5 research outputs found
Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles
Plotkin, Rao, and Smith (SODA'97) showed that any graph with edges and
vertices that excludes as a depth -minor has a
separator of size and that such a separator can be
found in time. A time bound of for
any constant was later given (W., FOCS'11) which is an
improvement for non-sparse graphs. We give three new algorithms. The first has
the same separator size and running time O(\mbox{poly}(h)\ell
m^{1+\epsilon}). This is a significant improvement for small and .
If for an arbitrarily small chosen constant
, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}).
The second algorithm achieves the same separator size (with a slightly larger
polynomial dependency on ) and running time O(\mbox{poly}(h)(\sqrt\ell
n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when . Our third algorithm has running time
O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when . It finds a separator of size O(n/\ell) + \tilde
O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when
is fixed and . A main tool in obtaining our results
is a novel application of a decremental approximate distance oracle of Roditty
and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor
fixes regarding the time bounds such that these bounds hold also for
non-sparse graph
Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications
Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator
theorem and showed that a -minor free graph with vertices has a
separator of size at most . They gave an algorithm that, given
a graph with edges and vertices and given an integer ,
outputs in time such a separator or a -minor of .
Plotkin, Rao, and Smith gave an time algorithm to find a
separator of size . Kawarabayashi and Reed improved the
bound on the size of the separator to and gave an algorithm that
finds such a separator in time for any constant , assuming is constant. This algorithm has an extremely large
dependency on in the running time (some power tower of whose height is
itself a function of ), making it impractical even for small . We are
interested in a small polynomial time dependency on and we show how to find
an -size separator or report that has a -minor in
O(\poly(h)n^{5/4 + \epsilon}) time for any constant . We also
present the first O(\poly(h)n) time algorithm to find a separator of size
for a constant . As corollaries of our results, we get improved
algorithms for shortest paths and maximum matching. Furthermore, for integers
and , we give an time algorithm that
either produces a -minor of depth or a separator of size
at most . This improves the shallow minor algorithm
of Plotkin, Rao, and Smith when . We get a
similar running time improvement for an approximation algorithm for the problem
of finding a largest -minor in a given graph.Comment: To appear at FOCS 201
Graph Partitioning of Transportation Networks under Disruption
This research is concerned with providing a solution capable of treating network complexity and scalability effectively so that it overcomes administrative, environmental and technique boundaries. One good approach dealing with this matter is applying graph partitioning techniques. Graph partitioning is an optimization problem with the aim of dividing a large geographical network into manageable size districts called sub-networks with less complexity in favor of balancing the workload and minimizing the communication among them, with the aim of maximizing their independency as much as possible. Over the past decades various models have been developed in such a way to satisfy a multi-objective problem such as delivery time and managerial cost. In real life, due to inevitable changes during network’s lifetime, it is vital to offer survivability and resilience in the existence of network failure and disruption. Further, it is essential to maintain functionality in critical facilities and high priority connections in the time of crisis. This paper suggests four partitioning techniques namely “Hierarchical recursive progression1^+ “(HRP1^+) and “Hierarchical recursive progression2^+ “(HRP2^+) and their extensions called “HRP1^+control” and “HRP2^+control” to solve the scalability as well as complexity of a network. For this matter, the initial balanced partition is produced on a predefined network. Furthermore two different approaches namely “complete failure update “and “partial failure update” are proposed and demonstrated in the occurrence of network disruption.
In sum, the three main objectives of this thesis are as follows:
Modeling disruption on logistics networks
Assuring and strengthen connectivity in the disrupted network for routing purposes
Developing partitioning approaches in favor of generating roughly equal sized and balanced partitions in the disrupted network
Separators in Graphs with Negative and Multiple Vertex Weights
A separator theorem for a class of graphs asserts that every graph in the class can be divided approximately in half by removing a set of vertices of specified size. Nontrivial separator theorems hold for several classes of graphs, including graphs of bounded genus and chordal graphs. We show that any separator theorem implies various weighted separator theorems. In particular, we show that if the vertices of the graph have real-valued weights, which may be positive or negative, then the graph can be divided exactly in half according to weight. If k unrelated sets of weights are given, the graph can be divided simultaneously by all k sets of weights. These results considerably strengthen earlier results of Gilbert, Lipton, and Tarjan: (1) for k = 1 with the weights restricted to be nonnegative, and (2) for k > 1, nonnegative weights, and simultaneous division within a factor of (1 + ffl) of exactly in half
Separators in Graphs with Negative and Multiple Vertex Weights
A separator theorem for a class of graphs asserts that every graph in the class can be divided approximately in half by removing a set of vertices of specified size. Nontrivial separator theorems hold for several classes of graphs, including graphs of bounded genus and chordal graphs. We show that any separator theorem implies various weighted separator theorems. In particular, we show that if the vertices of the graph have real-valued weights, which may be positive or negative, then the graph can be divided exactly in half according to weight. If k unrelated sets of weights are given, the graph can be divided simultaneously by all k sets of weights. These results considerably strengthen earlier results of Gilbert, Lipton, and Tarjan: (1) for k = 1 with the weights restricted to be nonnegative, and (2) for k ? 1, nonnegative weights, and simultaneous division within a factor of (1 + ffl) of exactly in half. Keywords: graph separator, cutset, divide-and-conquer algorithm, separator the..