A Case Study in Network Design JeanCharles Regin

1
A Case Study in Network DesignJean-Charles Regin
2
Plan

• Real world Problems
• Modeling principles
• Network Design application the ROCOCO project
• Presentation
• Methods
• Integration
• Conclusion
• Lessons

3
Real world problems

• Almost all are over-constrained
• Not pure
• Decomposable
• Small or Large number of variables (10,000) and
  constraints (100,000)
• Scalable but not solvable for n/10 or n/100
• ad-hoc efficient heuristic
• everything is possible
• need flexibility
• Want optimized not optimal

4
Real world problems

• Flexibility causes no problem add new
  constraints to the solver
• Be careful with objective functions relaxation
  is not good for CP (the constraints are weak and
  no pruning occurs)

5
Plan

• Real world Problems
• Modeling principles
• Network Design application the ROCOCO project
• Presentation
• Methods
• Integration
• Conclusion
• Lessons

6
Modeling principles

• What a good model is?
• Symmetries
• Implicit constraints
• Global constraints
• Relevant and redundant constraints
• Back propagation
• Dominance rules

7
Good Model?

• A good model is a model that leads to an
  efficient resolution of a given problem

8
Good Model?

• A good model is a model that leads to an
  efficient resolution of a given problem
• Deals with several notions

Symmetries Catalyzing constraints Global
constraints Relevant and redundant
constraints Back propagation Dominance rules
9
Symmetries

• Tutorial on this topic at CP04
• The complexity of a problem can often be reduced
  by detecting intrinsic symmetries
• When two or more variables have identical
  characteristics, it is pointless to differentiate
  them artificially
• The initial domains of these variables are
  identical
• These variables are subject to the same
  constraints
• The variables can be permuted without changing
  the statement of the problem
• Usually symmetries are removed by introducing an
  order between variables

10
Catalyzing constraints

• An catalyzing constraint makes explicit a
  property that satisfies any solution implicitly.
• D(x1)D(x2)D(x3)D(x4)a,b,c,d
• Constraints b,c and d have to be taken at least 1

11
Catalyzing constraints

• An catalyzing constraint makes explicit a
  property that satisfies any solution implicitly.
• D(x1)D(x2)D(x3)D(x4)a,b,c,d
• Constraints b,c and d have to be taken at least
  1
• Filtering algorithm if b is not assigned and if
  there is only one variable x that contains b in
  its domain then xb

12
Catalyzing constraints

• A catalyzing constraint makes explicit a property
  that satisfies any solution implicitly.
• D(x1)D(x2)D(x3)D(x4)a,b,c,d
• Constraints b,c and d have to be taken at least
  1
• Filtering algorithm if b is not assigned and if
  there is only one variable x that contains b in
  its domain then xb
• Problem if x1a and x2a then nothing is deduced

13
Catalyzing constraints

• A catalyzing constraint makes explicit a property
  that satisfies any solution implicitly.
• D(x1)D(x2)D(x3)D(x4)a,b,c,d
• Constraints b,c and d have to be taken at least
  1
• Filtering algorithm if b is not assigned and if
  there is only one variable x that contains b in
  its domain then xb
• Problem if x1a and x2a then nothing is deduced
• Implicit constraints a can be taken at most
  1b,c,d can be taken at most 2
• From the simultaneous presence of some
  constraints implicit constraints can be deduced

14
Global constraints

• A global constraint is a conjunction of
  constraints. This conjunction often takes into
  account implicit constraint deduced from the
  simultaneous presence of the other constraints
• This is the case for the previous example with
  the global cardinality constraint
• Use the strongest filtering algorithm as you can
  at the beginning
• It is rare to be able to solve a problem with
  weak FA and not to be able to solve it with
  strong FA

15
Global constraint Alldiff results

• Color the graph with cliquesc0 0, 1, 2, 3,
  4 c1 0, 5, 6, 7, 8c2 1, 5, 9, 10,
  11c3 2, 6, 9, 12, 13 c4 3, 7, 10, 12,
  14 c5 4, 8, 11, 13, 14
• clique size27 Global fails 0 cpu time
  1.212 s Local fails 1 cpu time 0.171
  sclique size31 Global fails 4 cpu time
  2.263 s Local fails 65 cpu time 0.37
  sclique size51 Global fails 501 cpu
  time 25.947 s Local fails 24512 cpu
  time 66.485 sclique size61 Global fails
  5 cpu time 58.223 s Local ?????????????

