Annual Report 2010 - Fachgruppe Informatik an der RWTH Aachen ...

Combinatorial Optimization and Recognition of Graph Classes with
Applications to Related Models
George B. Mertzios
We mainly investigate the structure of some classes of perfect graphs that have been widely
studied, due to both their interesting structure and their numerous applications. By exploiting
the structure of these graph classes, we provide solutions to some open problems on them (in
both the affirmative and negative), along with some new representation models that enable
the design of new efficient algorithms.
In particular, we first investigate the classes of interval and proper interval graphs, and
especially, path problems on them. These classes of graphs have been extensively studied and
they find many applications in several fields and disciplines such as genetics, molecular
biology, scheduling, VLSI design, archaeology, and psychology, among others. Although the
Hamiltonian path problem is well known to be linearly solvable on interval graphs, the
complexity status of the longest path problem, which is the most natural optimization version
of the Hamiltonian path problem, was an open question. We present the first polynomial
algorithm for this problem with running time O(n 4 ). Furthermore, we introduce a matrix
representation for both interval and proper interval graphs, called the Normal Interval
Representation (NIR) and the Stair Normal Interval Representation (SNIR) matrix,
respectively. The whole information of both NIR and SNIR matrices for a graph with n
vertices can be captured in O(n) space. We illustrate the use of this succinct matrix
representation (SNIR) for proper interval graphs to solve in optimal O(n) time the k-fixedendpoint
path cover problem, which is another optimization variant of the Hamiltonian path
problem.
Next, we investigate the classes of tolerance and bounded tolerance graphs, which generalize
in a natural way both interval and permutation graphs. This class of graphs has attracted many
research efforts since its introduction by Golumbic and Monma in 1982, as it finds many
important applications in bioinformatics, constrained-based temporal reasoning, resource
allocation, and scheduling, among others. We present the first non-trivial intersection model
for tolerance graphs, given by three-dimensional parallelepipeds. Apart of being important on
its own, this new intersection model enables the design of efficient algorithms on tolerance
graphs. Namely, given a tolerance graph G with n vertices, we present optimal O(n log n)
time algorithms for the minimum coloring and the maximum clique problems, as well as an
improved O(n 2 ) time algorithm for the maximum weighted independent set problem on G.
In spite of the extensive study of these classes, the recognition of both tolerance and bounded
tolerance graphs have been the most fundamental open problems since their introduction.
Therefore, all existing efficient algorithms assumed that the input graph is given along with a
tolerance or a bounded tolerance representation, respectively. We prove that both recognition
problems are NP-complete, thereby settling a long standing open question. These hardness
results are surprising, since it was expected that the recognition of these graph classes is
polynomial.
Finally, we investigate a scheduling model, which is closely related to the concept of interval
and tolerance graphs. Namely, we deal with the scheduling of weighted jobs with release
times and with equal processing time each on a single machine. In our model, the scheduling
of the jobs is preemptive, i.e., the processing of a job can be interrupted by another one. Our
goal is to find a schedule of the given jobs with the minimum weighted sum of completion
times. The complexity status of this problem has been stated as an open question. We present
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