Modeling and solving of a radio antennas
deployment support application with discrete
and interval constraints
Michael Heusch - IntCP 2006
Outline of the talk
 Presentation of the application
 Modeling with discrete and interval constraints
 Defining search heuristics
 Modeling the problem with the distn constraint
 Experimental results on solving the progressive
deployment problem
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Michael Heusch - IntCP 2006
Presentation of the LocRLFAP
Informal description of the de radio antennas deployment problem :
Constraints involved :

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Michael Heusch - IntCP 2006
Distance between
frequencies depends on
distance between
antennas
Presentation of the LocRLFAP
Informal description of the de radio antennas deployment problem :
Constraints involved :


Distance between
frequencies depends on
distance between
antennas
Minimal and maximal
distances between
antennas
Difficulties :
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Michael Heusch - IntCP 2006

Hybrid combinatorial
optimisation problem

non-linear continuous
constraints
Specification of the problem
Formulation as a constrained optimisation problem:
 Data

Fixed set of antennas (transmitter-receiver)

Dispatched on n sites {P1, … , Pn}

The links to establish is known in advance
 Variables of the problem:

A solution associates one frequency to each antenna and a position to
each site

Pi = (Xi,Yi): Position of a site

fi,j : frequency allocated to the link from Pi to Pj
 Optimisation problem:

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Minimise the maximal frequency used
Michael Heusch - IntCP 2006
Constraints of the problem
Constraints of the problem
 discrete constraints:

Compatibility between antennas

Forbidden frequencies
 continuous constraints

Maximum distance between antennas (range)

Minimum distance between the antennas (security, interference)
 mixed constraints

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Compatibility between the allocation and the deployment
Michael Heusch - IntCP 2006
Comparing the RLFAP/LocRLFAP with 5 sites
RLFAP
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Michael Heusch - IntCP 2006
LocRLFAP
Comparing the RLFAP/LocRLFAP with 5 sites
RLFAP
LocRLFAP
dist² (Si,Sj) = Σi (Xi - Xj)²
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Michael Heusch - IntCP 2006
Comparing the RLFAP/LocRLFAP
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Michael Heusch - IntCP 2006
Hybrid solving with collaborating solvers
Original approach
 Modeling with the finite domain constraint solver Eclair
 Full discretization of the problem
Modeling three types of constraints
 Discrete constraints
 Continuous constraints
 Mixed constraints
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Michael Heusch - IntCP 2006
Discrete constraints
 Co-site transmitter-receiver interference constraints:
 Duplex distance constraints for each bidirectional link
 Forbidden portions in the frequency range
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Michael Heusch - IntCP 2006
Continuous and mixed constraints
 Elementary continuous constraints:
dist²(Pi,Pj)

> mij² , for all i<j
dist²(Pi,Pj) < Mij² , if there exists a radio link between Pi and Pj
 Mixed constraints:


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Compatibility constraints

If dist(Pi,Pj)< d1, great interference

If d1 <= dist(Pi,Pj)< d2, limited interference
Expression with elementary constraints

{ dist(Pi,Pj)< d1 } v { |fik-fjl| > Δ1 },

{ dist(Pi,Pj)< d2 } v { |fik-fjl| > Δ2 },
Michael Heusch - IntCP 2006
d2
 (i,j,k), i≠j, i≠k, j≠k
 (i,j,k), i≠j, i≠k, j≠k
d1
Test set
Full deployment of networks with 5 to 10 sites
RLFAP
LocRLFAP
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Michael Heusch - IntCP 2006
Progressive deployment of networks with 9 and 10 sites
P
P
P
P
P
P
P
P
P
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Michael Heusch - IntCP 2006
P
Solving with elementary constraints
Full deployment in both models
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Michael Heusch - IntCP 2006
Improvements to the search algorithm
Usage of a naïve Branch & Bound with:
 Distinction of the type of variables

The problem is under-constrained on positions
 Branch on disjunctions?