16
Relevant Constraints

• At first glance it seems that adding a constraint
  which removes some symmetries, or which is an
  implicit or a global constraint improves the
  current model. This is FALSE
• Because
• The new filtering algorithm can delete no value,
  because everything is already deduced by the
  combination of constraints
• The new filtering algorithm can remove some
  values and impacts the variable-value strategy
  (more backtracks can be needed to reach the first
  solution)

17
Relevant constraints

• A constraint is relevant w.r.t. a model if the
  introduction of this constraint
• Is needed by the definition of the problem
• Or if it permits to remove some symmetries, or it
  is an implied or a global constraint, and the
  introduction of this constraint improves the
  search for the solution in term of performance
• A constraint is redundant w.r.t. a model if the
  constraint is not relevant w.r.t. the model.

18
Back propagation

• Consider an optimization problem with an
  objective variable obj.
• The back propagation is the consequences of the
  modifications of the variable obj
• Example?x obj. Back propagation
  modification of the x variable when obj is
  modified

19
Back propagation

• Try to improve the back propagation, because when
  a solution with a cost c is found the constraint
  obj lt c is added and a new solution is sought.
• It is important to use constraints involving cost
  variable. For instance gcc with cost

20
Dominance rules

• A dominance rule is a rule that eliminates some
  solutions that are not optimal, or some optimal
  solutions but not all
• This is a kind of symmetry breaking in regards to
  the optimality
• An example is given in the resolution of the next
  problem

21
A bad model?

• Golomb ruler (see CSP lib)A Golomb ruler may
  be defined as a set of n integers 0x1 lt x2 lt lt
  xn s.t. the n(n-1)/2 differences (xj - xi) are
  distinct. Goal minimize xn.
• with CP difficult for n gt 13.
• x1,,xn variables (xi-xj) variables. Alldiff
  involving all the variables.

22
Alldiff
1 2 3 4 5 6 7
x1-x2 x1-x3 x2-x3 x1 x2 x3
Not a good solution Bad incorporationof
constraintxi xj in alldiff
23
Plan

• Real world Problems
• Modeling principles
• Network Design application the ROCOCO project
• Presentation
• Methods
• Integration
• Conclusion
• Lessons

24
A Case Study in Network Design

• Very good example and benchmark
• To illustrate the advantages and drawbacks of
  different optimization techniques
• To illustrate the improvements that can be
  thought of when things do not work well
• To test new ideas
• To think about what we do introspection

25
The ROCOCO Project (1)

• France Telecom RD ISE
• Problem and benchmark definition
• Algorithm validation
• Research laboratories INRIA Numopt, LRI Orsay,
  PRiSM Versailles, Evry,
• Lower bounds Lagrangean relaxation, column
  generation, cuts
• Optimization techniques genetic algorithms
• ILOG
• Optimization techniques constraint programming,
  mixed integer programming, column generation

26
The ROCOCO Project (2)

• Two cycles
• Solution to the  basic  problem
• Improvements and study of the influence of
  additional constraints

27
The Problem (1)

• Routing of Communications
• Mono-routing each demand from a point p to a
  point q must follow a unique path
• Dimensioning of Links
• The capacity of each link must exceed the sums of
  the demands going through the link
• Additional Constraints
• Depend on the customer for whom the network is
  designed

28
The Problem (2)

• Data
• Customer traffic demands
• Possible links, capacities and costs

27Kb/s
S2
S1
S4
115Kb/s
S3

• Result
• Minimal cost network able to simultaneously
  respond to all the demands
• Route for each demand

