276 research outputs found
Fast Dynamic Graph Algorithms for Parameterized Problems
Fully dynamic graph is a data structure that (1) supports edge insertions and
deletions and (2) answers problem specific queries. The time complexity of (1)
and (2) are referred to as the update time and the query time respectively.
There are many researches on dynamic graphs whose update time and query time
are , that is, sublinear in the graph size. However, almost all such
researches are for problems in P. In this paper, we investigate dynamic graphs
for NP-hard problems exploiting the notion of fixed parameter tractability
(FPT).
We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion
parameterized by the solution size . These dynamic graphs achieve almost the
best possible update time and the query time
, where is the time complexity of any static
graph algorithm for the problems. We obtain these results by dynamically
maintaining an approximate solution which can be used to construct a small
problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a
corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm
for Cluster Vertex Deletion. Until now, only quadratic time kernelization
algorithms are known for this problem.
We also give a dynamic graph for Chromatic Number parameterized by the
solution size of Cluster Vertex Deletion, and a dynamic graph for
bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming
the parameter is a constant, each dynamic graph can be updated in
time and can compute a solution in time. These results are obtained by
another approach.Comment: SWAT 2014 to appea
Parameterized Algorithms and Data Reduction for Safe Convoy Routing
We study a problem that models safely routing a convoy through a transportation network, where any vertex adjacent to the travel path of the convoy requires additional precaution: Given a graph G=(V,E), two vertices s,t in V, and two integers k,l, we search for a simple s-t-path with at most k vertices and at most l neighbors. We study the problem in two types of transportation networks: graphs with small crossing number, as formed by road networks, and tree-like graphs, as formed by waterways. For graphs with constant crossing number, we provide a subexponential 2^O(sqrt n)-time algorithm and prove a matching lower bound. We also show a polynomial-time data reduction algorithm that reduces any problem instance to an equivalent instance (a so-called problem kernel) of size polynomial in the vertex cover number of the input graph. In contrast, we show that the problem in general graphs is hard to preprocess. Regarding tree-like graphs, we obtain a 2^O(tw) * l^2 * n-time algorithm for graphs of treewidth tw, show that there is no problem kernel with size polynomial in tw, yet show a problem kernel with size polynomial in the feedback edge number of the input graph
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