Outline

1. Impact of Structure on Complexity Carla Gomesgomes@cs.cornell.eduBart Selmanselman@cs.cornell.eduCornell UniversityIntelligent Information Systems InstituteKickoff MeetingAFOSR MURIMay 2001

2. Outline • I - Overview of our approach • II - Structure vs. complexity - • results on a abstract domain • III - Examples of Application Domains • IV - Conclusions

3. Overview of Approach • Overall theme --- exploit impact of structure on computational complexity • Identification of domain structural features • tractable vs. intractable subclasses • phase transition phenomena • backbone • balancedness • … • Goal: • Use findings in both the design and operation of distributed platform • Principled controlled hardness aware systems

4. Part I Structure vs. Complexity

5. Quasigroup Completion Problem (QCP) Given a matrix with apartial assignment of colors(32%colors in this case), can it be completed so that each color occurs exactly once in eachrow / column (latin square or quasigroup)? Example: 32% preassignment

6. Structural featuresof instances provide insights intotheir hardness namely: • Phase transition phenomena • Backbone • Inherent structure and balance

7. Are all the Quasigroup Instances (of same size) Equally Difficult? Time performance: 1820 165 150 What is the fundamental difference between instances?

8. Are all the Quasigroup Instances Equally Difficult? Time performance: 150 Fraction of preassignment: 35% 1820 165 50% 40%

9. Complexity of Quasigroup Completion Critically constrained area Underconstrained area Overconstrained area 20% 42% 50% Median Runtime (log scale) Fraction of pre-assignment

10. Complexity Graph Phase transition from almost all solvable to almost all unsolvable Almost all solvable area Almost all unsolvable area Phase Transition Fraction of unsolvable cases Fraction of pre-assignment

11. Quasigroup Patterns and Problems Hardness Hardness is also controlled by structure of constraints, not just percentage of holes Rectangular Pattern Aligned Pattern Balanced Pattern Tractable Very hard

12. Bandwidth Bandwidth:permute rows and columns of QCP tominimize the width of the diagonal bandthat covers all the holes. Fact: can solve QCP in time exponential in bandwidth swap

13. Random vs Balanced Balanced Random

14. After Permuting Random bandwidth = 2 Balanced bandwidth = 4

15. Structure vs. Computational Cost Balanced QCP Computational cost QCP Aligned/ Rectangular QCP % of holes Balancing makes the instances very hard - it increases bandwith!

16. Backbone Total number of backbone variables: 2 Backbone Backbone is the shared structure of all the solutions to a given instance. This instance has 4 solutions:

17. Phase Transition in the Backbone (only satisfiable instances) • We have observed a transition in the backbone from a phase where the size of the backbone is around 0% to a phase with backbone of size close to 100%. • The phase transition in the backbone is sudden and it coincides with the hardest problem instances. (Achlioptas, Gomes, Kautz, Selman 00, Monasson et al. 99)

18. Sudden phase transition in Backbone New Phase Transition in Backbone % Backbone % of Backbone Computational cost Fraction of preassigned cells

19. Why correlation between backbone and problem hardness? • Small backbone is associated with lots of solutions, widely distributed in the search space, therefore it is easy for the algorithm to find a solution; • Backbone close to 1 - the solutions are tightly clustered, all the constraints “vote” to push the search into that direction; • Partial Backbone - may be an indication that solutions are in different clusters that are widely distributed, with different clausespushing the search in different directions.

20. Structural Features The understanding of the structural properties that characterize problem instances such as phase transitions, backbone,balance, and bandwith provides new insights into the practical complexity of computational tasks.

21. Examples of Application Domains

22. Fiber Optic Networks • Wavelength Division Multiplexing (WDM) is the most promising technology for the next generation of wide-area backbone networks. • WDM networks use the large bandwidth available in optical fibers by partitioning it into several channels,each at adifferent wavelength.

23. Fiber Optic Networks Nodes connect point to point fiber optic links

24. Each fiber optic link supports a large number of wavelengths Nodes are capable of photonic switching --dynamic wavelength routing -- which involves the setting of the wavelengths. Fiber Optic Networks Nodes connect point to point fiber optic links

25. preassigned channels Routing in Fiber Optic Networks Input Ports Output Ports 1 1 2 2 3 3 4 4 Routing Node How can we achieve conflict-free routing in each node of the network? Dynamic wavelength routing is a NP-hard problem.

26. 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 QCP Example Use: Routers in Fiber Optic Networks Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Quasigroup Completion Problem. (Barry and Humblet 93, Cheung et al. 90, Green 92, Kumar et al. 99)

27. ANTs Challenge Problem IISI, Cornell University • Multiple doppler radar sensors track moving targets • Energy limited sensors • Communication constraints • Distributed environment • Dynamic problem

28. Domain Models IISI, Cornell University • Start with a simple graph model • Successively refine the model in stages to approximate the real situation: • Static weakly-constrained model • Static constraint satisfaction model with communication constraints • Static distributed constraint satisfaction model • Dynamic distributed constraint satisfaction model • Goal: Identify and isolate the sources of combinatorial complexity

29. Initial Assumptions IISI, Cornell University • Each sensor can only track one target at a time • 3 sensors are required to track a target

30. Initial Graph Model IISI, Cornell University • Bipartite graph G = (S U T, E) • S is the set of sensor nodes, T the set of target nodes, E the edges indicating which targets are visible to a given sensor • Decision Problem: Can each target be tracked by three sensors?