S2
S1
S4
Rented capacity 256Kb/s
S3
29
The Problem (3)

• Cost minimization principle
• Traffic demands share link capacities

S2
S1
512Kb/s
128Kb/s
S3
30
The Problem (4)

• Demands share links
• ? demandsi?j ? capacityi?j
• Technological constraints

31
The Problem (5)
64Kb/s
64Kb/s
64Kb/s
64Kb/s

• Side constraints
• Quality of service
• Reuse of existing equipment (limit on the number
  of ports, maximal traffic at a node)
• Commercial and legal constraints
• Possible future network evolution
• Network management (e.g., traffic concentration)

32
Benchmark Elaboration (1)
33
Optional Constraints

• Security some commodities to be secured cannot
  go through unsecured nodes and links
• No line multiplication at most one line per arc.
• Symmetric routing demands from node p to node q
  and demands from node q to node p are routed on
  symmetric paths.
• Number of bounds (hops) the number of arcs of
  the path used to route a given demand is limited.
• Number of ports the number of links entering
  into or leaving from a node is limited.
• Maximal traffic the total traffic managed by a
  given node is limited.

64Kb/s
64Kb/s
64Kb/s
64Kb/s
34
Numerical Characteristics (1)
35
Particular Cases

• Loop network (C10)
• Unidirectional lines (C11)
• Extension of an existing network (C16)
• Several commodities (with different  security 
  and  number of hops  constraints) between
  different sites and a central site (C20)

36
Comparison / Other Benchmarks
37
Mathematical Model

• Develop model on the board?
• Anticipate difficulties?

38
Mixed Integer Programming (1)

• Explicit problem definition
• Linear constraints
• Real (floating point) and integer variables
• One optimization criterion
• Separation between problem definition and problem
  solving
• The continuous relaxation (when fractional values
  are allowed for integer variables) is solvable in
  polynomial time
• Bound on optimization criterion
• Guidance for solving the MIP problem by
  branch-and-bound (rounding, cuts, )

39
Mixed Integer Programming (2)

• Routing variables xijd (DM 0-1 variables)
• Dimensioning variables capacity levels yijk
  (MK 0-1 variables)
• Main constraints
• Sj xijd 1 if i is the origin of d
• Si xijd 1 if j is the destination of d
• Si xijd Sk xjkd if j is neither the origin nor
  the destination of d
• Sk yijk 1
• Sd Demd xijd Sk Capaijk yijk

40
Mixed Integer Programming (3)
41
Constraint Programming (1)

• Explicit problem definition
• Separation between problem definition and problem
  solving
• Systematic deduction of the consequences of made
  decisions (constraint propagation)
• Partial constraint propagation è development of
  heuristic search algorithms to generate and
  optimize solutions
• Incremental constraint propagation (with global
  fix point semantics)
• Localized definition of the constraint
  propagation process (each constraint propagates
  independently of other constraints) è many
  different types of constraints viewed as as many
  sub-problems

42
Constraint Programming (5)

• Routing variables paths (D set variables)
• A set of arcs joining the origin to the
  destination of the demand
• Basic functions impose or forbid an arc (or a
  node)
• Dimensioning variables chosen capacity levels (M
  enumerated variables)
• Specific constraints and constraint propagation
  algorithms

43
Constraint Programming (6)
44
Path representation in CP

• Classical modelGraph represented by the
  nodesOne variable per nodeValue possible
  neighboor
• Path from s to t alldiff on nodes.

