Solving Finite Domain Hierarchical Constraint Optimization Problems By Lua Seet Chong Supervised By A.P. Martin Henz 9th March 2001

Solving Finite Domain Hierarchical Constraint Optimization

Problems

By

Lua Seet Chong

Supervised By A.P. Martin Henz

9th March 2001

Outline

• Motivation, Project Background

• Constraint Hierarchies

• Tree Search

• Local Search

• Experimental Results:– Gate Allocation Problem– Sports Scheduling Problem

• Conclusion

“Integrate” Project Background

• Solve the gate allocation problem

• Domain knowledge provided by CAAS and Changi Airport

• KRDL provides the management support

• Sponsored by NSTB

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

Previous Works• Constraint Hierarchies

Hierarchical Constraint Logic Programming, Alan Borning, 1992

• Over-constrained Integer Programming

WSAT(OIP), Joachim Paul Walser, 1997

Problem Encoding

• Propositional Satisfiability Problems (SAT)– represent the problem in CNF

• Constraint Satisfaction Problems (CSP)– allow many types of formulation– example: linear programming

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.

Over-Constrained Problems

• Constraint problems where conflicting constraints exist

• Hierarchical Constraints

Constraint Hierarchies

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 }

Feasible Region Boundedby Cb and Cb

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}

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}

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

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


• 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( , )}


• 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)


• 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

Tree Search

Finite DomainConstraint Programming

• Successful technique for solving combinatorial problems.

• 3 main components:– Propagation Algorithms– Branching Algorithms– Exploration Algorithms

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

Enumeration Algorithms

• Enumeration algorithm if

c is in the form of x = v

• 2 heuristics:– variable selection heuristic– value selection heuristic

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

Cost Driven Value Selection

Hierarchical Cost Driven Value Selection

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

Local Search

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]

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

Generalization of WSAT(OIP)

• Any constraints (i.e non-linear, symbolic)– select-partial-repair

• Constraint hierarchy – improve– select-unsatisfied-constraint

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

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

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)

Gate Allocation Problem

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

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

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

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

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

Example: FD Model of No Overlapping

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

Objectives of Experiments

• Comparing 0/1 model vs finite domain model

• Comparing the performance of proposed constraint selection scheme

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

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

Benchmark Problems

• Small Problem (P1 - P6)– ranging from 10 - 30 flights

• Bigger Problem (P7 - P15)– ranging from 50 -257 flights

Comparison of Solving Time

Performance of Finite Domain vs 0/1 Model using Best select-unsatisfied-constraint

Performance of Finite Domain vs 0/1 Model for Bigger Test Cases

Performance of Different Constraint Selection Scheme on FD Model

Performance of Different Constraint Selection Scheme on 0/1 Model

Performance of Different Constraint Selection Schemeon FD Model for Bigger Test Cases

Performance of Different Constraint Selection Scheme on 0/1 Modelfor Bigger Test Cases

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]

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

Sports Scheduling Problem

Number of Moves to Solve ACC Problem

WalkSearch

WSAT(OIP)

CPU Time Taken to Solve ACC Problem

WalkSearch

WSAT(OIP)

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

Conclusion

Conclusion

• Empirical Study on Changi Airport Gate Allocation Problem

• Hierarchical Cost Driven Descent is able to solve Gate Allocation Problem

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

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.

Thank You

Documents

Solving Finite Domain Hierarchical Constraint Optimization Problems By Lua Seet Chong Supervised By A.P. Martin Henz 9th March 2001