Solving Finite Domain Hierarchical Constraint Optimization Problems

1
Solving Finite Domain Hierarchical Constraint
Optimization Problems

• By
• Lua Seet Chong
• Supervised By A.P. Martin Henz
• 9th March 2001

2
Outline

• Motivation, Project Background
• Constraint Hierarchies
• Tree Search
• Local Search
• Experimental Results
• Gate Allocation Problem
• Sports Scheduling Problem
• Conclusion

3
Integrate Project Background

• Solve the gate allocation problem
• Domain knowledge provided by CAAS and Changi
  Airport
• KRDL provides the management support
• Sponsored by NSTB

4
Motivation

• Gate Allocation Problems
• large combinatorial optimization problems with
  many complex soft and easy hard constraints
• Local Search
• Constraint Hierarchies
• Flexibility of using symbolic constraints

5
Previous Works

• Constraint Hierarchies
• Hierarchical Constraint Logic Programming, Alan
  Borning, 1992
• Over-constrained Integer Programming
• WSAT(OIP), Joachim Paul Walser, 1997

6
Problem Encoding

• Propositional Satisfiability Problems (SAT)
• represent the problem in CNF
• Constraint Satisfaction Problems (CSP)
• allow many types of formulation
• example linear programming

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Why CSP is successful

• Clean separation between problem encodings and
  problem solving techniques
• Flexibility to extend the problem encoding by
  adding new constraint type
• Synergy problem solving techniques for all
  constraint types work with each other.

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Over-Constrained Problems

• Constraint problems where conflicting constraints
  exist
• Hierarchical Constraints

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Constraint Hierarchies
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Constraint Hierarchy Example

• Constraints
• ca y -x 10
• cb y ? 4x
• cc y ? x 8
• cd y ? 5
• Constraint Hierarchy
• C0 ca
• C1 cb , cc
• C2 cd

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Feasible Region Bounded by Cb and Cb
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Constraint Hierarchy Example
P2x ? 4, y ? 6
P3x ? 1, y ? 9
e(cd ) 1
e(cd ) 4
e(cc ) 0
e(cc ) 0
e(cb ) 2.3
e(cb ) 0
e(ca ) 0
e(ca ) 0
Weighted-Sum-Better
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Constraint Hierarchy Example
P4x ? 2, y ? 8
P3x ? 1, y ? 9
e(cd ) 3
e(cd ) 4
e(cc ) 0
e(cc ) 0
e(cb ) 0
e(cb ) 0
e(ca ) 0
e(ca ) 0
Weighted-Sum-Better
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Constraint Hierarchy

• X set of variables
• For each x ? X,
• Dx finite set of values that x can take
• k-nary constraint over variables x1,, xk is a
  relation over Dx1 ? ? Dxk

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Constraint Hierarchy

• Constraint Hierarchy,
• CH is a vector ?C0 ,C1 ,,Cn?
• where for each 0 ? i ? n,
• Ci is a multiset constraints of rank i
• C0 contains required (hard) constraints
• C1,,Cn denote preferential (soft) constraints

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Constraint Hierarchy

• A valuation ? is a function that maps the
  variables in X to elements in the domain D
• Solution set S0 ? ?c ? C0, c? holds
• Optimal solution set,
• Sbetter ? ?S0 ?? ? S0 , ?better(? , ? )

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Constraint Hierarchy

• Comparator
• weighted-sum-better(? , ? ) ?
• ?k. 1 ? k ? n such that
• ?i ?1k-1.
• rank-sum(? ,Ci) rank-sum(? , Ci)
• ? rank-sum(? ,Ck) lt rank-sum(? , Ck)

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Constraint Hierarchy

• rank-sum(? ,Ci) ? ?c?Ciw(c)e(c? )
• where w(c) is a real number weight for
  constraint c and e(c? ) is an error function

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Tree Search
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Finite DomainConstraint Programming

• Successful technique for solving combinatorial
  problems.
• 3 main components
• Propagation Algorithms
• Branching Algorithms
• Exploration Algorithms

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Branching Algorithms

• Assume
• a stable constraint store s and
• a branching constraint c
• A branching algorithm make use of a branching
  constraint c looking into 2 new constraint stores
  s ? c and s ? ?c

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Enumeration Algorithms

• Enumeration algorithm if
• c is in the form of x v
• 2 heuristics
• variable selection heuristic
• value selection heuristic

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Cost Driven Value Selection

• A value selection heuristic that order the values
  using the cost variable
• 2 Variant of Search
• Cost Driven Search
• Cost Driven Descent

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Cost Driven Value Selection
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Hierarchical Cost Driven Value Selection

• Order the values within a variable according to
  the hierarchical comparator instead of a integer
  comparator

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Local Search
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Walk Search Background

• GSAT, WSAT ? WSAT(OIP)
• Bart Selman and Henry Kautz, 1993
• WSAT(OIP) generalized the SAT problem solving
  techniques to solve over-constrained integer
  programming problems Walser 1997

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WalkSearch Algorithm

• Proc WalkSearch(C, X, Max_moves, Max_tries)
• for i 1 to Max_tries do
• ? an initial assignment
• ?_best ?
• for j 1 to Max_moves do
• if ? meets solution stopping condition
• then return ?
• if ? is feasible ? improve(?, ?_best, C)
• then ?_best ?
• c select-unsatisfied-constraint(C, X,
  ?)
• ltxk,vgt select-partial-repair(C, X, c,
  ?)
• ? ?xk ? v
• end
• end
• return ?_best
• end

