Solving a Challenging Quadratic 3D Assignment Problem

Transcription

1 Solving a Challenging Quadratic 3D Assignment Problem Hans Mittelmann Arizona State University Domenico Salvagnin DEI - University of Padova

2 Quadratic 3D Assignment Problem

3 Quadratic 3D Assignment Problem

4 Quadratic 3D Assignment Problem

5 Quadratic 3D Assignment Problem

6 Quadratic 3D Assignment Problem

7 Our Instance 16-PSK digital communication retransmission protocol

8 Our Instance 16-PSK digital communication retransmission protocol

9 Our Instance 16-PSK digital communication retransmission protocol

10 Our Instance 16-PSK digital communication retransmission protocol extremely dense objective: >12M coefficients

11 Our Instance 16-PSK digital communication retransmission protocol extremely dense objective: >12M coefficients high dynamism 3.6 x 10 12

12 Our Instance 16-PSK digital communication retransmission protocol extremely dense objective: >12M coefficients high dynamism 3.6 x symmetric: group of order

13 Parallel mode: deterministic, using up to 16 threads. Root relaxation solution time = sec. ( ticks)! Nodes Cuts/! Node Left Objective IInf Best Integer Best Bound ItCnt Gap * % % Cuts: % Cuts: % Heuristic still looking % Elapsed time = sec. ( ticks, tree = 0.01 MB, solutions = 1) % e % cutoff e % e % e % Elapsed time = sec. ( ticks, tree = MB, solutions = 1) Nodefile size = MB ( MB after compression) e % e % e % e % e % cutoff e %

14 Parallel mode: deterministic, using up to 16 threads. Root relaxation solution time = sec. ( ticks)! Nodes Cuts/! Node Left Objective IInf Best Integer Best Bound ItCnt Gap * % % Cuts: % Cuts: % Heuristic still looking % Elapsed time = sec. ( ticks, tree = 0.01 MB, solutions = 1) % e % cutoff e % e % e % Elapsed time = sec. ( ticks, tree = MB, solutions = 1) Nodefile size = MB ( MB after compression) e % e % e % e % e % cutoff e % Extremely Challenging for MIP solvers

15 1. lightweight MIP model

16 1. lightweight MIP model 2. cutting planes

17 1. lightweight MIP model 2. cutting planes 3. symmetry handling

18 Lightweight MIP model

19 Lightweight MIP model

20 Lightweight MIP model

21 Lightweight MIP model variables and constraints

22 Lightweight MIP model variables and constraints superweak dual bound

23 Cutting Planes I

24 Cutting Planes I computed by solving a LINEAR 3D assignment problem

25 Cutting Planes I computed by solving a LINEAR 3D assignment problem NP-hard in theory, quite cheap in practice

26 Cutting Planes I computed by solving a LINEAR 3D assignment problem NP-hard in theory, quite cheap in practice can exploit additional constraints (if available) both global and local

27 Cutting Planes II

28 Cutting Planes II computed by solving a MIP

29 Cutting Planes II computed by solving a MIP increase consistency between pairs of artificial variables

30 Cutting Planes II computed by solving a MIP increase consistency between pairs of artificial variables not significantly harder than family 1

31 Cutting Planes II computed by solving a MIP increase consistency between pairs of artificial variables not significantly harder than family 1 need to be conservative with separation

32 Symmetry Handling binary variables can be partitioned into 6 orbits

33 Symmetry Handling binary variables can be partitioned into 6 orbits

34 Symmetry Handling binary variables can be partitioned into 6 orbits sums within orbits stay the same

35 Symmetry Handling binary variables can be partitioned into 6 orbits sums within orbits stay the same aggregated variables as first level decisions

36 Symmetry Handling

37 Symmetry Handling symmetry decomposition based on orbital shrinking

38 Symmetry Handling symmetry decomposition based on orbital shrinking we enumerate all possible aggregated solutions

39 Symmetry Handling symmetry decomposition based on orbital shrinking we enumerate all possible aggregated solutions only 85!

40 Symmetry Handling symmetry decomposition based on orbital shrinking we enumerate all possible aggregated solutions only 85! for each we solve a MIP subproblem to find the best solution therein

41 Symmetry Handling symmetry decomposition based on orbital shrinking we enumerate all possible aggregated solutions only 85! for each we solve a MIP subproblem to find the best solution therein isomorphism pruning within sub-mips exploit symmetry twice!!!

42 symmetry decomposition MIP model cutting planes

43 Is it enough?

44 Is it enough? NO!

45 Is it enough? NO! primal heuristics

46 Is it enough? NO! primal heuristics branching order

47 Is it enough? NO! primal heuristics branching order parameter tuning

48 Is it enough? NO! primal heuristics branching order parameter tuning scaling

49 Is it enough? NO! primal heuristics branching order parameter tuning scaling indicator constraints

50 Computational Results subproblems count time nodes

51 Computational Results subproblems count time nodes easy 45 2,

52 Computational Results subproblems count time nodes easy 45 2, medium 25 53, ,000

53 Computational Results subproblems count time nodes easy 45 2, medium 25 53, ,000 hard ,900 2,240,000

54 Computational Results subproblems count time nodes easy 45 2, medium 25 53, ,000 hard ,900 2,240,000 total ,200 2,477,950

55 Computational Results subproblems count time nodes easy 45 2, medium 25 53, ,000 hard ,900 2,240,000 total ,200 2,477,950 less than one week on a desktop PC!

56 Conclusions

57 Conclusions solved biggest Q3AP instance to date

58 Conclusions solved biggest Q3AP instance to date had fun :-)

59 Conclusions solved biggest Q3AP instance to date had fun :-) developed (extended) techniques that can be used for other Q3APs and (more importantly) QAPs and beyond

60 Thanks for your attention!! Questions?