Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
A network-based integer program for physical cellidentifier assignment in 4G
Jamie Fairbrother1 Adam Letchford2,in collaboration with Keith Briggs
1STOR-i Centre for Doctoral Training, Lancaster University
2Department of Management Science, Lancaster University
MoN15, Bath, 23rd September 2016
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Introduction
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Femtocells
Femtocells:
• Femtocells are small home radio deviceswhich use the LTE (4G) network
• Supplement for home WiFi networks
• Serve BT roaming users to provide fasterand more reliable wireless coverage
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Femtocells
Femtocells:
• Femtocells are small home radio deviceswhich use the LTE (4G) network
• Supplement for home WiFi networks
• Serve BT roaming users to provide fasterand more reliable wireless coverage
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Femtocells
Femtocells:
• Femtocells are small home radio deviceswhich use the LTE (4G) network
• Supplement for home WiFi networks
• Serve BT roaming users to provide fasterand more reliable wireless coverage
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Physical cell ID (PCI) assignment
• Each cell is assigned a Physical Cell ID (PCI) between 1 and504
• Neighbouring cells must not have the same PCI
• It is desirable for neighbouring devices to have PCIs differentup to modulo 3
Problem:
• Cells are currently assigned a PCI using a distributed dynamicheuristic
• It is not known how well this performs in practice
• We wish to assess performance via mathematical optimization
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Physical cell ID (PCI) assignment
• Each cell is assigned a Physical Cell ID (PCI) between 1 and504
• Neighbouring cells must not have the same PCI
• It is desirable for neighbouring devices to have PCIs differentup to modulo 3
Problem:
• Cells are currently assigned a PCI using a distributed dynamicheuristic
• It is not known how well this performs in practice
• We wish to assess performance via mathematical optimization
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Physical cell ID (PCI) assignment
• Each cell is assigned a Physical Cell ID (PCI) between 1 and504
• Neighbouring cells must not have the same PCI
• It is desirable for neighbouring devices to have PCIs differentup to modulo 3
Problem:
• Cells are currently assigned a PCI using a distributed dynamicheuristic
• It is not known how well this performs in practice
• We wish to assess performance via mathematical optimization
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Physical cell ID (PCI) assignment
• Each cell is assigned a Physical Cell ID (PCI) between 1 and504
• Neighbouring cells must not have the same PCI
• It is desirable for neighbouring devices to have PCIs differentup to modulo 3
Problem:
• Cells are currently assigned a PCI using a distributed dynamicheuristic
• It is not known how well this performs in practice
• We wish to assess performance via mathematical optimization
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Physical cell ID (PCI) assignment
• Each cell is assigned a Physical Cell ID (PCI) between 1 and504
• Neighbouring cells must not have the same PCI
• It is desirable for neighbouring devices to have PCIs differentup to modulo 3
Problem:
• Cells are currently assigned a PCI using a distributed dynamicheuristic
• It is not known how well this performs in practice
• We wish to assess performance via mathematical optimization
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Physical cell ID (PCI) assignment
• Each cell is assigned a Physical Cell ID (PCI) between 1 and504
• Neighbouring cells must not have the same PCI
• It is desirable for neighbouring devices to have PCIs differentup to modulo 3
Problem:
• Cells are currently assigned a PCI using a distributed dynamicheuristic
• It is not known how well this performs in practice
• We wish to assess performance via mathematical optimization
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Formulation
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Modelling the problem
• The system is modelled as a (possibly planar) sparse graphG = (V ,E ) where the nodes correspond to femtocells and anedge is present between two nodes if they are neighbours
• PCI assignment can bethought of as graphcolouring
• Each PCI (up to modulo 3)represents a colour and onemust assign each node acolour in such a way whichavoids neighbouringfemtocells using the samecolour
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Modelling the problem
• The system is modelled as a (possibly planar) sparse graphG = (V ,E ) where the nodes correspond to femtocells and anedge is present between two nodes if they are neighbours
• PCI assignment can bethought of as graphcolouring
• Each PCI (up to modulo 3)represents a colour and onemust assign each node acolour in such a way whichavoids neighbouringfemtocells using the samecolour
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Modelling the problem
• The system is modelled as a (possibly planar) sparse graphG = (V ,E ) where the nodes correspond to femtocells and anedge is present between two nodes if they are neighbours
• PCI assignment can bethought of as graphcolouring
• Each PCI (up to modulo 3)represents a colour and onemust assign each node acolour in such a way whichavoids neighbouringfemtocells using the samecolour
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Minimizing clashes
• We would like to find a 3-colouring which minimizes the edgeswhose incident nodes use the same colour
• Can this be done by enumeration?