45
Path representation in CP
s
a
d
e
b
f
c
t
D(s)a,b, D(a)s,b,c,d, D(b)s,a,c,
D(c)a,b D(d)a,e,f, D(e)d,t, D(f)d,t,
D(t)s
46
Path representation in CP
s
a
d
e
b
f
c
t
D(s)a,b, D(a)s,b,c,d, D(b)s,a,c,
D(c)a,b D(d)a,e,f, D(e)d,t, D(f)d,t,
D(t)s
47
Path representation in CP
s
a
d
e
b
f
c
t
Problem if some variables do not belong to the
pathWhat is the value assigned to these
variables?
48
Path representation in CP
s
a
d
e
b
f
c
t
A dummy value is added to each domain BAD IDEA
D(s)a, D(a)c, D(c)b, D(b)dummyb, D(d)
e, D(e)t, D(f)dummyf, D(t)s
49
Path representation in CP
s
a
d
e
b
f
c
t
Loops are allowed (var links to itself) GOOD IDEA
D(s)a, D(a)c, D(c)b, D(b)b, not
possible b has been already taken by c
50
Path representation in CP

• Classical model- One var per node- Alldiff
  constraint cost for the matching O(m) per
  modification

51
New model

• General point of view we search for a subgraph.
  Two entities- Digraph class - DigraphVar class
• A DigraphVar is a subgraph of a digraph w.r.t.
  properties, for instance path.It is defined
  from a Digraph

52
New model

• General point of view we search for a subgraph.
  Two entities- Digraph class - DigraphVar class
• A DigraphVar is a subgraph of a digraph w.r.t.
  properties, for instance path.It is defined
  from a Digraph
• API similar to setvar API

53
Digraph

• class IlcDigraph IlcDigraph(IlcInt
  nbNodes,IlcIntArray from,IlcIntArray to)IlcInt
  getNbNodes()constIlcInt getNbArcs()constIlcInt
  getNbOutgoingArcs(const IlcInt node)
  constIlcInt getNbIncomingArcs(const IlcInt
  node) constIlcInt getEmanatingNode(const IlcInt
  arc)constIlcInt getTerminatingNode(const IlcInt
  arc)constIlcInt getFirstOutgoingArc(const
  IlcInt node)constIlcInt getNextOutgoingArc(const
  IlcInt node, const IlcInt arc)constIlcInt
  getFirstIncomingArc(const IlcInt
  node)constIlcInt getNextIncomingArc(const
  IlcInt node, const IlcInt arc)const

54
Digraph Variable

• Class DigraphVarIlcDigraphVar(IlcManager m,
  IlcDigraph g)IlcIntSetVar getNodesVar()const
  IlcIntSetVar getArcsVar()constIlcIntSetVar
  getSourcesVar()const IlcIntSetVar
  getSinksVar()constIlcBool isBound()constIlcBoo
  l isAPath()const // accessors IlcBool
  isArcRequired(IlcInt arc)const IlcBool
  isArcPossible(IlcInt arc)const IlcBool
  isNodeRequired(IlcInt node)const IlcBool
  isNodePossible(IlcInt node)const IlcBool
  isSourceRequired(IlcInt node)const IlcBool
  isSourcePossible(IlcInt node)const IlcBool
  isSinkRequired(IlcInt node)const IlcBool
  isSinkPossible(IlcInt node)const

55
Digraph Variable

• Class DigraphVar // modificators void
  removeAllOutgoingArcs(IlcInt node)constvoid
  removeAllIncomingArcs(IlcInt node)constvoid
  removeAllOutgoingArcsButArc(IlcInt node, IlcInt
  arc)constvoid removeAllIncomingArcsButArc(IlcInt
  node, IlcInt arc)constvoid removeArcPossible(Il
  cInt arc)const void addArcRequired(IlcInt
  arc)constvoid removeNodePossible(IlcInt
  node)constvoid addNodeRequired(IlcInt
  node)const void removeSinkPossible(IlcInt
  node)const void addSinkRequired(IlcInt
  node)constvoid removeSourcePossible(IlcInt
  node)constvoid addSourceRequired(IlcInt
  node)const

56
Digraph Variable

• Class DigrapVar// for iterationsIlcInt
  getFirstOutgoingArc(IlcInt node)constIlcInt
  getNextOutgoingArc(IlcInt node, IlcInt
  arc)constIlcInt getFirstIncomingArc(IlcInt
  node)constIlcInt getNextIncomingArc(IlcInt
  node, IlcInt arc)const IlcDigraph
  getDigraph()constIlcInt getNbIncomingArcs(IlcInt
  node)constIlcInt getNbOutgoingArcs(IlcInt
  node)const

