Computational strategies for
a multi-period network
design and routing problem
Bernard Fortz, Enrico Gorgone (ULB)
Dimitri Papadimitriou (Nokia - Bell Labs)
Journée SDN 2016, Orange Gardens, 23 novembre 2016
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Routing functions and algorithms
Routing function
For each destination t, determines for each node u the adjacent
node (next-hop) along a computed loop-free routing path from
u to t.
Routing algorithm
Computes a (set of) loop-free paths such that any node can
route traffic towards any destination node.
Distributed routing functions
and algorithms
Distributed routing function
Routing function fu executed locally at each node u ∈ V (G) and
independently of other nodes.
Distributed routing algorithm
Executed locally at each node u that computes a (set of)
loop-free paths used by the routing function fu , such that
node u can route traffic towards any destination node.
Algorithm performs locally, independently of other nodes,
and asynchronously (no synchronization during
computation)
Distributed routing protocols today
Information exchange
Captures network/link state or path state information
Exchanges this information throughout the network
Routing decision
Routing algorithm: uses this information (input) to locally
compute loop-free routing paths
Locally determines the next-hop per destination (output)
from the appropriate routing path (selected among among
feasible paths)
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Routing protocols today
Best effort approach, local routing path computation and
next-hop decisions.
Protocols based on the hypothesis that routers have very
limited memory and computing power.
Computational effort limited to shortest (constrained) paths
computation.
Much more power available today
Can we use it for better a better local decision process?
Optimization models
Protocols were not developed with optimization in mind.
Resulting problems are large-scale and NP-hard.
Often solved centrally, with deterministic information.
Recently: uncertainty/robustness issues taken into
account, leading to even larger scale optimization
problems.
Routing protocol design and routing optimization are
decoupled
How to modify the routing decision process to include
optimization objectives?
Optimization methods
Large scale problems intractable by state-of-the-art
solvers.
Heuristics or decomposition methods (e.g. Benders or
Lagrangian decomposition)
Heuristics: sub-optimal solutions.
Decomposition techniques: still used for centralized
decision making, independently of the distributed nature of
protocols.
Decomposition methods naturally lead to parallelization
How to adapt protocols and decomposition techniques to
collaborate and distribute the optimization work efficiently?
Our objectives
Go beyond the assumption that routers have limited
memory and computing power.
Develop new routing paradigms taking into account
optimization objectives.
Develop multi-agent optimization and routing techniques
based on properties of optimization models and structured
decomposition.
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Model overview
Features of problems per period
Dimension
Per destination
Per set of destinations
Per neighbor
Design problem
Link capacity
(re-)allocation
Link capacity increase/decrease
(on installed link)
Link installation
Routing problem
(Re-)routing decision
(next-hop selection)
Link cost variation
Routing adjacency creation
Features of problems over multiple periods
Dimension
Design problem
Per destination
Link capacity (re-)allocation
Per set of destinations
Link capacity increase/decrease
(on installed link)
Link maintenance
Per neighbor
Routing problem
Next-hop decision change
Routing state change
Next-hop decision unchanged
Routing state maintenance
Link cost variation
Routing adjacency maintenance
Multi-period network
design and routing optimization
Cost of a solution
Combination of
link installation and maintenance costs
the routing or capacity allocation costs
Network design problem
Time dimension: link maintenance cost
Routing decision problem
Upper bound on the number of allowed routing changes
Input data
Directed graph G = (V , A)
P periods
For each period p, demand matrix D p
For each arc (i, j),
nominal maximum capacity κij ,
installation cost cij
maintenance cost mij (mij < cij )
Bound M on the number of allowed routing table changes
Variables
yijp ∈ {0, 1}: indicates if arc (i, j) is newly opened at period
p
zijp ∈ {0, 1}: indicates if arc (i, j) is maintained at period p
xijtp ∈ {0, 1}: indicates if node j is the next hop for node i to
destination t at period p
Aggregated formulation:
fijtp ≥ 0: amount of flow on arc (i, j) destined to t at period p
Extended formulation:
fijstp ≥ 0: amount of flow on arc (i, j) going from source s to
destination t at period p
Routing cost
The routing cost per unit of flow for each arc (i, j) and each
period p is defined by an increasing convex function of its
utilization
300
200
100
0
0
0.2
0.4
0.6
0.8
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Aggregated formulation (1)
min
X X
p
cij yij
+
p
mij zij
p∈P (i,j)∈A
t∈V
subject to:
X
tp
fij
−
j:(i,j)∈A
X
tp
fjt
=
X
j:(j,i)∈A
X
s∈V
j:(j,t)∈A
p
p
xtp
ij ≤ yij + zij
p
p
yij
+ zij
≤1
p
zij
≤
p−1
yij
1
zij
=0
+ φ κij ,
X
+
p−1
zij
tp
fij
!!
tp
fji
= Dp (i, t) i, t ∈ V, i 6= t, p ∈ P
Dp (s, t)
t ∈ V, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P, p ≥ 2
(i, j) ∈ A
Aggregated formulation (2)
tp
tp tp
fij
≤ Cij
xij
(i, j) ∈ A, t ∈ V, p ∈ P
X tp
p
p
p
fij ≤ Cij (yij + zij ) (i, j) ∈ A, p ∈ P
t∈V
X
xtp
ij = 1
j:(i,j)∈A
xtp
ij ∈ {0, 1}
p
p
yij
, zij
∈
tp
fij ≥ 0
{0, 1}
i, t ∈ V, i 6= t, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
Model improvement
Introduce
oijp = yijp + zijp
with oijp binary.
