Impact of Structure on Complexity Carla Gomes

gomes@cs.cornell.eduBart Selman

selman@cs.cornell.eduCornell University

Intelligent Information Systems Institute

Kickoff MeetingAFOSR MURI

May 2001

Outline

• I - Overview of our approach• II - Structure vs. complexity -

– results on a abstract domain

• III - Examples of Application Domains• IV - Conclusions

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

Part I Structure vs. Complexity

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

Are all the Quasigroup Instances(of same size) Equally Difficult?

1820150Time performance:

165

What is the fundamental difference between instances?

Are all the Quasigroup Instances Equally Difficult?

1820 165

40% 50%

150Time performance:

35%Fraction of preassignment:

Complexity of Quasigroup Completion

Fraction of pre-assignment

Med

ian

Run

time

(log

scal

e)

Critically constrained area

Overconstrained areaUnderconstrained

area

42% 50%20%

Phase Transition

Almost all unsolvable area

Fraction of pre-assignmentFrac

tion

of u

nsol

vabl

e ca

ses

Almost all solvable area

Complexity Graph

Phase transition from almost all solvableto almost all unsolvable

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

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

BalancedRandom

After Permuting

Balanced bandwidth = 4

Random bandwidth = 2

Structure vs. Computational Cost

Balanced QCP

QCP

% of holes

Com

puta

tiona

lco

st

Balancing makes the instances very hard - it increases bandwith!

Aligned/ Rectangular QCP

Backbone

This instance has4 solutions:

BackboneTotal 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 Bac

kbon

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 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.

Examples of Application Domains

• 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

Fiber Optic Networks

Nodesconnect point to point

fiber optic links

Fiber Optic Networks

Nodesconnect point to point

fiber optic links

Each fiber optic link supports alarge number of wavelengths

Nodes are capable of photonic switching --dynamic wavelength routing --

which involves the setting of the wavelengths.

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 Ports Output Ports1

2

3

4

1

2

3

4

preassigned channels

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 FREELATIN ROUTER

Inpu

t por

ts

Output ports

3

12

4

Input Port Output Port12

43

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

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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 ModelIISI, Cornell University

Target visibilityGraph Representation

Sensornodes

Targetnodes

Initial Graph ModelIISI, Cornell University

The initial model presented is a bipartite graph, and this problem can be solved using a maximum flow algorithm in polynomial time

Sensornodes

Targetnodes

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

mun

icatio

n ed

ges

possible solution

Complexity and Phase Transition Phenomena of

Sensor Domain

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

Prob

abilit

y( a

ll ta

rget

s 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

Prob

abilit

y( a

ll ta

rget

s tra

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 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

x x0 1s1

s2

s4

t1 t2 t3 t4

s3x xx 1

1 x0 0

x xx 1

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

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

Conclusions and Research Directions

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;

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

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

The End