• No! There are 3|V | possible 3-colourings
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Integer programming formulation
min∑
e∈E ye
s.t.∑k
c=1 xvc = 1 (v ∈ V )
yuv ≥ xuc + xvc − 1 ({u, v} ∈ E , c = 1, . . . , k)
xvc ∈ {0, 1} (v ∈ V , c = 1, . . . , k)
yuv ∈ {0, 1} ({u, v} ∈ E ).
where
xvc =
{1 if node v uses colour (modulo 3) c
0 otherwise
yuv =
{1 if nodes u, v use the same colour (modulo 3)
0 otherwise
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Complexity and solution
• The integer programming formulation given above is a specialcase of the k-partition problem (k-PP)
• The k-PP is well known to be NP-hard in the strong sense
• We use the following approach:
1. Preprocessing (reducing and decomposing graph)2. Cutting-plane and branch-and-bound
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Graph construction and preprocessing
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Random disk graph• Locations of radio devices are uniformly randomly generated
on unit torus• Two radio devices can hear each other if they are within a
given radius• Conflict also occurs if two femtocells with a common
neighbour use the same PCI modulo 3• The larger the radius the more dense the graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Random disk graph• Locations of radio devices are uniformly randomly generated
on unit torus• Two radio devices can hear each other if they are within a
given radius• Conflict also occurs if two femtocells with a common
neighbour use the same PCI modulo 3• The larger the radius the more dense the graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Random disk graph• Locations of radio devices are uniformly randomly generated
on unit torus• Two radio devices can hear each other if they are within a
given radius• Conflict also occurs if two femtocells with a common
neighbour use the same PCI modulo 3• The larger the radius the more dense the graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Random disk graph• Locations of radio devices are uniformly randomly generated
on unit torus• Two radio devices can hear each other if they are within a
given radius• Conflict also occurs if two femtocells with a common
neighbour use the same PCI modulo 3• The larger the radius the more dense the graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Random disk graph• Locations of radio devices are uniformly randomly generated
on unit torus• Two radio devices can hear each other if they are within a
given radius• Conflict also occurs if two femtocells with a common
neighbour use the same PCI modulo 3• The larger the radius the more dense the graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Preprocessing
• Given the difficulty of problem it is essential the graph is smallas possible
• The neighbourhood graph can be reduced by two operations:k-core reduction and biconnected component decomposition
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
k-core reduction
• The k-core of a graph is the largest induced subgraph whereall vertices have at least k neighbours:
Figure : Full graph
• The k-core yields the same optimal solution value to the k-PPas the original graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
k-core reduction
• The k-core of a graph is the largest induced subgraph whereall vertices have at least k neighbours:
Figure : 2-core reduction
• The k-core yields the same optimal solution value to the k-PPas the original graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
k-core reduction
• The k-core of a graph is the largest induced subgraph whereall vertices have at least k neighbours:
Figure : 3-core reduction
• The k-core yields the same optimal solution value to the k-PPas the original graph
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Biconnected components
• A biconnected component is a maximal subgraph whichcannot be disconnected by the removal of a single node
Figure : Full graph
• The optimal solution value to the k-PP is the sum of thosefor all the biconnected components
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Biconnected components
• A biconnected component is a maximal subgraph whichcannot be disconnected by the removal of a single node
Figure : Biconnected components
• The optimal solution value to the k-PP is the sum of thosefor all the biconnected components
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Graph decomposition for PCI assignment
• k-core reduction can result in graph which is not connected orbiconnected, and biconnected components may not bereducible
• 3-core reductions and biconnected componentsdecompositions can be applied iteratively
• The optimal solution value for k-PP is the sum of that for allcomponents
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Power of decomposition
• The reduction achieved depends the sparsity of the graph
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Disk Radius
0.0
0.2
0.4
0.6
0.8
1.0
Edge R
educt
ion
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Disk Radius
0
20
40
60
80
100
Maxim
um
Vert
ices
Graph Decomposition, Original Graph: 200 vertices
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Cutting-Plane Algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Integer programs and linear relaxations
minimizex
cT x
s.t. Ax ≥ b
x ∈ Zn
• The linear relaxation of an integer program is the problemwithout integer constraints
• A cutting-plane is an inequality which is satisfied by allfeasible integer solutions but violated by at least one solutionof the linear relaxation
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Integer programs and linear relaxations
minimizex
cT x
s.t. Ax ≥ b
x ∈ Rn
• The linear relaxation of an integer program is the problemwithout integer constraints
• A cutting-plane is an inequality which is satisfied by allfeasible integer solutions but violated by at least one solutionof the linear relaxation
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Recap: Cutting-planes algorithm
Figure : Cutting-plane algorithm
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Clique inequalities
Theorem ([Chopra and Rao, 1993])
For a clique C ⊂ V , the following inequality is valid:∑u,v∈C
yuv ≥(
t + 1
2
)r +
(t
2
)(k − r)
where t = b |C |k c and r mod k. Moreover, it is facet-defining if
r 6= 0.
