4,460 research outputs found
Potential Maximal Clique Algorithms for Perfect Phylogeny Problems
Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard
combinatorial optimization problems related to minimal triangulations, are
broken into subproblems by block subgraphs defined by minimal separators. These
ideas were expanded on by Bouchitt\'e and Todinca, who used potential maximal
cliques to solve these problems using a dynamic programming approach in time
polynomial in the number of minimal separators of a graph. It is known that
solutions to the perfect phylogeny problem, maximum compatibility problem, and
unique perfect phylogeny problem are characterized by minimal triangulations of
the partition intersection graph. In this paper, we show that techniques
similar to those proposed by Bouchitt\'e and Todinca can be used to solve the
perfect phylogeny problem with missing data, the two- state maximum
compatibility problem with missing data, and the unique perfect phylogeny
problem with missing data in time polynomial in the number of minimal
separators of the partition intersection graph
Advanced Multilevel Node Separator Algorithms
A node separator of a graph is a subset S of the nodes such that removing S
and its incident edges divides the graph into two disconnected components of
about equal size. In this work, we introduce novel algorithms to find small
node separators in large graphs. With focus on solution quality, we introduce
novel flow-based local search algorithms which are integrated in a multilevel
framework. In addition, we transfer techniques successfully used in the graph
partitioning field. This includes the usage of edge ratings tailored to our
problem to guide the graph coarsening algorithm as well as highly localized
local search and iterated multilevel cycles to improve solution quality even
further. Experiments indicate that flow-based local search algorithms on its
own in a multilevel framework are already highly competitive in terms of
separator quality. Adding additional local search algorithms further improves
solution quality. Our strongest configuration almost always outperforms
competing systems while on average computing 10% and 62% smaller separators
than Metis and Scotch, respectively
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
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