113 research outputs found
Linear kernels for outbranching problems in sparse digraphs
In the -Leaf Out-Branching and -Internal Out-Branching problems we are
given a directed graph with a designated root and a nonnegative integer
. The question is to determine the existence of an outbranching rooted at
that has at least leaves, or at least internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with vertices are known on general graphs. In this
work we show that -Leaf Out-Branching admits a kernel with vertices
on -minor-free graphs, for any fixed family of graphs
, whereas -Internal Out-Branching admits a kernel with
vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page
Bidimensionality and Kernels
Bidimensionality theory was introduced by [E. D. Demaine et al., J. ACM, 52 (2005), pp. 866--893] as a tool to obtain subexponential time parameterized algorithms on H-minor-free graphs. In [E. D. Demaine and M. Hajiaghayi, Bidimensionality: New connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2005, pp. 590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (resp., contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO) admits a linear kernel for classes of graphs that exclude a fixed graph (resp., an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems admit linear kernels on the corresponding graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work.publishedVersio
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc -Leaf Out-Branching}, which is to find an
oriented spanning tree with at least leaves, we obtain an algorithm solving
the problem in time on directed graphs
whose underlying undirected graph excludes some fixed graph as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to . The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
-Internal Out-Branching}, which is to find an oriented spanning tree with at
least internal vertices. We obtain an algorithm solving the problem in time
on directed graphs whose underlying
undirected graph excludes some fixed apex graph as a minor. Finally, we
observe that for any , the {\sc -Directed Path} problem is
solvable in time , where is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs
Explicit linear kernels via dynamic programming
Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for -Dominating Set and -Scattered Set
on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs
excluding a fixed (topological) minor in the case where all the graphs in
\mathcal{F} are connected.Comment: 32 page
Bidimensionality and EPTAS
Bidimensionality theory is a powerful framework for the development of
metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to
obtain sub-exponential time parameterized algorithms for problems on H-minor
free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for
bidimensional problems, and subsequently improved these results to EPTASs.
Fomin et. al related the theory to the existence of linear kernels for
parameterized problems. In this paper we revisit bidimensionality theory from
the perspective of approximation algorithms and redesign the framework for
obtaining EPTASs to be more powerful, easier to apply and easier to understand.
Two of the most widely used approaches to obtain PTASs on planar graphs are
the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and
Hajiaghayi strengthened both approaches using bidimensionality and obtained
EPTASs for a multitude of problems. We unify the two strenghtened approaches to
combine the best of both worlds. At the heart of our framework is a
decomposition lemma which states that for "most" bidimensional problems, there
is a polynomial time algorithm which given an H-minor-free graph G as input and
an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n
X is f(e). Here, OPT is the objective function value of the problem in question
and f is a function depending only on e. This allows us to obtain EPTASs on
(apex)-minor-free graphs for all problems covered by the previous framework, as
well as for a wide range of packing problems, partial covering problems and
problems that are neither closed under taking minors, nor contractions. To the
best of our knowledge for many of these problems including cycle packing,
vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no
EPTASs on planar graphs were previously known
Towards a Theory of Parameterized Streaming Algorithms
Parameterized complexity attempts to give a more fine-grained analysis of the complexity of problems: instead of measuring the running time as a function of only the input size, we analyze the running time with respect to additional parameters. This approach has proven to be highly successful in delineating our understanding of NP-hard problems. Given this success with the TIME resource, it seems but natural to use this approach for dealing with the SPACE resource. First attempts in this direction have considered a few individual problems, with some success: Fafianie and Kratsch [MFCS\u2714] and Chitnis et al. [SODA\u2715] introduced the notions of streaming kernels and parameterized streaming algorithms respectively. For example, the latter shows how to refine the Omega(n^2) bit lower bound for finding a minimum Vertex Cover (VC) in the streaming setting by designing an algorithm for the parameterized k-VC problem which uses O(k^{2}log n) bits.
In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms. We first define a natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS, and then obtain tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy (see Figure 1 and Table 1). On the algorithmic side, our parameterized streaming algorithms use techniques from the FPT world such as bidimensionality, iterative compression and bounded-depth search trees. On the hardness side, we obtain lower bounds for the parameterized streaming complexity of various problems via novel reductions from problems in communication complexity. We also show a general (unconditional) lower bound for space complexity of parameterized streaming algorithms for a large class of problems inspired by the recently developed frameworks for showing (conditional) kernelization lower bounds.
Parameterized algorithms and streaming algorithms are approaches to cope with TIME and SPACE intractability respectively. It is our hope that this work on parameterized streaming algorithms leads to two-way flow of ideas between these two previously separated areas of theoretical computer science
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
09511 Abstracts Collection -- Parameterized complexity and approximation algorithms
From 14. 12. 2009 to 17. 12. 2009., the Dagstuhl Seminar 09511
``Parameterized complexity and approximation algorithms \u27\u27 was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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