Allows to remove integrality requirements on y , z.
Simplification of the constraint structure.
Improves CPLEX solving time by a factor of 2.
Improved Aggregated formulation (1)
min
X X
p
cij yij
+
p
mij zij
p∈P (i,j)∈A
t∈V
subject to:
X
tp
fij
−
j:(i,j)∈A
X
tp
fjt
j:(j,t)∈A
p
xtp
ij ≤ oij
p
zij
1
zij
≤
op−1
ij
=0
+ φ κij ,
X
=
X
j:(j,i)∈A
X
s∈V
tp
fij
!!
tp
fji
= Dp (i, t) i, t ∈ V, i 6= t, p ∈ P
Dp (s, t)
t ∈ V, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P, p ≥ 2
(i, j) ∈ A
Improved Aggregated formulation (2)
tp
tp tp
fij
≤ Cij
xij
X tp
p p
oij
fij ≤ Cij
t∈V
X
xtp
ij = 1
j:(i,j)∈A
xtp
ij ∈ {0, 1}
opij ∈ {0, 1}
p
p
tp
yij
, zij
, fij
≥
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P
i, t ∈ V, i 6= t, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P
0
(i, j) ∈ A, t ∈ V, p ∈ P
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
GUB constraints
Bottleneck in solving the LPs: large number of Generalized
Upper Bound (GUB) constraints.
fijtp ≤ Cijtp xijtp (i, j) ∈ A, s, t ∈ V , p ∈ P
Easily separable by inspection
Separation as CPLEX lazy cuts only or more aggressively
as user cuts.
Connectivity constraints
Removal of GUB constraints makes it much more difficult
for CPLEX to find a feasible solution.
The problem: routing variables do not induce a connected
graph if GUBs are removed.
Addition of cut constraints:
X
xijtp ≥ 1, S ⊆ V , t ∈ S, p ∈ P
(i,j)∈δ + (S)
Fast heuristic separation, only added at the root node.
Primal heuristic
In the Branch-and-Cut setting, CPLEX has difficulties in
finding good feasible solutions
Simple primal heuristic based on solving the problem by
period.
Test instances
Topology
Nodes
Links
Atlanta
DFN-BWin
Di-Yuan
Geant
PDH
15
10
11
22
11
22
45
42
36
34
Min,Max,Avg
Degree
2;4;2.93
9;9;9.00
7;9;7.64
2;8;3.27
4;8;6.18
Diameter
3
1
3
5
3
Traffic
Traffic demands: drawn from exponentially distributed
service times and quasi-randomly distributed inter-arrival
times
Per time period: demand for each period has an average
proportional to (demand) value reported in the SNDlib
database
Results
CPLEX
Basic
Threads
Atlanta
DFN-BWin
Di-Yuan
Geant
PDH
1
9297
304
15.21%
89
892
8
7805
649
8.76%
222
1216
1
6.03%
569
13.40%
97
601
8
8.56%
329
11.36%
85
3159
Branch-and-cut
Lazy
Heuristic
1
8
1
8
4.09% 0.08% 3.68% 2.91%
165
427
350
1859
12.93% 9.10% 12.31% 9.40%
48
72
72
85
2609
3222
1161
2124
Lazy Heuristic
1
8
1.16%
5.35%
392
1213
11.46% 10.64%
85
161
2125
4345
No clear winning strategy.
CPLEX performs better on difficult instances. Some work
needed to improve the formulation (valid inequalities).
Parallelism provides little/no gain.
The current heuristic does not help.
It’s good to be lazy!
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Extended formulation (1)
min
X X
p∈P (i,j)∈A
p
p
cij yij
+ mij zij
+ φ κij ,
subject to:
X
stp
fij
−
j:(i,j)∈A





X
X
s,t∈V

stp 
fij
stp
fji
j:(j,i)∈A
p
D (s, t) if i = s
=
−Dp (s, t) if i = t


0 otherwise
p
p
xtp
≤
y
+
z
ij
ij
ij
p
p
yij
+ zij
≤1
p
p−1
p−1
zij
≤ yij
+ zij
1
zij
=0
i, s, t ∈ V, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P, p ≥ 2
(i, j) ∈ A
Extended formulation (2)
stp
stp tp
(i, j) ∈ A, s, t ∈ V, p ∈ P
≤ Cij
xij
fij
X stp
p
p
p
fij ≤ Cij
(yij
+ zij
) (i, j) ∈ A, p ∈ P
s,t∈V
X
xtp
ij = 1
j:(i,j)∈A
xtp
ij ∈ {0, 1}
p
p
yij
, zij
∈ {0, 1}
stp
fij
≥0
i, t ∈ V, i 6= t, p ∈ P
(i, j) ∈ A, t ∈ V, p ∈ P
(i, j) ∈ A, p ∈ P
(i, j) ∈ A, s, t ∈ V, p ∈ P
Lagrangian relaxation
The extended formulation has a much stronger
LP relaxation but is too heavy to be handled efficiently by
LP solvers.
