Transcript A Graph Model for Studying the Complexity of the ANTs problem

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)
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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
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Identification of domain structural features
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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
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Computational Cost
Fraction of unsolvable cases
Graph
PhaseComplexity
Transition
Standard Phase transition
from almost all solvable
to almost all unsolvable
Almost all solvable
area
Almost all unsolvable
area
Fraction of preassignment
Quasigroup Patterns and
Problems Hardness
Hardness is also controlled by structure of
constraints, not just percentage of holes
Rectangular Pattern
Aligned Pattern
Tractable
Balanced Pattern
Very hard
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
Random
Balanced
After Permuting
Random
bandwidth = 2
Balanced
bandwidth = 4
Structure vs. Computational Cost
Computational
cost
Balanced QCP
QCP
Aligned/ Rectangular
QCP
% of holes
Balancing makes the instances very hard - it increases bandwith!
Backbone
Backbone is the shared structure of all the
solutions to a given instance.
This instance has
4 solutions:
Backbone
Total number of backbone variables: 2
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
% of Backbone
% Backbone
Sudden phase transition in Backbone
Computational
cost
Fraction of preassigned cells
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
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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:
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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
Target visibility
Sensor
nodes
Graph Representation
Target
nodes
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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
Results incorporated into framework
developed by Milind Tambe’s group at ISI,
USC
Joint work in progress
Sensor
Target
nodes
nodes
Sensor Communication
Constraints
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initial model
+ communication edges
Possible solution
In the graph model, we now have additional edges between
sensor nodes
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Constrained Graph Model
communication edges
sensors
targets
possible solution
Complexity and Phase
Transition Phenomena
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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 NPcomplete, by reduction from the
problem of partitioning a graph into
isomorphic subgraphs
Average Case complexity
and Phase Transition
Phenomena
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Phase Transition w.r.t.
Communication Level:
Probability( all targets tracked )
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
Communication edge probability p
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Phase Transition w.r.t.
Radar Detection Range
Probability( all targets tracked )
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
Normalized Radar Range R
Distributed Model
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Distributed CSP Model
In a distributed CSP (DCSP) variables
and constraints are distributed among
multiple agents. It consists of:
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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:
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One with each sensor as a distinct agent
One with a distinct tracker agent for each
target
Sensor Agents
Binary variables associated with each target
Intra-agent constraints :
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Sensor must track at most 1 visible target
Inter-agent constraints:
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3 communicating sensors should track each target
t1
t2
t3
t4
s1
x
0
x
1
s2
x
x
x
1
s3
x
x
x
1
s4
1 0
x
0
Target Tracker Agents
Binary variables associated with each sensor
Intra-agent constraints :
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Each target must be tracked by 3 communicating
sensors to which it is visible
Inter-agent constraints:
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A sensor can only track one target
s1
s2
s3
s4
s5
s6
s7
s8
s9
t1
1 0
1
x
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x
1
t2
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x
1
1
1
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t3
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1
1
0
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 0
1
x
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t2
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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
s1
s2
s3
s4
s5
s6
s7
s8
s9
t1
1 0
1
x
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t2
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1
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t3
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1
0
Future Work
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Structure
Further study structural issues as they
occur in the Sensor domain e.g.:
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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?
Summary
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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:
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additional structural issues on the sensor
domain
complexity issues in distributed and dynamic
settings
Collaborations / Interactions
ISI: Analytic Tools to Evaluate Negotiation
Difficulty

Design and evaluation of SAT encodings for
CAMERA’s scheduling task.
ISI: DYNAMITE
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Formal complexity analysis DCSP model (e.g.,
characterization of tractable subclasses).
UMASS: Scalable RT Negotiating Toolkit
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Analysis of complexity of negotiation protocols.
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The End