57
Digraph Variable

• Class DigraphVar
• // graph functions
• IlcInt getFirstArcPossibleOnShortestPath(IlcIntDis
  tanceFunctionI d, const IlcInt source,
  const IlcInt sink, IlcInt
  dem1)const
• IlcInt computeShortestPathDistance(IlcIntDistanceF
  unctionI dist, const IlcInt source,
  const IlcInt sink, IlcInt
  dem1)const
• IlcIntArray computeShortestPath(IlcIntDistanceFunc
  tionI dist, const IlcInt source,
  const IlcInt sink, IlcInt dem1)const

58
Distance Function

• class IlcIntDistanceFunctionI IlcIntDistanceFunc
  tionI(IlcDigraph g,IlcInt maxCost) virtual
  IlcInt getCost(IlcDigraphVar var,
  IlcInt arc, IlcInt dem)0

59
Path Constraints

• IlcConstraint IlcSimplePath(IlcDigraphVar g,
  IlcInt source, IlcInt sink)
• IlcConstraint IlcShortestPath(IlcDigraphVar g,
• IlcInt source, IlcInt sink,
  IlcIntVar obj, IlcIntDistanceFunctionI
  dist)

60
Element constraint

• enum IlcGraphProperty IlcNodeRequired0L,
  IlcSourceRequired1,
  IlcSinkRequired2, IlcEmanatingRequired
  3, IlcTerminatingRequired4,
  IlcTraversedRequired5,
  IlcArcRequired20
• IlcConstraint IlcGraphElement(IlcInt item,
  IlcDigraphVarArray gvs,
  IlcIntSetVar var,
  IlcGraphProperty pte)

61
Selectors

• class IlcDigraphSelectDigraphVarI
  IlcDigraphSelectDigraphVarI() virtual
  IlcInt select(IlcDigraphVarArray vars)0
• class IlcDigraphSelectArcI IlcDigraphSelectArcI
  () virtual IlcInt select(IlcDigraphVarArray
  vars, IlcInt index)0

62
Selectors (contd)

• class IlcDigraphSelectShortestPathArcI
  IlcDigraphSelectArcI IlcDigraphSelectShortestPa
  thArcI(IlcIntDistanceFunctionI fn) virtual
  IlcInt select(IlcDigraphVarArray vars, IlcInt
  index) virtual IlcInt getSource(IlcDigraphVarArr
  ay vars, IlcInt index)0 virtual
  IlcInt getSink(IlcDigraphVarArray vars, IlcInt
  index)0 virtual IlcInt getDemand(IlcDigraphVarA
  rray vars, IlcInt index)0

63
Goals

• IlcGoal IlcDigraphRequireArc(IlcManager m,
  IlcDigraphVar digraph,
  IlcInt arcIndex)
• IlcGoal IlcDigraphRemoveArc(IlcManager m,
  IlcDigraphVar digraph,
  IlcInt arcIndex)
• IlcGoal IlcDigraphAddArc(IlcManager m,
  IlcDigraphVarArray vars, IlcInt index,
  IlcDigraphSelectArcI selectArc)

64
Goals (contd)

• IlcGoal IlcDigraphInstantiate(IlcManager m,
  IlcDigraphVarArray vars, IlcInt index,
  IlcDigraphSelectArcI selectArc)
• IlcGoal IlcDigraphGenerate(IlcManager m,
  IlcDigraphVarArray vars,
  IlcDigraphSelectDigraphVarI selectD,
  IlcDigraphSelectArcI selectArc)

65
Why not PathVar?

• Path is a property of a graph. We prefer to
  express properties by constraint
• In any cases, we need to be able to test if an
  object is a path/tree/cycle

66
Rococo in CP

• Advantages of digraph variables
• Simple
• Open to many additional constraints
• Much more efficient than basic constraint
  programming (combines constraint programming with
  optimization algorithms on graphs)

