Diego Delle Donne & Javier Marenco Computer Science department, FCEN, University of Buenos Aires.

A polyhedral study of the minimum-adjacency vertex coloring problem

Outline

Diego Delle Donne & Javier Marenco

Computer Science department, FCEN, University of Buenos Aires.

Sciences Institute, National University of General Sarmiento.

A polyhedral study of the minimum-adjacency vertex

coloring problem

Modelling the problem

ILP model

Polyhedral study

Branch & Cut

Computational results

Introduction

Final remarks


Introduction

Cellular GSM networks: ¿How do communications work?

Hand-over


Introduction

Possible problems:

Co-channel interference (same channel)

Adjacent-channel interference


Introduction

Covering an area using a network:

… but the number of available channels is limited…


Introduction

Possible schema for a frequency assignment:

1. Assign one channel to each antenna avoiding co-channel interference.

2. Minimize adjacent-channel interference.


Introduction

Considerations:

Most common antennas cover 120º and there is often more than one antenna within a sector

Control channels (BCCH) Vs. transmit channels (TCH)

There are many studies on this problem, but little work on exact approaches.

We ommit other characteristics of the problem: blocked channels, minimum distances, etc.



Introduction

ILP model

Polyhedral study

Branch & Cut


Final remarks



G = (V, E)

C = {c1, c2, … , ct}

Goal: Find a coloring of G using colors from C, minimizing the number of adjacent vertices receiving adjacent colors (NP-H).


ILP model


Polyhedral study

Branch & Cut


Final remarks


ILP model

Considered models:

Orientation model (Grötschel et. al., 1998)

Distance model

Representatives model (Campêlo, Corrêa and Frota, 2004)

Stable model (Méndez Díaz and Zabala, 2001)


ILP model

1vcc C

x

v V

xvc represents whether color c is assigned to vertex v or not

zvw asserts whether vertices v and w receive adjacent colors or not

min vwvw E

z

1vc wcx x , ,vw E v w c C

1 21 vc wc vwx x z ,vw E v w

1 2 1 2,| - | 1c c C c c

{0,1}vcx , v V c C {0,1}vwz ,vw E v w

: Stable model


Polyhedral study

ILP model

Branch & Cut


Final remarks


Theorem: The Consecutive Colors Clique Inequalities are valid for

PS(G,C) and, if |C| > , |C|>|Q|, |C|>|K|+ 4 and |K|> , they are

also facet-defining for PS(G,C).

Definition: Let be a clique of G and let Q = {c1,…,cq} be a set

of consecutive colors. We define the Consecutive Colors Clique

Inequality to be:

Polyhedral study

Q =

K V

1

1

,,

2 ( - 1) +q

q

vc vc vc vwv K c Q v w K

c c c

x x x q z

c1 cqc2 ……x x x x x x x

q-1 adjacencies

Removes 2 adjacencies

Removes 1 adjacency

( )G2

|Q|


Theorem: The Multi Consecutive Colors Clique Inequalities are

valid for PS(G,C).

Definition: Let be a clique of G and ,

Polyhedral study

C =

K V

1

1

1 1 ,,

2 ( - 1) +h hqh h

h hqh

p p

vc h vwvc vch v K h v w Kc Q

c c c

x x x q z

1

1 1 11{ ,..., }qQ c c

p non-adjacent sets of

consecutive colors. We define the Multi Consecutive Colors Clique

Inequality to be:

2

2 2 21 1{ ,..., },..., { ,..., }

p

p p pq qQ c c Q c c

1Q 2Q pQ

…


Theorem: The 3-Colors Inner Clique Inequalities are valid for

PS(G,C) and, if |C| > and |C| > |K| + 4, they are also facet-

defining for PS(G,C).

