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GRASP and Path Relinking for 2Layer Straight Line Crossing
Minimization
MANUEL LAGUNA
Graduate School of Business, University of Colorado, Campus Box 419, Boulder, CO 80309
Manuel.Laguna@Colorado.Edu
RAFAEL MARTÍ
Departamento de Estadística e Investigación Operativa
Facultad de Matemáticas, Universidad de Valencia
Dr. Moliner 50, 46100 Burjassot (Valencia) Spain
Rafael.Marti@uv.es
Submitted: May 21, 1997
First revision: February 24, 1998
Second revision: June 10, 1998
Final version: August 21, 1998
ABSTRACT — In this paper, we develop a greedy randomized adaptive search procedure (GRASP)
for the problem of minimizing straightline crossings in a 2layer graph.
The procedure is fast and is
particularly appealing when dealing with lowdensity graphs.
When a modest increase in
computational time is allowed, the procedure may be coupled with a path relinking strategy to search
for improved outcomes.
Although the principles of path relinking have appeared in the tabu search
literature, this search strategy has not been fully implemented and tested.
We perform extensive
computational experiments with more than 3,000 graph instances to first study the effect of changes in
critical search parameters and then to compare the efficiency of alternative solution procedures.
Our
results indicate that graph density is a major influential factor on the performance of a solution
procedure.
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View Full DocumentLaguna and Martí / 2
The problem of minimizing straightline crossings in layered graphs has been the subject of study for
at least 17 years, beginning with the Relative Degree Algorithm introduced by Carpano
[2]
.
The
problem consists of aligning the two shores
V
1
and
V
2
of a bipartite graph
G
= (
V
1
,
V
2
,
E
) on two
parallel straight lines (layers) such that the number of crossing between the edges in
E
is minimized
when the edges are drawn as straight lines connecting the endnodes (Jünger and Mutzel, 1997).
The
problem is also known as the
bipartite drawing problem
(or BDP).
In the BDP the problem consists of
finding an optimal ordering for the vertices in both layers, which differ from the
layer permutation
problem
(LPP) that seeks the optimal ordering of one layer only.
Table 1 summarizes some of the
relevant work in the area to the present.
The research listed in Table 1 combines procedures
specifically designed for both 2layer and multilayer graphs.
In some instances, however, LPP
procedures have been extended to the BDP case, in a similar way that 2layer graph methods have
been adapted to the multilayer case.
Table 1
Summary of relevant literature.
Reference
Procedure
Comments
Carpano
[2]
Relative degree algorithm
Sugiyama, et al.
[21]
Barycenter
Similar to Carpano’s.
Eades and Kelly
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