Identify Phase Transitions In Problem Hardness Leverage Randomization In Computation

1. Combinatorial Problems in Cooperative Control: Complexity and Scalability Carla P. Gomes and Bart SelmanCornell UniversityMuri MeetingJune 2002

2. Overview Overall Approach Goal • Identify Phase Transitions In Problem Hardness • Leverage Randomization In Computation Develop procedures that recognize and react to Structure in Problem Hardness Hardness Aware Systems (Computationally) • Principled dynamic control of communication and computational resources in large distributed autonomous systems, allowing for: • Scalability • Time critical applications • Robustness guarantees • - • Use findings in both the design and operation of complex (distributed) systems

3. Outline • ROBOFLAG Drill – Computational Issues • Capturing Structure in Combinatorial Problems • Randomization and Approximations • Conclusions

4. ROBOFLAG Drill • Problem is hybrid, combining discrete and continuous components, with multiple constraints. • Overall the Roboflag control problem provides an • excellent test bed for the development of scalable • techniques for complex optimization.

5. Problem Representation • ROBOFLAG Drill • Formulation by Raff D’Andrea and Matt Earl. • Represented as a mixed logical system (MLD) in which the objective is to compute optimal control policies that minimize the total score of the game. • Mathematical Formulation of the Optimization Problem • Mixed Integer Linear Program

6. . • We are investigating how to scale up solutions • of the ROBOFLAGDrill focusing on: • - Mixed Integer Program (MIP) formulations • - Randomization and Approximation methods • - Combining MIP and constraint search • techniques. • - Portfolios of Algorithms

7. Scaling Up Mixed Integer Linear Program Formulations (MILP) • Standard approach for solving MILP: • Branch and Bound • How can we improve upon Branch and Bound strategies? • Ideas: • Different search strategies for node selection • Randomization • Portfolios of algorithms

8. Branch & Bound:Depth First vs. Best bound • Critical to performance of Branch & Bound is the way • in which the next node to be expanded is selected. • Standard approach: • Best-bound --- select the node with the best LP bound • Alternative: • Depth-first --- often quickly reaches an integer solution • (may take longer to produce an overall optimal value) • Tradeoffs between these choices depend on underlying • problem stucture (Gomes et al. 2001).

9. ROBOFLAG Testbed • Hybrid node selection - Best Bound and Depth First • Depth First search works well. • Problems that could not be solved before with best bound using were solved with depth first. • Current largest problem solved with CPLEX using Depth First Search (8 attackers and 3 defenders): • Integer variables = 4040 • Continuous variables 400 • Constraints - 13580 constraints • Time - 244 secs • (Matt Earl 2002)

10. Much room for improvement… • We are not yet using other problem formulations, • Nor are we yet exploiting randomization and parallelism. • Doing so should allow us to solve problems at • least one or two orders of magnitude larger. • (100,000 to 500,000 vars and 1,000,000+ • constraints) • Also, we should be able to include more complex constraints.

11. Capturing Structure in Combinatorial Problemsthe importance of problem representation…

12. Completing Latin Squares:An Abstraction for Real World Applications A Latin Square is an n-by-n matrix such that each row and column is a permutation of the same n colors Latin Square (Order 4) 32% preassignment Gomes and Selman 96

13. each channel cannot be repeated in the same input port (row constraints); • each channel cannot be repeated in the same output port (column constraints); Input Port Output Port Output ports 1 1 2 2 3 3 Input ports 4 4 CONFLICT FREE LATIN ROUTER Switches in Fiber Optic Networks Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Latin Square Problem. (Barry and Humblet 93, Cheung et al. 90, Green 92, Kumar et al. 99)

14. LP Based Formulations

15. Assignment Formulation Rows Colors Columns Cubic representation of QCP

16. QCPAssignment Formulation Max number of colored cells Row/color line Column/color line Row/column line

17. Using a MIP formulation and Branch and Bound we can only find solutions for Latin Squares up to Order 15 (15 x 15) •  we can do better, even with an LP based formulation using a less obvious encoding

18. Packing formulation Families of patterns (partial patterns are not shown) Max number of colored cells in the selected patterns s.t. one pattern per family a cell is covered at most by one pattern Gomes and Shmoys 2002

19. QCPPacking Formulation Max number of colored cells one pattern per color at most one pattern covering each cell

20. Any feasible solution to the packing LP relaxation is • also a solution to the assignment LP relaxation • The value of the assignment relaxation is at least the bound implied by the packing formulation => the packing formulation provides a tighter upper bound than the assignment formulation • Limitation – size of formulation is exponential in n. (one may apply column generation techniques)

21. ApproximationBased on Packing Formulation • Randomization scheme: • for each color K choose a pattern with probability (so that some matching is selected for each color) • As a result we have a pattern per color. • Problem: some patterns may overlap, even though in expectation, the constraints imply that the number of matchings in which a cell is involved is 1.