67
Constraint Programming (10)

• Search strategy select the most important demand
  and the path for which the additional (marginal)
  cost for routing this demand is minimal
• Shortest path problem with constraints
• Successive constraints impose the last arc, then
  the previous arc, ..., and finally the first arc
  of the shortest path
• Each of these added constraints leads to creating
  a choice point upon backtracking, the imposed
  arc is forbidden and a new shortest path, taking
  this interdiction into account, computed

68
Constraint Programming (11)

• New solution obtained when all the paths are
  fully instantiated
• Search continues with an additional constraint
  forcing the cost of new solutions to be strictly
  smaller than the cost of the current solution

69
Constraint Programming (12)

• Slice-Based Search
• Search for the first solution given by the
  heuristic
• Search for solutions obtained by obeying the
  heuristic always except once
• Search for solutions obtained by obeying the
  heuristic always except twice

70
Constraint Programming (13)

• Variants of slice-based search
• Incrementing little by little the number of
  authorized discrepancies
• SBS 1 0 discrepancy, then 1, then 2, then 3,
• SBS 2 0 discrepancy, then 2, then 4, then 6,
• SBS 3 0 discrepancy, then 3, then 6, then 9,
• SBS N1 makes less redundant search than SBS N,
  but at the beginning SBS N is more focused on
  solutions close to the heuristic chosen as a basis

71
Constraint Programming (14)

• Test results (23 instances, 2 minutes)
• Equivalent results 13 instances
• SBS2 with the best CPU times
• SBS1 ( 28 / SBS2)
• SBS3 ( 74 / SBS2)
• SBS1 SBS2 better than SBS3 3 instances
• SBS1 SBS3 better than SBS2 2 instances
• SBS1 better than SBS2 better than SBS3 2
  instances
• SBS1 better than SBS3 better than SBS2 1
  instance
• SBS2 better than SBS1 better than SBS3 1
  instance
• SBS3 better than SBS1 better than SBS2 1
  instance (the most difficult)

72
Constraint Programming (15)

• Test results (23 instances, 10 minutes)
• Equivalent results 16 instances
• SBS2 with the best CPU times
• SBS1 ( 16 / SBS2)
• SBS3 ( 55 / SBS2)
• SBS1 SBS2 better than SBS3 2 instances
• SBS2 better than SBS1 better than SBS3 4
  instances
• SBS3 better than SBS1 better than SBS2 1
  instance (the most difficult) 
• Summary SBS1 is never the worst, SBS2 is the
  best for 10 minutes of CPU time, SBS3 is the best
  only on the most difficult instance

73
Column Generation (1)

• Principle
• Generate candidate solution components
• Find the optimal combination of these components
• Infer characteristics of  good  solution
  components to add
• Iterate
• Examples
• Train scheduling
• Crew scheduling
• Routing

74
Column Generation (2)

• Column Generation
• Routing variables paths xpd (a 0-1 variable per
  possible path)
• Total number lt D e (N!)
• Start with all paths with at most 2 arcs
• Dimensioning variables capacity levels yijk
  (MK 0-1 variables)
• Possibility to start with few capacity levels

75
Column Generation (3)

• Master Problem
• With integer variables to obtain feasible
  solutions
• With floating point variables to obtain lower
  bounds and dual values
• Sub-Problem
• To integrate new routing variables

76
Column Generation (4)

• Optional constraints in column generation
• Maximal number of arcs in the shortest path
  sub-problem
• Maximal number of ports and traffic per node in
  the master problem
• Security constraints in both the master and the
  sub-problem

77
Column Generation (5)

• Repeat
• Once every 5 iterations, search for an integer
  solution (MIP)
• Search for the optimal solution of the continuous
  relaxation
• Get dual values
• Add columns
• Paths the best path for each demand if this path
  has a reduced cost smaller than the current gap
• Links
• If some paths have been added the best capacity
  level for each arc if its reduced cost is
  negative
• Otherwise the best capacity level for each arc
  if its reduced cost is smaller than the current
  gap
• Final optimization with integer variables (MIP)

