
tp
V4
Vp
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4. The 2-edge case 4.2. Polyhedral
results
Theorem The separation problem for the
rooted-partition inequalities when L2, p odd,
and Viti for i1,,p, can be solved in
polynomial time.
Proof (Outline) By reduction to the minimization
of a submodular function.
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4. The 2-edge case 4.3. BranchCut
4.3. BranchCut algorithm
(Huygens, Labbé, M., Pesneau (2005))
THNDP, L2,3.
Used constraints trivial inequalities st-cut
inequalities L-st-path-cut inequalities doubl
e cut inequalities rooted-partition
inequalities (and other inequalities)
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4. The 2-edge case 4.3. BranchCut
Some computational results
- Random and real instances, - Max runtime 5
hours. - The double cut and rooted partition
inequalities are separated heuristically.
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4. The 2-edge case 4.3. BranchCut
Results for random instances for L2, 3 and
rooted demands
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4. The 2-edge case 4.3. BranchCut
Results for real instances for L2, 3 (arbitrary
demands)
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4. The 2-edge case 4.4. Formulation for
L4
4.4. Formulation for L4
G(V,E)
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4. The 2-edge case 4.4. Formulation for
L4
4.4. Formulation for L4
G(V,E)
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4. The 2-edge case 4.4. Formulation for
L4
4.4. Formulation for L4
G(V,E)
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4. The 2-edge case 4.4. Formulation for
L4
The 2-layred 4-path-cut inequalities generalize
the so-called jump inequalities (Dahl and
Gouveia (2004))
Theorem (Huygens, M., (2005)) The 2-layred
4-path-cut inequalities are valid for the THNDP
polytope.
Theorem (Huygens, M., (2005)) For L4, the
trivial, st-cut, L-st-path cut inequalities,and
the 2-layred 4-path-cut inequalities together
with the integrality constraints suffice to
formulate the THNDP as an integer program.
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4. The 2-edge case 4.4. Formulation for
L4
L5
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4. The 2-edge case 4.4. Formulation for
L4
L6
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4. The 2-edge case 4.4. Formulation for
L4
Node case
For the node case we may also define the - node
st-cut inequalities
xG-v(?(W))1, for every node v, and st-cut
?(W))
- node L-st-path-cut inequalities
xG-v(T)1, for every node v, and
L-st-path-cut T
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4. The 2-edge case 4.4. Formulation for
L4
Theorem (Huygens, M., (2005)) For L4, the
trivial, st-cut, node st-cut, L-st-path-cut, and
node L-st-path-cut inequalities, together with
the integrality constraints, suffice to
formulate the node-THNDP as an integer program.

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5. Open Questions
5. Open questions and concluding remarks

• Separation algorithm for the 2-layred 4-path-cut
  inequalities,
• for fractional solutions (it is polynomial for
  0-1 solutions)?
• Generalization of the (edge and node)
  formulations for L5.

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5. Open Questions

• - The Survivable network design problems are
  difficult to
• solve (even special cases).
• - The problems with length constraints remains
  the most
• complicated SNDP. A better knowledge of their
  facial structure would be usefull to establish
  efficient cutting plane techniques.
• Would it be popssible to develop usefull cutting
  plane and column generation techniques for the
  very general model with length constraints,
  dimensioning and routing...?