2-edge connected subgraphswith bounded rings
B. FortzInstitut de Gestion et d’Administration, Louvain la Neuve, Belgique
A. R. MahjoubLIMOS, CNRS, Université Blaise Pascal, Clermont-Ferrand, France
S. T. McCormickFaculty of Commerce, Vancouver, Canada
P. PesneauLIMOS, CNRS, Université Blaise Pascal, Clermont-Ferrand, France
Pierre Pesneau 2
Outline
• Presentation of the problem
• Polyhedral study
• Branch&Cut algorithm
• Computational results
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Network design
• Designing the network topology• Network survivability
2-edge connectivity
• Network performance in case of failure
bounded ring constraints
• Problem :
Find a 2-edge connected subgraph at minimum cost such that each edge belong to a cycle of length bounded by an integer K.
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2-node connectivity(Fortz, Labbé, Maffioli 2000)
• Formulation in terms of edges and cycles• Valid inequalities and necessary and sufficient
conditions to be facet defining• Separation algorithms• Branch&Cut algorithm• Heuristics.
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Cut inequalities
Let . Pose
Let
EF
V.W ,VW
.
F, e exF
otherwise0
if1)(
2))(( Wx W W\V
)(W
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Cycle inequalities
e
T
• Let be a partition
such that ,• let ,
• let
• Every solution must verify :
p10 V , ,V ,V K p
p,VVe 0
.1
1
0
e,VV\ ET ii
p
i
0 x(e) x(T) Cycle configuration
1V
0V
2V
pV
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Formulation
., 0,1
,1, 0
ion,configurat cycle ) ,(,0 )( )(
,,2 ))((
Eex(e)
Eex(e)
eTexTx
VWV WWx
cx Minimize
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Cycle inequalities : facets
• Let G=(V,E) be a complete graph.• Let be a partition of V and• The associated cycle inequality is facet defining if
and only if :– ,– and
– for all
p, V, , VV 10 . ,VV e p0
Kp 10 V ,1KV
31 ii VV .1,,0 Ki
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Cycle inequalities : separation
Let x be a solution.
The separation problem of cycle inequalities
for an edge e=st
The bounded (s,t)-path cut problem with B=K-1.
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Bounded path cut problem
• Let G=(V,E) be a graph, s and t two nodes and B an integer.
• Bounded (s,t)-path cut : set of edges that cut all (s,t)-path of length
• Problem : Find a minimum cost bounded (s,t)-path cut (BPCP).
.B
Pierre Pesneau 11t
vtw ztw
' N'u 'v
'z
uvw
uvw
N u v z
suw szw
BPCP• If B=2 the problem is trivial.• If B=3 :
– The problem is polynomial.– It can be reduced to find a minimum cut in a particular
directed graph.s
t
N u v z
suw szw
vtw ztw
uvw
s
G 'G
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BPCP
• If :Heuristic based on the Primal-Dual method :
While C is not a bounded (s,t)-path cut do
Find an (s,t)-path P of bounded length
Increase until an edge verifies
Improve C by removing useless edges of C
4B
;;0 Cy
Py Pf .
:f
QfQQ wy
;fCC
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Cycle inequalities : separation
• Let G=(V,E) be a graph and e=st be an edge.• Calculate a bounded (s,t)-path cut C with B=K-1.• If x(C) < x(e), the we get a violated cycle
inequality and the associated partition is obtained by a breadth-first search from s in the graph G\C.
• We strengthen the partition by reducing to a single node.
KV
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Cyclomatic inequalities
• Introduced by Fortz, Labbé (1999).• Let be a partition of V.• Every solution must verify :
pVV ,, 0
.KKp))VV((x p
1 ,, 0
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Cyclomatic inequalities : separation
1st heuristic :Based on the separation of partition inequalities (Cunningham) :
Consider :
1)) ,, (( 0
KKpVVx p
.100with V
p
pVVx p )),,(( 0
We apply Barahona’s algorithm for the separation of the partition inequalities.
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Cyclomatic inequalities : separation
2nd heuristic :Let G=(V,E) be a graph and x a solution
While and |V|>2 do
Find an edge e with the greatest value in x
Contract edge e in the graph G
If |V|>2 then we have a violated cyclomatic inequality and each element of the associated partition is given by the expansion of the nodes of the graph.
1
)(KpKEx
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Cycle partition inequalities
• Let be a partition of V such that• Let T be the chords of the partition and C be the
other edges of the partition.• Every solution must verify :
p, V , V 0
.p)C(x)T(xK
pp 2 1
1
.Kp
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Cycle partition inequalities : facets
• Let G=(V,E) be a complete graph.• Let be a partition of V.• The cycle partition inequality is facet defining if
and only if :– – there is at most one such that
– for
p, V , V 0
,Kp p...,,i 0 ,VV ii 11
2iV .p...,,i 0
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Cycle partition inequalities : separation
Same idea than the separation of cyclomatic inequalities :
Let G=(V,E) be a graph and x be a solution
While |V|>K+1 do
Contract edge e with the greatest value in x
Search the order of the nodes of the final graph such that
is minimum.
If this value is <2K then, the expansion of the nodes of the final graph give a paretition inducing violated cycle partition inequality.
)()( 1
1 CxTxK
pp
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Computational results
• Branch&Cut algorithm.• Tree manager : BCP (IBM).• Linear solver : CPLEX 7.1.• PC PIV 1,7 GHz, 1 Go RAM.• Random and real data.• • Complete graphs.• Time limit : 2 hours.