31. IISI, Cornell University Target visibility Graph Representation Target nodes Sensor nodes Initial Graph Model

32. IISI, Cornell University Sensor nodes Target nodes Initial Graph Model • The initial model presented is a bipartite graph, and this problem can be solved using a maximum flow algorithm in polynomial time

33. IISI, Cornell University initial model initial model initial model + communication edges + communication edges Possible solution Sensor Communication Constraints • In the graph model, we now have additional edges between sensor nodes

34. IISI, Cornell University Constrained Graph Model sensors targets communication edges possible solution

35. Complexity and Phase Transition Phenomena of Sensor Domain

36. Complexity IISI, Cornell University • Decision Problem: Can each target be tracked by three sensors which can communicate together ? • We have shown that this constraint satisfaction problem (CSP) is NP-complete, by reduction from the problem of partitioning a graph into isomorphic subgraphs

37. Average Case complexity and Phase Transition Phenomena

38. Phase Transition w.r.t. Communication Level: IISI, Cornell University Experiments with a random configuration of 9 sensors and 3 targets such that there is a communication channel between two sensors with probability p Insights into the design and operation of sensor networks w.r.t. communication level Probability( all targets tracked ) Communication edge probability p

39. Phase Transition w.r.t. Radar Detection Range IISI, Cornell University Experiments with a random configuration of 9 sensors and 3 targets such that each sensor is able to detect targets within a range R Insights into the design and operation of sensor networks w.r.t. radar detection range Probability( all targets tracked ) Normalized Radar Range R

40. Distributed Model

41. Distributed CSP Model IISI, Cornell University • In a distributed CSP (DCSP) variables and constraints are distributed among multiple agents. It consists of: • A set of agents 1, 2, … n • A set of CSPs P1, P2,…Pn , one for each agent • There are intra-agent constraints and inter-agent constraints

42. DCSP Model IISI, Cornell University • We can represent the sensor tracking problem as DCSP using dual representations: • One with each sensor as a distinct agent • One with a distinct tracker agent for each target

43. Sensor Agents x x x x x x 1 1 t1 t2 t3 t4 s1 x 0 x 1 s2 s3 s4 1 0 x 0 • Binary variables associated with each target • Intra-agent constraints : • Sensor must track at most 1 visible target • Inter-agent constraints: • 3 communicating sensors should track each target

44. Target Tracker Agents s1 s2 s3 s4 s5 s6 s7 s8 s9 t1 1 0 1 x x x x x 1 t2 x x x 1 1 1 x x x t3 x x x 1 x x 1 1 0 • Binary variables associated with each sensor • Intra-agent constraints : • Each target must be tracked by 3 communicating sensors to which it is visible • Inter-agent constraints: • A sensor can only track one target

45. Implicit versus Explicit Constraints • Explicit constraint:(correspond to the explicit domain constraints) • no two targets can be tracked by same sensor (e.g. t2, t3 cannot share s4 and t1, t3 cannot share s9) • three sensors are required to track a target (e.g. s1,s3,s9 for t1) • Implicit constraint:(due to a chain of explicit constraints: (e.g. implicit constraint between s4 for t2 and s9 for t1 ) s1 s2 s3 s4 s5 s6 s7 s8 s9 t1 1 0 1 x x x x x 1 t2 x x x 1 1 1 x x x t3 x x x 1 x x 1 1 0

46. Communication Costs for Implicit Constraints s1 s2 s3 s4 s5 s6 s7 s8 s9 t1 1 0 1 x x x x x 1 t2 x x x 1 1 1 x x x t3 x x x 1 x x 1 1 0 • Explicit constraints can be resolved by direct communication between agents • Resolving Implicit constraints may require long communication paths through multiple agents  scalability problems

47. Conclusions and Research Directions

48. Future directions • Study structural issues and inpact on complexity, as they occur in the distributed environments e.g.: • effect of balancing; • backbone (insights into critical resources); • refinement of phase transition notions considering additional parameters;

49. DCSP Model • With the DCSP model, we plan to study both per-node computational costs as well as inter-node communication costs • We are in the process of applying known DCSP algorithms to study issues concerning complexity and scalability

50. Summary We have made considerable progress in our understanding of the nature of hard computational problems - structure matters! We have harnessed a variety of mechanisms with proven impact on time-critical problem solving. A rich spectrum of applications taking advantage of these new methods are on the horizon in planning, scheduling and many other areas. Future focus on Dynamic Distributed models