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
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
Modelling the problem
Introduction
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
ILP model
Modelling the problem
Polyhedral study
Branch & Cut
Computational results
Final remarks
A polyhedral study of the minimum-adjacency vertex coloring problem
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)
A polyhedral study of the minimum-adjacency vertex coloring problem
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
A polyhedral study of the minimum-adjacency vertex coloring problem
Polyhedral study
ILP model
Branch & Cut
Computational results
Final remarks
A polyhedral study of the minimum-adjacency vertex coloring problem
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|
A polyhedral study of the minimum-adjacency vertex coloring problem
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
…
A polyhedral study of the minimum-adjacency vertex coloring problem
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
A polyhedral study of the minimum-adjacency vertex coloring problem
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).
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
A polyhedral study of the minimum-adjacency vertex coloring problem
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).
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
A polyhedral study of the minimum-adjacency vertex coloring problem
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
A polyhedral study of the minimum-adjacency vertex coloring problem
Branch & Cut
Polyhedral study
Computational results
Final remarks
A polyhedral study of the minimum-adjacency vertex coloring problem
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)
A polyhedral study of the minimum-adjacency vertex coloring problem
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
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
Computational results
Branch & Cut
Final remarks
A polyhedral study of the minimum-adjacency vertex coloring problem
Computational results
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
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
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.
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
Final remarks
Computational results
A polyhedral study of the minimum-adjacency vertex coloring problem
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:
A polyhedral study of the minimum-adjacency vertex coloring problem
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).
A polyhedral study of the minimum-adjacency vertex coloring problem
Final remarks
Thank you!
Diego Delle Donne & Javier Marenco
Computer Science department, FCEN, Univeristy of Buenos Aires.
Sciences Institute, National University of General Sarmiento.