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Generalization of WSAT(OIP)

• Any constraints (i.e non-linear, symbolic)
• select-partial-repair
• Constraint hierarchy
• improve
• select-unsatisfied-constraint

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select-partial-repair

• Generate the VarValue pairs
• Remove tabued VarValue pairs
• Choose VarValue pair that give the highest score.
  Tie breaking using i) least frequently ii)
  longest time ago
• If the chosen VarValue pair does not improve the
  global score, choose any pair from VarValue with
  probability pnoise

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select-unsatisfied-constraint

• hardOrSoft, select a violated constraint in C0
  with probability Phard
• topOrRest, select a violated constraint in top
  most unsatisfied rank with probability Ptop
• rankProb, select a violated constraint in Ci with
  probability Pi
• consProb, select a violated constraint in Ci with
  a dynamic probability Dpi which is
• Pi Civiolated
• ?j?0,,n Pj Cjviolated

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Why consProb?
Prob. for selecting a constraint in rank i
?Ci?
Rank i
Pi
rankProb
consProb
100
0
1000
100/111 ? 1000 0.0009009
100
(10)
10/111 ? 10 0.009009
10
1
10
10
(100)
1/111 ? 100 0.00009009
1
2
100
1
(1)
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Gate Allocation Problem
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Gate Allocation Problem (GAP)

• Allocating gates to arriving and departing
  aircrafts, Haghani, 1998, Yu Cheng, 1998
• Minimizing Transfer Walking Distance
• Work on instances from Changi Airport

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GAP constraints

• No Overlapping - No two aircraft can be allocated
  to the same gate simultaneously
• Aircraft Type - Particular gates can be
  restricted to admit only certain aircraft types
• Push Back - An aircraft leaving a gate
  (push-back) will restrict other operations in
  close temporal and spatial vicinity
• 22 more constraints

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GAP 0/1 Model

• GAP with m aircrafts and n gates
• m?n 0/1 variables Yij are introduced where
• 1 ? i ? m and 1 ? j ? n
• Yij 1 iff aircraft i is allocated to gate j

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Example 0/1 Model of No Overlapping

• If aircraft i and k has overlapping ground time,
  for every gate j where 1 ? j ? n
• Yij Ykj ? 1

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GAP Finite Domain Model

• GAP with m aircrafts and n gates
• For each aircraft i, Xi is introduced to
  represent the gate aircraft i uses
• The domain of Xi is 1 to n
• Xi j iff aircraft i is allocated to gate j

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Example FD Model of No Overlapping

• For each maximal set of aircraft S whose ground
  time overlaps, a symbolic constraint alldiff(S)
  is introduced

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Objectives of Experiments

• Comparing 0/1 model vs finite domain model
• Comparing the performance of proposed constraint
  selection scheme

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Setup of Experiments

• For each problem model, benchmark problem and
  constraint selection scheme
• pNoise (0.0, 0.1, , 0.5)
• 5 probability distributions among ranks
• 8421
• 90.330.330.33
• 80.50.51
• 6112
• 1000100101

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Setup of Experiments

• Small benchmarks allow optimality comparison
• use CPLEX to find optimal solution
• count how often optimal solution is reached
• Large benchmarks
• compare scaling behavior
• use relative solution quality to compare

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Benchmark Problems

• Small Problem (P1 - P6)
• ranging from 10 - 30 flights
• Bigger Problem (P7 - P15)
• ranging from 50 -257 flights

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Comparison of Solving Time
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Performance of Finite Domain vs 0/1 Model using
Best select-unsatisfied-constraint
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Performance of Finite Domain vs 0/1 Model for
Bigger Test Cases
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Performance of Different Constraint Selection
Scheme on FD Model
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Performance of Different Constraint Selection
Scheme on 0/1 Model
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Performance of Different Constraint Selection
Scheme on FD Model for Bigger Test Cases
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Performance of Different Constraint Selection
Scheme on 0/1 Model for Bigger Test Cases
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Hierarchical Cost Driven Descent on GAP
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Experimental Result Summary

• Finite Domain model allows WalkSearch solver to
  works better than 0/1 model
• Constraint hierarchy specific
• select-unsatisfied-constraint perform better
  (ConsProb, RankProb)
• Hierarchical Cost Driven Descent is able to find
  reasonably good solution

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Sports Scheduling Problem
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Number of Moves to Solve ACC Problem
WalkSearch
WSAT(OIP)
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CPU Time Taken to Solve ACC Problem
WalkSearch
WSAT(OIP)
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Experimental Result Summary

• WalkSearch solver works for a non-hierarchical
  constraint problem
• WalkSearch solver works as good as WSAT(OIP)
• WalkSearch find solution more frequently than
  WSAT(OIP) but it runs slower

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Conclusion
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Conclusion

• Empirical Study on Changi Airport Gate Allocation
  Problem
• Hierarchical Cost Driven Descent is able to solve
  Gate Allocation Problem

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Conclusion

• Adapted Contraint Hierarchies to Finite Domain
  Problem
• Hierarchy-specific constraint selection scheme
  helps to find better solutions
• WalkSearch Solver can solve both 0/1 and Finite
  Domain Hierarchical Constraint Problems

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Acknowledgement

• Integrate project members
• Roland H. C. Yap
• J. Paul Walser
• Lim Yun Fong
• Shi Xiao Ping
• Hu You Lan
• We thank Civil Aviation Authority of Singapore,
  Kent Ridge Digital Labs for providing documents
  and test data sets on the Changi Airport gate
  allocation problem.

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? Thank You ?