Conflict Analysis

1
Conflict Analysis

• SCIP Workshop at ZIB
• October 2007

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General Idea

• Problem is divided into smaller subproblems
• branching tree
• Some subproblems are infeasible
• Analyzing the infeasibilities can yield
  information about the problem
• Information can be used at other subproblems to
  prune the search tree

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Example
x1 1
x2 0
x3 0

• a subproblem is infeasible

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Example
x1 1
x2 0
x3 0

• a subproblem is infeasible
• conflict analysis yields feasible constraint,
  that cuts off other nodes in the tree

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Outline

• Conflict Analysis in SAT
• Conflict Analysis in MIP
• Implementation
• Computational Results

6
Outline

• Conflict Analysis in SAT
• Conflict Analysis in MIP
• Implementation
• Computational Results

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Satisfiability Problem (SAT)

• Boolean variables x1,...,xn ? 0,1
• Clauses
• Task
• find assignment x ? 0,1n that satisfies all
  clauses, or prove that no such assignment exists

with literals lik xj or lik ?xj 1 xj
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Binary Constraint Propagation

• clause

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Binary Constraint Propagation

• clause
• fixings

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Binary Constraint Propagation

• clause
• fixings
• deduction

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Conflict Graph

• decision variables
• deduced variables
• conflict

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Conflict Graph
conflict detecting clause

• decision variables
• deduced variables
• conflict

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Conflict Graph Conflict Cuts

• choose cut that separates decision variables from
  conflict vertex

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Conflict Graph Conflict Cuts
reason side
conflict side

• choose cut that separates decision variables from
  conflict vertex

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Conflict Graph Conflict Cuts
reason side
conflict side

• choose cut that separates decision variables from
  conflict vertex
• conflict clause

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Conflict Graph Trivial Cuts

• ?-cut
• decision cut

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Conflict Graph First-UIP

• Unique Implication Point lies on all paths from
  the last decision vertex to the conflict vertex

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Conflict Graph First-UIP

• Unique Implication Point lies on all paths from
  the last decision vertex to the conflict vertex
• First UIP the UIP closest to ? (except ? itself)

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Conflict Graph First-UIP Cut

• Put all vertices fixed after First-UIP to the
  conflict side, remaining vertices to the reason
  side
• First-UIP cut

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Conflict Analysis Algorithm (FUIP)

• BCP detected a conflict

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Conflict Analysis Algorithm (FUIP)

• Initialize conflict queue with the variables
  involved in the conflict

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Conflict Analysis Algorithm (FUIP)

• Initialize conflict queue with the variables
  involved in the conflict

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

26
Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

27
Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

28
Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

29
Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

30
Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

31
Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• If there is only one variable of the last depth
  level left, stop

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Conflict Analysis Algorithm (FUIP)

• If there is only one variable of the last depth
  level left, stop
• The remaining variables define the conflict set

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Conflict Analysis Algorithm (FUIP)

• The conflict clause consists of all (negated)
  assignments in the conflict set

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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• Implementation
• Computational Results

36
Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• Implementation
• Computational Results

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Mixed Integer Program

• linear objective function c
• linear constraints Ax ? b
• real or integer valued variables x

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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint

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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint

technical issue
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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint
• main reason for infeasibility LP relaxation
• conflict graph has no link from the decision and
  deduction vertices to the conflict vertex

technical issue
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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint
• main reason for infeasibility LP relaxation
• conflict graph has no link from the decision and
  deduction vertices to the conflict vertex

technical issue
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Back to SAT Conflict Graph
conflict detecting clause

• One clause detected theconflict
• Only a few variables linked to the conflict vertex

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Infeasible LP Conflict Graph

• The LP as a whole is responsible for the conflict
• All local bound changes are linked to the
  conflict vertex

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Infeasible LP Conflict Graph

• LP analysis selects some of these local bounds

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Infeasible LP Conflict Graph