78
Column Generation (6)

• Variant Use the previous algorithm with a
  systematic limit on the number of arcs of a path
• Start with paths with at most 4 arcs
• When the problem is solved with a given limit,
  increment the allowed number of arcs and iterate

79
Pros and Cons of Different Techniques (1)

• Constraint Programming
• Global constraints on paths
• The overall cost is a sum of many step functions
  (almost no propagation)
• Mixed Integer Programming
• Sum objective handled with a global view
• No good model for mono-routing (in the
  relaxation, the LP solver provides a flow)
• Bad continuous relaxation of the step functions
• Column Generation
• Sum objective handled with a global view
• A column is a path
• Bad continuous relaxation of the step functions

80
Pros and Cons of Different Techniques (2)

• Security
• CP Easy to model with logical constraints but no
  global propagation
• MIP, CG Leads to lots of fractional values in
  the relaxation (e.g., routing a demand on two
  paths, each made of half-secure and
  half-unsecured links)
• No line multiplication
• CP, MIP, CG Smaller problem
• MIP, CG Impact on the continuous relaxation of
  the step functions
• Symmetric routing
• CP, MIP, CG Smaller problem

81
Pros and Cons of Different Techniques (3)

• Number of bounds (arcs)
• CG Much less potential paths and paths much
  easier to generate (especially when the number of
  bounds is really small)
• CP More propagation but with more complex
  algorithm
• MIP Easy to model (sum of 0-1 variables
  representing the presence of each arc in a path)
  but more fractional values in the relaxation
• Number of ports
• CP Easy to model with logical constraints but no
  global propagation
• MIP, CG Requires additional integer variables
  (with fractional values in the relaxation)
• Maximal traffic
• MIP, CG Linear constraints
• CP Linear constraints with no global propagation

82
Improvements of MIP (1)

• Adding cuts
• For each demand and each node, at most one arc
  entering (or leaving) the node can be used.
• For each node, the sum of the capacities of the
  arcs entering (or leaving) the node must be
  greater than or equal to the sum of the demands
  arriving at (or starting from) the node plus the
  sum of the demands traversing the node.
• For each demand and each arc, the routing of the
  demand through the arc excludes, for this arc,
  the multiplied capacity levels strictly inferior
  to the demand.
• Impact mostly on the lower bound.

83
Improvements of MIP (2)

• Cumulative formulation
• Rather than using a 0-1 variable yk for each
  capacity level k, use a 0-1 variable dk to
  represent the decision to go from a capacity
  level to the next, i.e., dk yk1 yk.
• Impact mostly on the lower bound
• Specialized search strategy
• More solutions found in 10 minutes or less but of
  poorer quality
• Fixing heuristic
• Fix all variables with integer values in the
  current relaxation
• Solve a sub-MIP on the other variables
• Large neighborhood search

84
Improvements of CG (1)

• Varying the time allocated to the generation of
  integer solutions
• More solutions found in 10 minutes or less but of
  poorer quality
• Use Branch-and-Price
• Branching on capacity levels
• Branching on paths
• Keep a pool of  promising  columns
• Keep regular calls to a MIP solver to generate
  integer solutions, but keep the number of nodes
  explored  small 

85
Improvements of CG (2)

• Use Branch-and-Price-and-Cut
• Set cuts lower bound on the total capacity of
  the links that cross the frontier of some given
  set of nodes.
• BigDemand cuts lower bound on the number of
  links opened with a given minimal capacity that
  cross the frontier of some given set of nodes.
• Connexity cuts lower bound on the number of open
  links that have at least one extremity in a given
  set of nodes.
• Link cuts constrain the capacity of an arc to be
  greater than or equal to the demand for a path
  going through this arc (only these have an impact
  on path pricing).
• On small instances, about 70 of the gap is
  closed.

