2. Impact of Structure on Complexity Carla Gomes gomes@cs.cornell.edu Bart Selman selman@cs.cornell.edu Cornell University Intelligent Information Systems Institute Kickoff Meeting AFOSR MURI May 2001

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

4. 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

5. Part I Structure vs. Complexity

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

7. Structural features of instances provide insights into their hardness namely: –Phase transition phenomena –Backbone –Inherent structure and balance

8. Are all the Quasigroup Instances (of same size) Equally Difficult? 1820150 Time performance: 165 What is the fundamental difference between instances?

9. Are all the Quasigroup Instances Equally Difficult? 1820165 40% 50% 150 Time performance: 35% Fraction of preassignment:

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

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

12. Quasigroup Patterns and Problems Hardness Rectangular PatternAligned PatternBalanced Pattern TractableVery hard Hardness is also controlled by structure of constraints, not just percentage of holes

13. Bandwidth Bandwidth: permute rows and columns of QCP to minimize the width of the diagonal band that covers all the holes. Fact: can solve QCP in time exponential in bandwidth swap

14. Random vs Balanced Balanced Random

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

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

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

18. 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)

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

20. 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 clauses pushing the search in different directions.

21. 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.

22. Examples of Application Domains

23. 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 a different wavelength. Fiber Optic Networks

24. Nodes connect point to point fiber optic links

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

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

27. 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) each channel cannot be repeated in the same input port (row constraints); each channel cannot be repeated in the same output port (column constraints); CONFLICT FREE LATIN ROUTER Input ports Output ports 3 1 2 4 Input PortOutput Port 1 2 4 3

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

29. Domain Models 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 IISI, Cornell University

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

31. Initial Graph Model 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? IISI, Cornell University

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

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

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

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

36. Complexity and Phase Transition Phenomena of Sensor Domain

37. Complexity 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 IISI, Cornell University

38. Average Case complexity and Phase Transition Phenomena

39. 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 Probability( all targets tracked ) Communication edge probability p Insights into the design and operation of sensor networks w.r.t. communication level

40. 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 Probability( all targets tracked ) Normalized Radar Range R Insights into the design and operation of sensor networks w.r.t. radar detection range

41. Distributed Model

42. Distributed CSP Model 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 P 1, P 2, … P n, one for each agent –There are intra-agent constraints and inter- agent constraints IISI, Cornell University

43. DCSP Model 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 IISI, Cornell University

44. Sensor Agents 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 xx01 s1 s2 s4 t1t2t3t4 s3 xxx1 1x00 xxx1

45. Target Tracker Agents 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 11xx10xxx xx1xxx1x1 t1 t2 xxx10xx11t3 s1s2s3s4s5s6s7s8s9

46. 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 ) 11xx10xxx xx1xxx1x1 t1 t2 xxx10xx11t3 s1s2s3s4s5s6s7s8s9

47. Communication Costs for Implicit Constraints Explicit constraints can be resolved by direct communication between agents Resolving Implicit constraints may require long communication paths through multiple agents  scalability problems 11xx10xxx xx1xxx1x1 t1 t2 xxx10xx11t3 s1s2s3s4s5s6s7s8s9

48. Conclusions and Research Directions

49. 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;

50. 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

51. 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

52. The End