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.