Impact of Structure on Complexity Carla Gomes
[email protected] Selman
[email protected] 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
IISI, Cornell University
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