.103 K
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Results : random instancesNode K Cut Cycle Metric Subset Cyclom. Cycle P Gap ro Gap
finCPU (s)
20 3 13.0 27.4 19.8 11.0 9.0 7.0 0.15 0.00 0.28
20 4 40.2 1000.4 496.8 118.2 56.8 28.8 3.46 0.00 54.28
20 5 52.0 2487.8 709.6 1068.6 81.8 28.0 4.36 0.00 276.30
20 6 61.6 1838.8 419.4 1656.0 62.8 20.6 4.47 0.00 173.64
20 10 38.8 484.2 124.4 1953.6 26.2 1.4 2.50 0.00 42.86
30 3 28.2 98.2 59.8 20.6 33.4 14.8 0.83 0.00 8.50
30 4 97.611029.
84768.2 646.4 457.8 125.0 5.99 1.97 7200.00
30 5 94.812905.
22462.0 2893.2 380.2 51.6 6.41 2.00 7200.00
30 6104.
412738.
01704.4 6560.8 207.4 18.0 7.86 3.85 7200.00
30 10121.
27326.4 647.8
23070.0
103.4 0.8 4.33 0.96 5504.88
40 3 53.4 311.0 175.8 40.0 192.4 53.6 1.31 0.00 362.92
40 4108.
87140.6 2824.2 453.0 658.4 109.0 5.98 3.12 7200.00
40 5120.
88141.8 1369.2 1540.6 395.2 41.0 8.21 5.54 7200.00
40 6118.
68776.6 932.6 3227.4 254.6 34.0 9.26 6.73 7200.00
40 10152.
08886.8 607.8
15942.2
127.0 0.5 4.32 2.04 7200.00
50 3 78.8 701.4 414.8 92.6 631.0 179.8 2.01 0.26 3522.34
50 4112.
24475.4 1592.6 253.6 902.6 83.6 7.21 5.54 7200.00
50 5111.
24982.6 787.0 928.4 497.4 45.0 9.34 7.55 7200.00
50 6112.
05420.2 527.4 1726.4 277.2 17.4 10.89 9.27 7200.00
50 10139.
47298.4 259.4
10923.2
102.6 1.6 6.23 4.58 7200.00
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Results : real instancesNode K Cut Cycle Metric Subset Cyclom. Cycle P Gap ro Gap
finCPU (s)
12 3 2 4 6 2 1 1 0.00 0.00 0.01
12 4 10 22 22 12 9 0 0.52 0.00 0.13
12 5 14 28 28 43 4 0 1.77 0.00 0.29
12 6 8 8 17 38 5 0 0.72 0.00 0.15
12 10 4 0 20 0 6 0 0.83 0.00 0.09
17 3 13 43 41 15 11 6 0.51 0.00 0.52
17 4 18 159 111 40 16 3 1.73 0.00 2.81
17 5 28 97 42 103 16 2 2.21 0.00 1.47
17 6 9 7 11 26 6 2 0.00 0.00 0.08
17 10 2 0 0 0 2 0 0.00 0.00 0.03
30 3 30 91 50 18 49 17 0.58 0.00 9.96
30 4 110 11373 5768 816 443 102 4.59 1.12 7200.00
30 5 161 13161 2835 4185 405 76 4.72 0.98 7200.00
30 6 144 12874 2415 7987 264 24 5.50 1.92 7200.00
30 10 69 617 164 1668 27 0 1.41 0.00 58.40
52 3 87 584 358 73 837 108 1.31 0.00 2092.77
52 4 107 3656 1398 175 867 60 6.15 4.60 7200.00
52 5 130 3867 728 745 607 40 9.11 7.88 7200.00
52 6 128 4506 477 1400 308 8 8.09 6.72 7200.00
52 10 110 3552 118 7273 90 1 7.60 6.42 7200.00
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Example of a solution
52 nodes
3K
35 minutes2047 constraints
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Perspectives
• Solve bigger instances
(particularly for K=3,4 and 5).• Improve separation routines.• Find new classes of valid inequalities.
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Subset inequalities
• Let T be an edge set such that G\T does not contain a feasible solution.
• We have :
• Separation : when we separate cycle inequalities, if two consecutive elements of the partition are reduced to a single node, then T induced a violated subset inequality.
.1)( Tx
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Cut inequalities : facets
• Let G=(V,E) be a complete graph.• Let • The cut inequality is facet defining if and only if :
– either and
– or and
.VW,VW
2 4 W,K
32 3 ,W,K ,W\V 2
.,W\V 32
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Cyclomatic inequalities : facets
• Let G=(V,E) be a complete graph.• Let be a partition of V.• The cyclomatic inequality is facet defining if and
only if and :– either and or and
for
– or for
pV,...,V 0
4K 2iV,p,...,i 0
32 3 ,V,K i .p,...,i 0
3iV 11 mod)1( Kp
Kp
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Metric inequalities• Introduced by Fortz, Labbé, Maffioli (2000).• Let G=(V,E) be a graph and• Let be a set of node potential satisfying :
• Every solution must verify :
where
Separation : heuristic of Fortz et al. (2000).
.Eije Vkk
.Kji 1
eeEf
ff xxv
. allfor 1
1,0max,1min eEklf
Kv
ji
klf