4,647 research outputs found
A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem
Given a directed graph , the Directed Maximum Leaf Spanning Tree
problem asks to compute a directed spanning tree (i.e., an out-branching) with
as many leaves as possible. By designing a Branch-and-Reduced algorithm
combined with the Measure & Conquer technique for running time analysis, we
show that the problem can be solved in time \Oh^*(1.9043^n) using polynomial
space. Hitherto, there have been only few examples. Provided exponential space
this run time upper bound can be lowered to \Oh^*(1.8139^n)
Minimum Cuts in Near-Linear Time
We significantly improve known time bounds for solving the minimum cut
problem on undirected graphs. We use a ``semi-duality'' between minimum cuts
and maximum spanning tree packings combined with our previously developed
random sampling techniques. We give a randomized algorithm that finds a minimum
cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We
also give a simpler randomized algorithm that finds all minimum cuts with high
probability in O(n^2 log n) time. This variant has an optimal RNC
parallelization. Both variants improve on the previous best time bound of O(n^2
log^3 n). Other applications of the tree-packing approach are new, nearly tight
bounds on the number of near minimum cuts a graph may have and a new data
structure for representing them in a space-efficient manner
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks
It is challenging to design large and low-cost communication networks. In
this paper, we formulate this challenge as the prize-collecting Steiner Tree
Problem (PCSTP). The objective is to minimize the costs of transmission routes
and the disconnected monetary or informational profits. Initially, we note that
the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to
improve suboptimal solutions to PCSTP. Based on these techniques, we propose
two fast heuristic algorithms: the first one is a quasilinear time heuristic
algorithm that is faster and consumes less memory than other algorithms; and
the second one is an improvement of a stateof-the-art polynomial time heuristic
algorithm that can find high-quality solutions at a speed that is only inferior
to the first one. We demonstrate the competitiveness of our heuristic
algorithms by comparing them with the state-of-the-art ones on the largest
existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we
generate new instances that are even larger (1 000 000 vertices and 10 000 000
edges) to further demonstrate their advantages in large networks. The
state-ofthe-art algorithms are too slow to find high-quality solutions for
instances of this size, whereas our new heuristic algorithms can do this in
around 6 to 45s on a personal computer. Ultimately, we apply our
post-processing techniques to update the bestknown solution for a notoriously
difficult benchmark instance to show that they can improve near-optimal
solutions to PCSTP. In conclusion, we demonstrate the usefulness of our
heuristic algorithms and post-processing techniques for designing large and
low-cost communication networks
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