22. Packing formulation 1 0.8 1 1 0.2 Max number of colored cells in the selected patterns s.t. one pattern per family a cell is covered at most by one pattern

23. (1-1/e)- ApproximationBased on Packing Formulation • Let’s assume that the PLS is completable • Z*=h • What is the expected number of cells uncolored by our randomized procedure due to overlapping conflicts? • From we can compute • So, the desired probability corresponds to the probability of a cell not be colored with any color, i.e.:

24. (1-1/e)- ApproximationBased on Packing Formulation • This expression is maximized when all the • are equal therefore: • So the expected number of uncolored cells is at most  at least holes are expected to be filled by this technique. 1- 1/e ~ 0.632 - This is a very good guarantee for a polynomial time algorithm!

25. Another Formulation • Constraint Satisfaction Problem

26. QCP as a CSP • Variables - • Constraints - row column

27. Exploiting Structure for Domain Reduction • A very successful strategy for domain reduction in CSP is to exploit the structureof groups of constraints and treat them as global constraints. • Example using Network Flow Algorithms: • All-different constraints

28. Matching on a Bipartite graph Two solutions: All-different constraint we can update the domains of the column variables Analogously, we can update the domains of the other variables Exploiting Structure in QCP ALLDIFF as Global Constraint

29. Pure CSP approaches solve QCP instances up • to order 33 (1089 variables) relatively well. • (LP based – only up to order 15 – 125 variables)

30. We are exploring more direct encodings for the ROBOFLAG DRILL • Representations avoiding discretization based on time. •  constraint based abstractions closer to the physical system, e.g., based movements / trajectories.

31. Randomization

32. Background • Stochastic strategies have been very successful in the area of local search. • Simulated annealing • Genetic algorithms • Tabu Search • Walksat and variants. • Limitation: inherent incomplete nature of local search methods.

33. Randomized backtrack search • Randomized variable and/or value selection – lots of different ways. • Example: randomly breaking ties in variable and/or value selection. • Compare with standard lexicographic tie-breaking. • Note: No problem maintaining the completeness of the algorithm!

34. Empirical Evidence of Heavy-Tails Time: 7 11 30 (*) (*) (*) no solution found - reached cutoff: 2000 Erratic Behavior of Mean Sample mean Number runs Easy instance – 15 % preassigned cells 3500 2000 Median = 1! 500 Gomes et al. 97

35. Decay of Distributions Power Law Decay Exponential Decay Standard Distribution (finite mean & variance) Standard Exponential Decay e.g. Normal: Heavy-Tailed Power Law Decay e.g. Pareto-Levy: Infinite variance, infinite mean

36. Exploiting Heavy-Tailed Behavior 70% unsolved 1-F(x) Unsolved fraction 0.001% unsolved 250 (62 restarts) Number backtracks (log) • Heavy Tailed behavior has been observed in several domains: QCP, Graph Coloring, Planning, Scheduling, Circuit synthesis, Decoding, etc. • Consequence for algorithm design: • Use restarts or parallel / interleaved runs to exploit the extreme variance performance. Restarts provably eliminate heavy-tailed behavior (Gomes et al. 2000)

37. Using randomization and restarts we can solve considerably larger instances up to order QCP instances up to order 40 (1600 variables). • Note: this problem is highly exponential – instances of order 40 are much more difficult than instances of order 33! • We are also experimenting with randomization in the ROBOFLAG DRILL

38. Hybrid MIP/CSP Approaches

39. CSP Model • LP Model + LP Randomized Rounding • Heavy-tails • We want to maintain completeness How do we combine all these ingredients? A HYBRID COMPLETE CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH

40. HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH • Central features of algorithm: • Complete Backtrack search algorithm • It maintains two formulations • CSP model • Relaxed LP model • LP Randomized rounding  for setting values at the top of the tree • CSP + LP inference

41. HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH • Populate CSP Model • Perform propagation • Populate LP solver • Solve LP Variable setting controlled by LP Randomized Rounding CSP & LP Inference %LP Interleave-LP Search & Inference controlled by CSP Adaptive CUTOFF

42. Empirical Results

43. Time Performance Order 35

44. Performance • With the hybrid strategy we also solve instances of order 40 in critically constrained area – out of reach for pure CSP; • We even solved a few balanced instances of order 50 in the critically constrained order! • more systematic experimentation is required to better understand limitations and strengths of approach.

45. Conclusions

46. Conclusions • Approximations based on LP randomized rounding • (variable/value setting) + constraint propagation --- very powerful. • Combatting heavy-tails of backtrack search through randomization. • Consequence: • New ways of designing algorithms --- • aim for strategies which have highly asymmetric distributions that • can be exploited using restarts, portfolios of algorithms, and • interleaved/parallel runs. • General approach --- holds promise for a range of hard • combinatorial problems.

47. Scaling up ROBOFLAG - Other Formulations for Solving the Control Optimization Problem • Encodings that provide “tighter” relaxations for the LP problem. • Approximate representations using abstractions (“synthesize larger • movements / trajectories”). Avoid discretization based on time. • Less compact representations may allow for more propagation • and scale up better. • Constraint Satisfaction Problem (CSP) formulations. • Hybrid CSP/LP formulations. • Approximations based on LP randomized rounding. • Goal: At least two orders of magnitude scale-up over • current state-of-the-art.

48. Demos, papers, etc. www.cs.cornell.edu/gomesCheck also:www.cis.cornell.edu/iisi