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Introduction. Modelling the problem. A polyhedral study of the minimum-adjacency vertex coloring problem. ILP model. Polyhedral study. Branch & Cut. Computational results. Final remarks. Diego Delle Donne & Javier Marenco Computer Science department, FCEN, University of Buenos Aires.

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Introduction

Modelling the problem

A polyhedral study of the minimum-adjacency vertex coloring problem

ILP model

Polyhedral study

Branch & Cut

Computational results

Final remarks

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


Introduction

Cellular GSM networks: ¿How do communications work?

Hand-over

A polyhedral study of the minimum-adjacency vertex coloring problem


Introduction

Possible problems:

Co-channel interference (same channel)

Adjacent-channel interference

A polyhedral study of the minimum-adjacency vertex coloring problem


Introduction

Covering an area using a network:

… but the number of available channels is limited…

A polyhedral study of the minimum-adjacency vertex coloring problem


Introduction

Possible schema for a frequency assignment:

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

2. Minimize adjacent-channel interference.

A polyhedral study of the minimum-adjacency vertex coloring problem


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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Introduction

Modelling the problem

ILP model

Polyhedral study

Branch & Cut

Computational results

Final remarks

A polyhedral study of the minimum-adjacency vertex coloring problem


Modelling the problem

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).

A polyhedral study of the minimum-adjacency vertex coloring problem


Modelling the problem

ILP model

Polyhedral study

Branch & Cut

Computational results

Final remarks

A polyhedral study of the minimum-adjacency vertex coloring problem


ILP model

Considered models:

Stable model (Méndez Díaz and Zabala, 2001)

Orientation model (Grötschel et. al., 1998)

Distance model

Representatives model (Campêlo, Corrêa and Frota, 2004)

A polyhedral study of the minimum-adjacency vertex coloring problem


: Stable model

ILP model

xvcrepresents whether color c is assigned to vertex v or not

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

A polyhedral study of the minimum-adjacency vertex coloring problem


ILP model

Polyhedral study

Branch & Cut

Computational results

Final remarks

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

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:

q-1 adjacencies

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).

x

x

x

x

x

x

x

Q =

c1

c2

……

cq

Removes 2 adjacencies

Removes 1 adjacency

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

Definition: Let be a clique of G and ,

p non-adjacent sets of consecutive colors. We define the Multi Consecutive Colors Clique Inequality to be:

Theorem: The Multi Consecutive Colors Clique Inequalities are valid for PS(G,C).

C =

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

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:

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).

Q = {c1, c2,c3}

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

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:

Theorem: The 3-Colors Outer Clique Inequalities are valid for PS(G,C) and, if |C| > and |C| > |K| + 4, they are also facet-defining for PS(G,C).

Q = {c1, c2,c3}

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

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:

Theorem: The 4-Colors Vertex Clique Inequalities are valid for PS(G,C) and, if |C| > and |C| > |K| + 4, they are also facet-defining for PS(G,C).

Q = {c1, c2,c3 ,c4}

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

Definition: Let be a clique of G and an available color. We define the Clique Inequality (Méndez Díaz and Zabala, 2001) to be:

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).

A polyhedral study of the minimum-adjacency vertex coloring problem


Polyhedral study

Branch & Cut

Computational results

Final remarks

A polyhedral study of the minimum-adjacency vertex coloring problem


Branch & Cut

Separation algorithms (searching for cliques):

Backtracking

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)

A polyhedral study of the minimum-adjacency vertex coloring problem


C =

Branch & Cut

Extending intervals for the MCCK Inequality:

A polyhedral study of the minimum-adjacency vertex coloring problem


Branch & Cut

Other used technics:

Primal bounds: Construction of feasible solutions by rounding techniques

Subproblem reduction by logical implications

Selection of the branching variable

A polyhedral study of the minimum-adjacency vertex coloring problem


Branch & Cut

Computational results

Final remarks

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

Context:

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.

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

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

Results:

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

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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

A polyhedral study of the minimum-adjacency vertex coloring problem


Computational results

Final remarks

A polyhedral study of the minimum-adjacency vertex coloring problem


Final remarks

Conclusions:

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.

A polyhedral study of the minimum-adjacency vertex coloring problem


Final remarks

Future work:

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

Deepen the study of the other models.

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

A polyhedral study of the minimum-adjacency vertex coloring problem


Thank you!

Diego Delle Donne & Javier Marenco

Computer Science department, FCEN, Univeristy of Buenos Aires.

Sciences Institute, National University of General Sarmiento.

A polyhedral study of the minimum-adjacency vertex coloring problem


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