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

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

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

Structure vs. Complexity New results

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

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

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

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

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

Random vs Balanced Balanced Random

After Permuting Balanced bandwidth = 4 Random bandwidth = 2

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

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.

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)

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

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.

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.

Ant’s Challenge Problem Sensor Domain

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

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

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

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 Results incorporated into framework developed by Milind Tambe’s group at ISI, USC Joint work in progress Sensor nodes Target nodes

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

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

Complexity and Phase Transition Phenomena

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

Average Case complexity and Phase Transition Phenomena

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

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

Distributed Model

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

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

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

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

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

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

Future Work

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; 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 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? 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 IISI, Cornell University

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.

The End IISI, Cornell University