• LP analysis selects some of these bounds
• cut yields conflict constraint

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Conflict Analysis for infeasible LPs

• if the LP relaxation is infeasible, the whole
  relaxation is involved in the conflict
• all constraints
• all global bounds
• all local bounds
• try to find a small subset of the local bounds
  that still leads to an infeasible LP relaxation
• variant of minimal infeasible subsystem
  problem(see Amaldi, Pfetsch, Trotter)
• heuristic use dual ray to relax local bounds

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Infeasible LP Dual Ray Heuristic

• LP relaxation

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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP

local bounds
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP
• dual ray

local bounds
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP
• dual ray

local bounds
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP
• dual ray
• relax bounds

local bounds
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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Implementation

• Which pruned subproblems should be analyzed?
• propagation conflicts yes
• infeasible LP conflicts yes
• bound-exceeding LP conflicts no
• strong branching conflicts no
• How many conflict constraints per conflict?
• all FUIP constraints 1-FUIP, 2-FUIP, ...
• How should conflict constraints be used?
• only for propagation, not as cutting planes
• How can useless conflict constraints be
  identified?
• "easy" for SAT due to depth-first-search
• open topic for best-first-search MIP, needs more
  ideas!
• currently aging mechanism

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How to Support Conflict Analysisin User Plugins

• affected plugin types
• constraint handlers (propagation)
• propagator
• special bound inference methods for propagation
• SCIPinferVarLbCons(), SCIPinferVarUbCons()
• SCIPinferVarLbProp(), SCIPinferVarUbProp()
• provide constraint and one additional integer
  (inferinfo)
• reverse propagation
• must provide the reason for a bound deduction
• can use the inferinfo to help finding the reason

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Reverse Propagation

• input constraint and tightened bound
• deduced

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Reverse Propagation

• input constraint and tightened bound
• deduced
• output bounds that are reason for deduction
• and

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Reverse Propagation Examples

• set covering constraints
• only one type of propagation
• reverse propagation easyall variables except
  the inferred variable are reason variables

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Reverse Propagation Examples

• set covering constraints
• only one type of propagation
• reverse propagation easyall variables except
  the inferred variable are reason variables
• linear constraints
• propagation of left and right hand side
• inferinfo array position of inferred variable
  and lhs/rhs indicator
• reverse propagationfind set of variables j ? k
  with bounds that trigger the deduction

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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
61
Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Computational Results

• Instances of MIPLIB 2003 and Mittelmann
• that could be solved by one of the three in 1
  hour
• for which one of the three needed at least 1000
  nodes

nodes -20 time -14
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Computational Results

• ALU instances up to 8 bits
• for which one of the three needed at least 1000
  nodes
• infeasible MIP instances

nodes -85 time -58
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Computational Results

• "Enlight" instances up to board size 10 ? 10
• for which one of the three needed at least 1000
  nodes
• 2 infeasible and 4 feasible MIP instances

nodes -78 time -59
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Computational Results

• instance "arki001" from MIPLIB 3.0 / MIPLIB 2003
• first solved by Balas and Saxena (Dec 2005)
• "Optimizing over the Split Closure"
• 54 hours rank-1 split cut generation 11 hours
  CPLEX 9.0
• Bob-tuned CPLEX
• "using a lot of strong branching"
• 628 hours, 103 million nodes
• SCIP
• using an insane amount of strong branching

• full conflict analysis (bound-exceeding LPs and
  strong branchings)

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Computational Results
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Computational Results
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Computational Results

• instance roll3000" from MIPLIB 2003
• SCIP
• using an insane amount of strong branching
• moderate conflict analysis
• 3.2GHz Pentium-IV

• Unfortunately...
• XPressMP 2006B with tuned settings 1 hour (3 GHz
  Athlon-64)
• Reported by Alkis Vazacopoulos (ISMP 2006)
• As far as I understood lots of local cuts