solving finite domain hierarchical constraint optimization problems by lua seet chong supervised by...
TRANSCRIPT
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Solving Finite Domain Hierarchical Constraint Optimization
Problems
By
Lua Seet Chong
Supervised By A.P. Martin Henz
9th March 2001
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Outline
• Motivation, Project Background
• Constraint Hierarchies
• Tree Search
• Local Search
• Experimental Results:– Gate Allocation Problem– Sports Scheduling Problem
• Conclusion
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“Integrate” Project Background
• Solve the gate allocation problem
• Domain knowledge provided by CAAS and Changi Airport
• KRDL provides the management support
• Sponsored by NSTB
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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
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Previous Works• Constraint Hierarchies
Hierarchical Constraint Logic Programming, Alan Borning, 1992
• Over-constrained Integer Programming
WSAT(OIP), Joachim Paul Walser, 1997
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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 Boundedby Cb and Cb
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Constraint Hierarchy Example
e(ca ) = 0
e(cb ) = 2.3e(cc ) = 0
e(cd ) = 1
e(ca ) = 0
e(cb ) = 0e(cc ) = 0
e(cd ) = 4
Weighted-Sum-Better
P2:{x 4, y 6} P3:{x 1, y 9}
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Constraint Hierarchy Example
e(ca ) = 0
e(cb ) = 0e(cc ) = 0
e(cd ) = 3
e(ca ) = 0
e(cb ) = 0e(cc ) = 0
e(cd ) = 4
Weighted-Sum-Better
P4:{x 2, y 8} P3:{x 1, y 9}
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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 {1…k-1}.
rank-sum( ,Ci) = rank-sum( , Ci)
rank-sum( ,Ck) < rank-sum( , Ck)
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Constraint Hierarchy
• rank-sum( ,Ci) cCiw(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, ); <xk,v> := 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 |Ci|violated
j{0,…,n} Pj |Cj|violated
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Why consProb?
Ci
1
10
100
Rank i Pi
2
1
0
100
10
1000
rankProb consProb
1/111 100= 0.00009009
10/111 10= 0.009009
100/111 1000= 0.0009009
1
10
100
Prob. for selecting a constraint in rank i
(10)
(100)
(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
• mn 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
• 8:4:2:1
• 9:0.33:0.33:0.33
• 8:0.5:0.5:1
• 6:1:1:2
• 1000:100:10:1
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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 Schemeon FD Model for Bigger Test Cases
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Performance of Different Constraint Selection Scheme on 0/1 Modelfor Bigger Test Cases
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Hierarchical Cost Driven Descent on GAP
Problems Pessimistic HCDS Optimistic HCDS
P1 [0 0 5 1442] [0 0 5 1442]
P2 [0 2 25 11096] [0 2 15 34168]
P3 [0 2 25 11096] [0 2 20 22442]
P4 [0 2 25 11096] [0 2 20 22442]
P5 [0 3 21 14214] [0 3 15 39386]
P6 - -
P7 [0 4 31 48630] [0 4 46 73942]
P8 [0 4 41 50107] [0 4 56 73942]
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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