Controlling Computational Cost: Structure and Phase Transition

1. Controlling Computational Cost: Structure and Phase Transition Carla Gomes, Scott Kirkpatrick, Bart Selman, Ramon Bejar, Bhaskar Krishnamachari Intelligent Information Systems Institute, Cornell University Autonomous Negotiating Teams Principal Investigators' Meeting, April 30-May 2

2. Outline • I - Overview of our approach • II - Structure vs. complexity - new results • III - Ants - Challenge Problem (Sensor Domain) • Graph Models • Results on average case complexity • Distributed CSP model • IV - Conclusions and Future Work

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. Structure vs. Complexity New results

5. 32% preassignment 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:

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

7. Complexity Graph Standard Phase transition from almost all solvable to almost all unsolvable Almost all solvable area Almost all unsolvable area Phase Transition Computational Cost Fraction of unsolvable cases Fraction of preassignment

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

9. 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

10. Random vs Balanced Balanced Random

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

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

13. 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:

14. 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)

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

16. 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.

17. 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.

18. Ant’s Challenge Problem Sensor Domain

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

20. IISI, Cornell University 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

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

22. IISI, Cornell University 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?

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

24. 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 Results incorporated into framework developed by Milind Tambe’s group at ISI, USC Joint work in progress

25. 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

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

27. Complexity and Phase Transition Phenomena

28. IISI, Cornell University 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

29. Average Case complexity and Phase Transition Phenomena

30. IISI, Cornell University Phase Transition w.r.t. Communication Level: 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

31. IISI, Cornell University Phase Transition w.r.t. Radar Detection Range 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

32. Distributed Model

33. IISI, Cornell University 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 P1, P2,…Pn , one for each agent • There are intra-agent constraints and inter-agent constraints

34. IISI, Cornell University 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

35. x x x x x x 1 1 t1 t2 t3 t4 s1 x 0 x 1 s2 s3 s4 1 0 x 0 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

36. 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 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

37. 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

38. 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 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

39. Future Work

40. IISI, Cornell University Structure • Further study structural issues as they occur in the Sensor domain e.g.: • effect of balancing; • backbone (insights into critical resources); • refinement of phase transition notions considering additional parameters;

41. IISI, Cornell University 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

42. IISI, Cornell University Dynamic DCSP Model • Further refinement of the model: incorporate target mobility • The graph topology changes with time • What are the complexity issues when online distributed algorithms are involved?

43. Summary

44. IISI, Cornell University Summary • Graph-based models which represent key aspects of the problem domain • Results on the complexity of computation and communication for the static model • Extensions: • additional structural issues on the sensor domain • complexity issues in distributed and dynamic settings

45. Collaborations / Interactions • ISI: Analytic Tools to Evaluate Negotiation Difficulty • Design and evaluation of SAT encodings for CAMERA’s scheduling task. • ISI: DYNAMITE • Formal complexity analysis DCSP model (e.g., characterization of tractable subclasses). • UMASS: Scalable RT Negotiating Toolkit • Analysis of complexity of negotiation protocols.

46. IISI, Cornell University The End