Branch first on constraints entailing a strong interdistance?
 Variable selection heuristics
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
minDomain

min(dom/deg)

minDomain+maxConstraints
Michael Heusch - IntCP 2006
Results with minDomain+maxConstraints
Progressive deployment in both models
9 sites
10 sites
99% of the backtracks are performed on the
continuous part of the search tree
A bit less backtracks on the hybrid model
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Michael Heusch - IntCP 2006
Hybrid solving is 1 to 3 times slower
Introducing the distn global constraint
 distn ([P1, … , Pn], V)
Pi = Xi x Yi : Cartesian coordinates of the point pi
V i,j : distance to maintain between Pi and Pj
 distn(p1, … , pn], v)
satisfied if and only if
dist(pi,pj) = vi,j
 Filtering algorithm uses geometric approximation techniques
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Michael Heusch - IntCP 2006
Applications of the constraint
 Molecular conformation
 Robotics
 Antennas deployment
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Michael Heusch - IntCP 2006
Using distn in the model
Second formulation of the problem with the global constraint:
 Simple continuous constraints
Introduction
of a matrix {Vi,j} of distance variables:
Domain(Vi,j)=[mi,j , Mi,j]

Expression
of the set of min and max distance constraints:
distn([P1, … , Pn], V)

 Expression of the mixed « distant compatibility » disjunctions
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
distn([P1, … , Pn], V)

{ Vij<d 1 } v { |fik-fjl| > Δ 1 },

{ Vij<d 2 } v { |fik-fjl| > Δ 2 },
Michael Heusch - IntCP 2006
 (i,j,k), i≠j, i≠k, j≠k
(i,j,k), i≠j, i≠k, j≠k
Results using distn (9 sites)
Simple heuristics
Advanced heuristics
hybrid model / discrete model comparison:
Similar performance of both models
1.8 times slower
wrt. simple model, distn divides by 2
the nb. of backtracks
1.5 times more backtracks
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Michael Heusch - IntCP 2006
Results using distn (10 sites)
Simple heuristics
Advanced heuristics
hybrid model / discrete model comparison:
Performance on the solved instances:
4 additional instances are solved
• 63% less backtracks
All instances are solved
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Michael Heusch - IntCP 2006
Quality of solutions
9 sites
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Michael Heusch - IntCP 2006
10 sites
Conclusion and perspectives
 We showed the relevance of coupling discrete and continuous
constraints

Obtain solution of greater quality

Better performance when solving

Independence w.r.t. the discretization step

Validation on one industrial application
 Key points
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
Definition of appropriate search heuristics

Usage of the distn global constraint
Michael Heusch - IntCP 2006
Perspectives on the application
 Validation on instances of greater size
 Take forbidden zone constraints into account
 Provide deployment zones using polygons
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Michael Heusch - IntCP 2006
Other approaches for solving the RLFAP
 Other approaches for solving the classical RLFAP

Graph coloring

Branch & Cut

CP


LDS [Walser – CP96]

Russian Doll Search [Schiex et. al - CP97]
Heuristics

Tabou [Vasquez – ROADEF 2001]

Simulated annealing, evolutionary algorithms…
 Motivations for an approach using CP

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Robustness wrt modification of the constraints of the problem
Michael Heusch - IntCP 2006
Sketch of distn’s filtering algorithm
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Michael Heusch - IntCP 2006
Filtering algorithm on polygons
Method using polygons for representing domains
 Theorem by K. Nurmela et P. Östergård (1999)
pi1
pi2
Pj
Pi
pik-1
pik
 M. Markót et T. Csendes: A New Verified Optimization Technique for
the ``Packing Circles in a Unit Square'' Problems.
SIAM Journal of Optimization, 2005
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Michael Heusch - IntCP 2006
Filtering algorithm on polygons
P2
P1
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Michael Heusch - IntCP 2006
Filtering algorithm on polygons
+
-
+
P2
-
P1
+
+
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Michael Heusch - IntCP 2006
Filtering algorithm on polygons
P2
P1
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Michael Heusch - IntCP 2006
Interval extension of the algorithm
P2
P1
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Michael Heusch - IntCP 2006
Filtering algorithm of distn
Adjusting bounds of the distance variables
P2
P1
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Michael Heusch - IntCP 2006