6 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
Dynamic Data Structures for Parameterized String Problems
We revisit classic string problems considered in the area of parameterized
complexity, and study them through the lens of dynamic data structures. That
is, instead of asking for a static algorithm that solves the given instance
efficiently, our goal is to design a data structure that efficiently maintains
a solution, or reports a lack thereof, upon updates in the instance.
We first consider the Closest String problem, for which we design randomized
dynamic data structures with amortized update times and
, respectively, where is the alphabet and
is the assumed bound on the maximum distance. These are obtained by
combining known static approaches to Closest String with color-coding.
Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can
easily infer a meta-theorem that provides dynamic data structures for
parameterized string problems with worst-case update time of the form
, where is the parameter in question and is
the length of the string. We showcase the utility of this meta-theorem by
giving such data structures for problems Disjoint Factors and Edit Distance. We
also give explicit data structures for these problems, with worst-case update
times and ,
respectively. Finally, we discuss how a lower bound methodology introduced by
Amarilli et al.~[ICALP'21] can be used to show that obtaining update time
for Disjoint Factors and Edit Distance is unlikely already
for a constant value of the parameter .Comment: 28 page
Efficient fully dynamic elimination forests with applications to detecting long paths and cycles
We present a data structure that in a dynamic graph of treedepth at most ,
which is modified over time by edge insertions and deletions, maintains an
optimum-height elimination forest. The data structure achieves worst-case
update time , which matches the best known parameter
dependency in the running time of a static fpt algorithm for computing the
treedepth of a graph. This improves a result of Dvo\v{r}\'ak et al. [ESA 2014],
who for the same problem achieved update time for some non-elementary
(i.e. tower-exponential) function . As a by-product, we improve known upper
bounds on the sizes of minimal obstructions for having treedepth from
doubly-exponential in to .
As applications, we design new fully dynamic parameterized data structures
for detecting long paths and cycles in general graphs. More precisely, for a
fixed parameter and a dynamic graph , modified over time by edge
insertions and deletions, our data structures maintain answers to the following
queries:
- Does contain a simple path on vertices?
- Does contain a simple cycle on at least vertices?
In the first case, the data structure achieves amortized update time
. In the second case, the amortized update time is . In both cases we assume access to a dictionary
on the edges of .Comment: 74 pages, 5 figure