Impa[e-mail][e-mail]%20Cornell%20University%20Intelligent%20Information%20Systems%20Institute%20Kickoff%20Meeting%20AFOSR%20MURI%20May%202001

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Impact of Structure on Complexity Carla
Gomesgomes_at_cs.cornell.eduBart
Selmanselman_at_cs.cornell.eduCornell
UniversityIntelligent Information Systems
InstituteKickoff MeetingAFOSR MURIMay 2001

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Outline

• I - Overview of our approach
• II - Structure vs. complexity -
• results on a abstract domain
• III - Examples of Application Domains
• IV - Conclusions

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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

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• Part I
• Structure vs. Complexity

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Quasigroup Completion Problem (QCP)
Given a matrix with a partial assignment of
colors (32colors 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
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• Structural features of instances provide
  insights into their hardness namely
• Phase transition phenomena
• Backbone
• Inherent structure and balance

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Are all the Quasigroup Instances (of same size)
Equally Difficult?
What is the fundamental difference between
instances?
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Are all the Quasigroup Instances Equally
Difficult?
1820
165
50
40
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Complexity of Quasigroup Completion
Median Runtime (log scale)
Fraction of pre-assignment
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Phase Transition
Fraction of unsolvable cases
Fraction of pre-assignment
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Quasigroup Patterns and Problems Hardness
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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
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Random vs Balanced


Balanced
Random
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After Permuting
Balanced bandwidth 4
Random bandwidth 2
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Structure vs. Computational Cost
Balanced QCP
Computational cost
QCP
Aligned/ Rectangular QCP
of holes
Balancing makes the instances very hard - it
increases bandwith!
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Backbone
Backbone is the shared structure of all the
solutions to a given instance.
This instance has 4 solutions
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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)
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New Phase Transition in Backbone
Backbone
of Backbone
Computational cost
Fraction of preassigned cells
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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.

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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.

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• Examples of Application Domains

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Fiber Optic Networks

• 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.

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Fiber Optic Networks
Nodes connect point to point fiber optic links
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Fiber Optic Networks
Nodes connect point to point fiber optic links
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Routing in Fiber Optic Networks
Input Ports
Output Ports
1
1
2
2
3
3
4
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Routing Node
How can we achieve conflict-free routing in each
node of the network?
Dynamic wavelength routing is a NP-hard problem.
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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)
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ANTs Challenge Problem
IISI, Cornell University

• Multiple doppler radar sensors track moving
  targets
• Energy limited sensors
• Communication
• constraints
• Distributed
• environment
• Dynamic problem

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Domain Models
IISI, Cornell University

• 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
IISI, Cornell University

• 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
IISI, Cornell University

• 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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IISI, Cornell University
Initial Graph Model
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IISI, Cornell University
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

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IISI, Cornell University
Sensor Communication Constraints

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

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IISI, Cornell University
Constrained Graph Model
sensors
targets
communication edges
possible solution
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• Complexity and Phase Transition Phenomena of
• Sensor Domain

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Complexity
IISI, Cornell University

• 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

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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
Insights into the design and operation of sensor
networks w.r.t. communication level
Probability( all targets tracked )
Communication edge probability p
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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
Insights into the design and operation of sensor
networks w.r.t. radar detection range
Probability( all targets tracked )
Normalized Radar Range R
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• Distributed Model

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Distributed CSP Model
IISI, Cornell University

• 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
IISI, Cornell University

• 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

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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

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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
1
x
x
1
0
x
x
x
x
x
1
x
x
x
1
x
1
t2
x
x
x
1
0
x
x
1
1
t3
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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

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• Conclusions and Research Directions

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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

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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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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

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The End