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

7. 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.

8. Over-Constrained Problems • Constraint problems where conflicting constraints exist • Hierarchical Constraints

9. Constraint Hierarchies

10. 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 }

11. Feasible Region Bounded by Cb and Cb

12. Constraint Hierarchy Example P2:{x 4, y  6} P3:{x 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

13. Constraint Hierarchy Example P4:{x 2, y  8} P3:{x 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

14. 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

15. Constraint Hierarchy • Constraint Hierarchy, CHis 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

16. 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( ,  )}

17. 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)

18. 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

19. Tree Search

20. Finite DomainConstraint Programming • Successful technique for solving combinatorial problems. • 3 main components: • Propagation Algorithms • Branching Algorithms • Exploration Algorithms

21. 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

22. Enumeration Algorithms • Enumeration algorithm if c is in the form of x = v • 2 heuristics: • variable selection heuristic • value selection heuristic

23. 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

24. Cost Driven Value Selection

25. Hierarchical Cost Driven Value Selection • Order the values within a variable according to the hierarchical comparator instead of a integer comparator

26. Local Search

27. 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]

28. 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

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

30. 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

31. select-unsatisfied-constraint • hardOrSoft, select a violated constraint in C0with 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

32. Why consProb? Prob. for selecting a constraint in rank i Pi Ci Rank i rankProb consProb 0 100 1000 100/111  1000 = 0.0009009 100 (10) 10/111  10 = 0.009009 1 10 10 10 (100) 1/111  100 = 0.00009009 2 1 100 1 (1)

33. Gate Allocation Problem

34. 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

35. 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

36. 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

37. 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

38. 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

39. Example: FD Model of No Overlapping • For each maximal set of aircraft S whose ground time overlaps, a symbolic constraint alldiff(S) is introduced

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

41. 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

42. 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

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

44. Comparison of Solving Time

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

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

47. Performance of Different Constraint Selection Scheme on FD Model

48. Performance of Different Constraint Selection Scheme on 0/1 Model

49. Performance of Different Constraint Selection Scheme on FD Model for Bigger Test Cases

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