By relaxing flow conservation constraints, the problem
becomes decomposable by origin node for routing and
design decisions.
Lagrangian relaxation:
Implementation
Lagrangian dual solved by a bundle method (code from A.
Frangionni).
Warm start:
Initial Lagrangian multipliers: dual of a MCF relaxation.
Solves the aggregated formulation for a limited time, then
switch to the extended formulation.
Subproblem solved as a MIP only when LP provides a
feasible solution.
Lagrangian heuristic: rolling-horizon fixing of variables
based on the current Lagrangian dual solution.
Lagrangian relaxation:
Numerical results (5 periods)
Network
polska
france
norway
sun
nobel-eu
india35
cost-266
giul39 (*)
pioro40
germany50
zib54
|V |
12
25
27
27
28
35
37
39
40
50
54
|A|
36
90
102
204
82
160
114
334
178
176
162
AF-CPLEX
BB
BI
8.97% 8.97%
0.65% 3.12%
0.28% 7.99%
-1.17% 25.74%
4.63% 18.34%
-4.80% 32.42%
-0.26% 6.28%
-1.27%
—–
12.78%
—–
5.64%
—–
-0.37% 14.77%
DF-CPLEX
BB
BI
8.97% 8.97%
1.39% 2.74%
0.58% 97.29%
1.91% 6.95%
3.36%
—–
0.15%
—–
0.00% 7.11%
—–
—–
0.00%
—–
0.00%
—–
0.00%
—–
BM / LH
BB
BI
0.00% 11.86%
0.00% 5.35%
0.00% 10.13%
-0.02% 10.35%
-0.09% 14.60%
-0.01% 20.05%
-0.04% 1.75%
0.00% 1.44%
-0.03% 23.65%
-1.76% 52.96%
-12.34% 3.23%
Lagrangian relaxation:
Numerical results (10 periods)
Network
polska
france
norway
sun
nobel-eu
india35
cost-266
giul39 (*)
pioro40
germany50
zib54
|V |
12
25
27
27
28
35
37
39
40
50
54
|A|
36
90
102
204
82
160
114
334
178
176
162
AF-CPLEX
BB
BI
3.91% 4.86%
0.35% 1.00%
-0.97%
—–
-3.17%
—–
4.68% 49.16%
-3.99%
—–
-1.51%
—–
3.37%
—–
5.69%
—–
0.03%
—–
-1.13%
—–
DF-CPLEX
BB
BI
3.85% 4.76%
0.86% 0.86%
0.00%
—–
0.01%
—–
2.99%
—–
0.01%
—–
0.00%
—–
—–
—–
0.00%
—–
0.00%
—–
0.00%
—–
BM / LH
BB
BI
0.00% 8.22%
-0.01% 10.28%
-0.02% 14.51%
-0.02% 14.09%
-0.22% 16.01%
-4.35% 20.04%
-0.07% 3.89%
0.00%
—–
-2.14% 19.84%
-6.42%
—–
-24.03% 3.12%
Lagrangian relaxation:
Numerical results (15 periods)
Network
polska
france
norway
sun
nobel-eu
india35
cost-266
giul39 (*)
pioro40 (*)
germany50 (*)
zib54 (*)
|V |
12
25
27
27
28
35
37
39
40
50
54
|A|
36
90
102
204
82
160
114
334
178
176
162
AF-CPLEX
BB
BI
2.33% 4.12%
0.28% 0.28%
-2.49%
—–
-2.79%
—–
2.05% 41.36%
-3.40%
—–
-2.29%
—–
3.06%
—–
3.25%
—–
6.57%
—–
37.78%
—–
DF-CPLEX
BB
BI
2.55% 4.09%
0.28% 0.28%
0.00%
—–
0.00% 42.90%
3.79%
—–
0.00%
—–
0.00%
—–
—–
—–
—–
—–
—–
—–
—–
—–
BM
BB
0.00%
-0.01%
-0.03%
-0.03%
-1.19%
-5.81%
-0.14%
0.00%
0.00%
0.00%
0.00%
/ LH
BI
6.86%
7.52%
33.89%
18.59%
20.21%
20.78%
8.76%
—–
—–
—–
45.49%
Outline
1
Preliminaries
2
Issues with current networking and optimization paradigms
3
A global model for multi-period design and routing
4
Formulations
5
Branch-and-cut strategies
6
Lagrangian relaxation
7
Perspectives
Future developments
Additional valid inequalities
Better separation procedures and primal heuristics
Distributed decomposition model
Model decision process inside each router and information
exchanges (distributed consensus)
Tradeoff between fully centralized (computationally
unscalable) and fully distributed (communication cost,
slower convergence)