Figure : A clique of size 5
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Solution Algorithm
1. Run an LP-based cutting-plane algorithm in y -space withclique inequalities
2. Add the x variables and associated constraints
3. Run branch-and-bound on the strengthened (x , y) formulation
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Modulo-6 Extension
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Modulo-6 Interference
• Adjacent devices which both use the same PCI modulo 6 cancause additional interference
• Taking this into account our problem becomes a 6-colouring
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Problem reformulation
min w∑
{u,v}∈E yuv +∑
{u,v}∈E zuv
s.t.∑6
c=1 xvc = 1 (v ∈ V )
yuv ≥ xuc + xu,c+3 + xvc + xv ,c+3 − 1 ({u, v} ∈ E , c = 1, 2, 3)
zuv ≥ xuc + xvc − 1 ({u, v} ∈ E , c = 1, . . . , 6)
xvc ∈ {0, 1} (v ∈ V , c = 1, . . . , 6)
yuv ∈ {0, 1} ({u, v} ∈ E )
zuv ∈ {0, 1} ({u, v} ∈ E ).
zuv =
{1 if nodes u, v same PCI modulo 6
0 otherwise
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
A new valid inequality
Theorem ([Fairbrother and Letchford, 2016])
For all C ⊆ V inducing a clique in G , with |C | ≥ 3, the following“(y , z)-clique” inequalities are valid:
2∑
u,v∈Czuv ≥
∑u,v∈C
yuv −⌊|C |2
⌋.
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
A new cutting-plane algorithm
1. Run an LP-based cutting-plane algorithm in y -space.
2. Add the z variables and associated constraints.
3. Run an LP-based cutting-plane algorithm in (y , z)-space.
4. Add the x variables and associated constraints.
5. Run branch-and-bound on the strengthened (x , y , z)formulation.
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Numerical results: generated graphs
Avg. Deg. # Components # Biconn. Components Clique No.
4.76 10 5 74.84 8 6 75.40 6 4 75.00 8 6 75.76 7 5 9
Table : Graph attributes for random disk graphs with 50 points andradius 0.14
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Numerical results: Optimality Gaps
• Lower bounds calculated after each cutting-plane phase forrandom neighbourhood graphs
phase 1 phase 2 opt. val.
16.700 17.700 18.00015.000 16.000 16.00018.286 19.286 20.00016.444 17.444 18.00023.000 26.000 27.000
Table : Lower bounds yielded by cutting plane algorithms
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Numerical results: Solution time
• Time to solve problem with and without preprocessing andcutting plane are recorded
Neither PP CP PP+CP
92.746 11.993 1.869 1.6679.324 12.462 0.287 0.341
103.921 88.772 11.091 0.8906.468 9.560 2.791 0.889
26.077 67.443 8.100 0.916
Table : Branch and bound times for different combinations of methods(PP=preprocessing, CP=cutting planes)
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Conclusions
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Conclusions
• The problem of assigning PCI to femtocell devices can beformulated as an integer program
• This problem is defined over an appropriately defined network
• Problem is difficult especially when one takes into accountmodulo 6 clashes
• Preprocessing is particularly effective at reducing problem formore sparse graphs
• Cutting-plane algorithms based on clique inequalities yieldgood lower bounds and drastically reduce solution time
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Future work
• Test performance of distributed heuristic with respect tooptimal solution value
• Use more realistic point processes, or even real data forconstruction of neighbourhood graph
• Consider other possible extensions, such as powerconfiguration
Introduction Formulation Graph construction and preprocessing Cutting-Plane Algorithm Modulo-6 Extension Conclusions
Chopra, S. and Rao, M. R. (1993).The partition problem.Mathematical Programming, 59(1-3):87–115.
Fairbrother, J. and Letchford, A. (2016).A hierarchical k-partition problem arising in mobile wirelesscommunications.Working paper.