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 approachII - 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)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 preassignmentFra

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f u

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le c

ases

Almost all solvable area

Complexity Graph

Standard Phase transition from almost all solvableto almost all unsolvable

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uta

tio

nal

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st

Quasigroup Patterns and Problems Hardness

Rectangular Pattern Aligned Pattern Balanced Pattern

Tractable Very hard

Hardness is also controlled by structure of constraints, not just percentage of holes

BandwidthBandwidth: 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

BalancedRandom

After Permuting

Balanced bandwidth = 4

Random bandwidth = 2

Structure vs. Computational Cost

Balanced QCP

QCP

% of holes

Com

pu

tati

on

al

cost

Balancing makes the instances very hard - it increases bandwith!

Aligned/ Rectangular QCP

Backbone

This instance has4 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

Computationalcost

% o

f B

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e

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 FeaturesStructural 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 ProblemSensor Domain

ANTs Challenge Problem

Multiple doppler radar sensors track moving targets Energy limited sensors Communication

constraints Distributed

environment Dynamic problem

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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

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Initial Assumptions

Each sensor can only track one target at a time 3 sensors are required to track a target

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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?

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Initial Graph Model

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Target visibilityGraph Representation

Sensornodes

Targetnodes

Initial Graph Model

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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

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initial modelinitial model + communication edgesinitial model + communication edges

Possible solution

In the graph model, we now have additional edges between sensor nodes

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Constrained Graph Modelsensors targets

com

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nic

ati

on e

dg

es

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

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Average Case complexity and Phase Transition Phenomena

Phase Transition w.r.t. Communication Level:

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Experiments with a random configuration of 9 sensors and 3 targets such that there is a communication channel between two sensors with probability p

Pro

babili

ty(

all

targ

ets

tra

cked )

Communication edge probability p

Insights into the designand operation of sensor networks w.r.t. communication level

Phase Transition w.r.t. Radar Detection Range

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Experiments with a random configuration of 9 sensors and 3 targets such that each sensor is able to detect targets within a range R

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all

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cked )

Normalized Radar Range R

Insights into the designand 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 P1, P2, … Pn , one for each

agent There are intra-agent constraints and

inter-agent constraints

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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

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Sensor AgentsBinary 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

x x0 1s1

s2

s4

t1 t2 t3 t4

s3

x xx 1

1 x0 0

x xx 1

Target Tracker Agents Binary variables associated with each sensorIntra-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

1 1 x x 10 x xx

x x 1 x xx 1 x1

t1

t2

x x x 1 0x x 11t3

s1 s2 s3 s4 s5 s6 s7 s8 s9

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 )

1 1 x x 10 x xx

x x 1 x xx 1 x1

t1

t2

x x x 1 0x x 11t3

s1 s2 s3 s4 s5 s6 s7 s8 s9

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

1 1 x x 10 x xx

x x 1 x xx 1 x1

t1

t2

x x x 1 0x x 11t3

s1 s2 s3 s4 s5 s6 s7 s8 s9

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;

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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

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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?

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Summary

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

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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

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