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Handbook of Graph Drawing and Visualization
book

Handbook of Graph Drawing and Visualization

by Roberto Tamassia
August 2013
Intermediate to advanced content levelIntermediate to advanced
862 pages
32h 20m
English
Chapman and Hall/CRC
Content preview from Handbook of Graph Drawing and Visualization
13.5. VERTEX ORDERING 439
Exact Methods
Following from their work on the linear ordering and OLCM problems J¨unger et
al. derived an integer linear programming (ILP) formulation for MLCM [JLMO97]. As
in the one-layer case, quadratic terms analogous to those appearing in equation (13.7)
prevent formulation as a linear program. In this case, however, neither layer is fixed so
a different strategy is called for. The proposed solution is to introduce crossing variables,
c
ijkl
, denoting whether the edges (i, j) and (k, l) cross when i < j, k < l, i < k and j 6= l.
Using this notation along with boolean variables x
ij
to denote π
1
(i) < π
1
(j) and y
ij
to
denote π
2
(i) < π
2
(j) TLCM can be formulated as
minimize
X
(i,j),(k,l)E
c
ijkl
(13.14)
c
ijkl
y
jl
x
ik
c
ijkl
(i, ...
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Publisher Resources

ISBN: 9781420010268