86
Improvements of CP (1)

• Changing the order in which demands are routed
• Weight 2 demand 1 reverse-demand
• Example
• D01 1800, D10 950, D23 1000, D32 1000,
  D45 1900, D54 50
• Symmetric routing D01 and D10, then D23 and D32,
  then D45 and D54
• Non-symmetric routing D45, D01, D23, D32, D10,
  D54
• After the modification
• Symmetric routing D01 and D10, then D45 and D54,
  then D23 and D32
• Non-symmetric routing D01, D45, D10, D23, D32,
  D54
• Gain on B10 3 on average
• Loss on the C series 1 on average

87
Improvements of CP (2)

• Specific (path-oriented) constraint propagation
  and heuristic
• Is there a path from p to q, with b arcs or less,
• that goes through n?
• that does not go through n?
• Heuristic find such a path with the smallest
  marginal cost
• Costly constraint propagation algorithm O(n4) in
  the worst case
• Gain on B12 3.5 on average over the 32 variants
  with limits on the number of bounds
• Loss on C12 0.4

88
Improvements of CP (3)

• Direct constraint between variables representing
  the paths and variables representing the traffic
  through each node
• Use of Parallel Solver
• A few lines of code
• Modification of the tree-search traversal
  strategy
• Branch more close to the root of the tree

89
Improvements of CP (4)

• Greedy local search
• Try to remove arcs from a solution without adding
  any new arc (allowing some increase in existing
  arc capacities)
• Large neighborhood search
• Randomization in the choice of the paths to
  re-instantiate
• Structured neighborhood randomly pick two arcs
  and re-instantiate all paths going through any of
  these arcs
• Randomization in the instantiation order
• Portfolio of heuristics to penalize paths (with
  weights assigned to each heuristic and adjusted
  during search)
• Parallel multi-point large neighborhood search
  different workers work in parallel on different
  fragments

90
Current Results (1)
91
Current Results (2)
92
Current Results (3)
93
Integration (1)
94
Integration (2)

• Select optimization algorithm

95
Integration (3)

• Submit data file

96
Integration (4)

• Display results

97
Integration (5)

• Partially done

bench.txt
98
Conclusion (1)

• Good results obtained by combining constraint
  programming (on graphs) and local optimization
• In the absence of optional constraints solutions
  comparable to those provided by current tools
• Capacity to take optional constraints into
  account
• Branch-and-Price-and-Cut also provides good
  results
• The benchmark remains challenge
• Optimum in general unknown
• Instances very different one from the other

99
Conclusion (2)

• Integration of optional constraints
• CP 5 to 20 lines of code added in a
  non-intrusive manner for each of the six optional
  constraint (thanks to the graph library)
• MIP 10 to 40 lines
• CG 10 to 40 lines, in multiple places in the
  code
• Other constraints could be harder to add
• Adjustments necessary in all cases to improve
  efficiency

100
Plan

• Real world Problems
• Modeling principles
• Network Design application the ROCOCO project
• Presentation
• Methods
• Integration
• Conclusion
• Lessons

101
Lessons

• What do we learn?
• Surprising things!
• Do not forget the essential

102
The key idea path variable!
103
Assume P ? NP

• Ok, we cannot avoid an exponential behavior
• For some instances, an NP Complete Problem will
  required an exponential time to be solved
• So, our only hope is to shift the exponential
  such that the problem is solvable for a size and
  a time that are acceptable

104
Shifting the exponential
105
Shifting the exponential
106
The key idea path variable!
107
We worked a lot!
108
General considerations

• When solving a problem in CP
• Potential performance gain
• data structure optimization (code) x 10
• search strategies x 1 000
• model x 1 000 000
• Repartition of effort for ROCOCO
• data structure optimization (code) 65
• search strategies 25
• model 10

109
Conclusion

• CP is a general technique can encapsulate a lot
  of work
• CP is an efficient method for solving some
  combinatorial problems small or large
• Filtering algorithms are quite important for non
  binary constraints
• CP allows the use of sophisticated strategies
• If you want to use CP think CP (avoid Boolean
  (0-1) variables). CP allows the use of symbolic
  representation

110
More information

• Go to www.ilog.com
• Go to my webpagewww.constraint-programming.com/p
  eople/reginor google my name Jean Charles Regin