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 c a : y = -x + 10 c b : y  4x c c : y  x + 8 c d : y  5 Constraint Hierarchy C 0 : { c a } C 1 : {c b, c c } C 2 : { c d }

Feasible Region Bounded by Cb and Cb

Constraint Hierarchy Example e(c a ) = 0 e(c b ) = 2.3 e(c c ) = 0 e(c d ) = 1 e(c a ) = 0 e(c b ) = 0 e(c c ) = 0 e(c d ) = 4 Weighted-Sum-Better P2:{x  4, y  6} P3:{x  1, y  9}

Constraint Hierarchy Example e(c a ) = 0 e(c b ) = 0 e(c c ) = 0 e(c d ) = 3 e(c a ) = 0 e(c b ) = 0 e(c c ) = 0 e(c d ) = 4 Weighted-Sum-Better P4:{x  2, y  8} P3:{x  1, y  9}

Constraint Hierarchy X : set of variables For each x  X, D x : finite set of values that x can take k-nary constraint over variables x 1,…, x k is a relation over D x 1  …  D x k

Constraint Hierarchy Constraint Hierarchy, C H is a vector  C 0,C 1,…,C n  where for each 0  i  n, C i is a multiset constraints of rank i C 0 contains required (hard) constraints C 1,…,C n denote preferential (soft) constraints

Constraint Hierarchy A valuation  is a function that maps the variables in X to elements in the domain D Solution set S 0 = {  |  c  C 0, c  holds} Optimal solution set, S better = {   S 0 |   S 0,  better( ,  )}

Constraint Hierarchy Comparator weighted-sum-better( ,  )   k. 1  k  n such that  i  {1…k-1}. rank-sum( ,C i ) = rank-sum( , C i )  rank-sum( ,C k ) < rank-sum( , C k )

Constraint Hierarchy rank-sum( ,C i )   c  Ci w(c)e(c  ) where w(c) is a real number weight for constraint c and e(c  ) is an error function

Tree Search

Finite Domain Constraint 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,  ); := select-partial-repair(C, X, c,  );  :=  [x k  v]; 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 p noise

select-unsatisfied-constraint hardOrSoft, select a violated constraint in C 0 with probability P hard topOrRest, select a violated constraint in top most unsatisfied rank with probability P top rankProb, select a violated constraint in C i with probability P i consProb, select a violated constraint in C i with a dynamic probability D p i which is P i |C i | violated  j  {0,…,n} P j |C j | violated

Why consProb? CiCi Rank i PiPi rankProb consProb 1/111  100 = /111  10 = /111  1000 = Prob. for selecting a constraint in rank i (10) (100) (1)(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 m  n 0/1 variables Y ij are introduced where 1  i  m and 1  j  n Y ij = 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 Y ij + Y kj  1

GAP Finite Domain Model GAP with m aircrafts and n gates For each aircraft i, X i is introduced to represent the gate aircraft i uses The domain of X i is 1 to n X i = 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 flights Bigger Problem (P7 - P15) –ranging from 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 Scheme on FD Model for Bigger Test Cases

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

Hierarchical Cost Driven Descent on GAP ProblemsPessimistic HCDSOptimistic HCDS P1[ ] P2[ ][ ] P3[ ][ ] P4[ ][ ] P5[ ][ ] P6-- P7[ ][ ] P8[ ][ ]

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

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