Definition: Let be a clique of G and a vertex from

the clique. Let be a set of consecutive colors. We

define the 3-Colors Inner Clique Inequality to be:

Polyhedral study

Q = {c1, c2, c3}

K V k K

1 2 3}Q {c ,c ,c

1 2 3 2 ( ) 1vk kc kc kc vc

v K v Kv k v k

z x x x x

k

( )G


Theorem: The 3-Colors Outer Clique Inequalities are valid for



Definition: Let be a clique of G and a vertex from

the clique. Let be a set of consecutive colors. We

define the 3-Colors Outer Clique Inequality to be:

Polyhedral study

Q = {c1, c2, c3}

K V k K

1 2 3}Q {c ,c ,c

1 2 3 1 3 ( 2 ) ( ) 2vk kc kc kc vc vc

v K v Kv k v k

z x x x x x

k

( )G


Theorem: The 4-Colors Vertex Clique Inequalities are valid for



Definition: Let be a clique of G and a vertex from the

clique. Let be a set of consecutive colors. We define

the 4-Colors Vertex Clique Inequality to be:

Polyhedral study

Q = {c1, c2, c3 ,c4}

K V k K

1 2 3 4}Q {c ,c ,c ,c

1 2 3 4 2 3( 2 2 ) ( ) 2vk kc kc kc kc vc vc

v K v Kv k v k

z x x x x x x

k

( )G


Theorem: The Clique Inequalities are valid for PS(G,C) and, if K is a

maximal clique and |C| > , they are also facet-defining for

PS(G,C).

Definition: Let be a clique of G and an available color.

We define the Clique Inequality (Méndez Díaz and Zabala, 2001) to

be:

Polyhedral study

K V c C

1vcv K

x

( )G


Branch & Cut

Polyhedral study


Final remarks


Branch & Cut

Backtracking

Separation algorithms (searching for cliques):

Limit on the exploration of the backtracking tree

Bounds for cutting whole branches

Returns: first N, best N, all

Greedy heuristic

Returns N cliques (at most one for each starting vertex)


Branch & Cut

Extending intervals for the MCCK Inequality:

1

1

1 1 ,,

2 ( - 1) +h hqh h

h hqh

p p

vc h vwvc vch v K h v w Kc Q

c c c

x x x q z

1Q

C =21c 2

q2c

2Q


Branch & Cut

Other used technics:

Primal bounds: Construction of feasible solutions by rounding techniques

Subproblem reduction by logical implications

Selection of the branching variable



Branch & Cut

Final remarks



Test instances extracted from EUCLID CALMA Project.

Test set of 30 instances (chosen from the preliminary experimentation)

Pentium IV with 1Gb of RAM memory with one microprocessor running at 1.8 Ghz.

Context:

Each family of inequalities doesn’t seem to work well when used individually.

Combining the MCCK and the K inequalities gives a very strong cutting plane phase for the branch & cut



In this chart we can see the average time taken by different combinations of families with a base combination of MCCK+K.

We also show the type of separation used to search for violating cliques.



If we zoom in the section corresponding to the backtracking separation we can see that the best combination for these test set is the base combination of inequalities (MCCK+K), searching cliques with the “best” parameter.



Next we test different values for the number of cliques returned by the backtracking separation (for MCCK+K with “N best cliques” separation).

We also show the node limit for the exploration of the backtracking tree: 150, 300, 450 and 600 nodes.



Zooming again we can see that the best times are achieved using a number between 14 and 22 cliques (actually the best time was achieved using 20 cliques) with a node limit of 150.



Adding other families to the MCCK+K combination only worsens the resolution times.

Results:

The best parametrization for the Branch & Cut was achieved using backtracking with a 150 node limit on the bactracking tree exploration and returning the best 20 found cliques.

Now with this combination of inequalities and the best parametrization found, we will compare our running times against CPLEX’s.






Final remarks



Final remarks

The associated polytope has a very interesting combinatorial structure.

The Branch & Cut seems to be very efficient and the proposed inequalities seem to help in a decisive manner.

The experimentation shows that “MCCK+K+best 20” would be the best parametrization for the branch & cut algorithm.

Conclusions:


Final remarks

Deepen the study of the other models.

Incorporate to this study the characteristics setted aside at the beginning.

Future work:

Conclude the experiments testing different values for other branch & cut parameters (skip factor, cutting phase iterations, etc).


Final remarks

Thank you!

Diego Delle Donne & Javier Marenco

Computer Science department, FCEN, Univeristy of Buenos Aires.

Sciences Institute, National University of General Sarmiento.

Diego Delle Donne & Javier Marenco Computer Science department, FCEN, University of Buenos Aires.

Documents

clique of g

adjacent colors

colors inner clique

colors outer clique

color c

maximal clique

set of consecutive colors

polyhedral studyq