136 research outputs found
A randomized polynomial kernel for Subset Feedback Vertex Set
The Subset Feedback Vertex Set problem generalizes the classical Feedback
Vertex Set problem and asks, for a given undirected graph , a set , and an integer , whether there exists a set of at most
vertices such that no cycle in contains a vertex of . It was
independently shown by Cygan et al. (ICALP '11, SIDMA '13) and Kawarabayashi
and Kobayashi (JCTB '12) that Subset Feedback Vertex Set is fixed-parameter
tractable for parameter . Cygan et al. asked whether the problem also admits
a polynomial kernelization.
We answer the question of Cygan et al. positively by giving a randomized
polynomial kernelization for the equivalent version where is a set of
edges. In a first step we show that Edge Subset Feedback Vertex Set has a
randomized polynomial kernel parameterized by with
vertices. For this we use the matroid-based tools of Kratsch and Wahlstr\"om
(FOCS '12) that for example were used to obtain a polynomial kernel for
-Multiway Cut. Next we present a preprocessing that reduces the given
instance to an equivalent instance where the size of
is bounded by . These two results lead to a polynomial kernel for
Subset Feedback Vertex Set with vertices
On Kernelization for Edge Dominating Set under Structural Parameters
In the NP-hard Edge Dominating Set problem (EDS) we are given a graph G=(V,E) and an integer k, and need to determine whether there is a set F subseteq E of at most k edges that are incident with all (other) edges of G. It is known that this problem is fixed-parameter tractable and admits a polynomial kernelization when parameterized by k. A caveat for this parameter is that it needs to be large, i.e., at least equal to half the size of a maximum matching of G, for instances not to be trivially negative. Motivated by this, we study the existence of polynomial kernelizations for EDS when parameterized by structural parameters that may be much smaller than k.
Unfortunately, at first glance this looks rather hopeless: Even when parameterized by the deletion distance to a disjoint union of paths P_3 of length two there is no polynomial kernelization (under standard assumptions), ruling out polynomial kernelizations for many smaller parameters like the feedback vertex set size. In contrast, somewhat surprisingly, there is a polynomial kernelization for deletion distance to a disjoint union of paths P_5 of length four. As our main result, we fully classify for all finite sets H of graphs, whether a kernel size polynomial in |X| is possible when given X such that each connected component of G-X is isomorphic to a graph in H
Smaller Parameters for Vertex Cover Kernelization
We revisit the topic of polynomial kernels for Vertex Cover relative to
structural parameters. Our starting point is a recent paper due to Fomin and
Str{\o}mme [WG 2016] who gave a kernel with vertices
when is a vertex set such that each connected component of contains
at most one cycle, i.e., is a modulator to a pseudoforest. We strongly
generalize this result by using modulators to -quasi-forests, i.e., graphs
where each connected component has a feedback vertex set of size at most ,
and obtain kernels with vertices. Our result relies
on proving that minimal blocking sets in a -quasi-forest have size at most
. This bound is tight and there is a related lower bound of
on the bit size of kernels.
In fact, we also get bounds for minimal blocking sets of more general graph
classes: For -quasi-bipartite graphs, where each connected component can be
made bipartite by deleting at most vertices, we get the same tight bound of
vertices. For graphs whose connected components each have a vertex cover
of cost at most more than the best fractional vertex cover, which we call
-quasi-integral, we show that minimal blocking sets have size at most
, which is also tight. Combined with existing randomized polynomial
kernelizations this leads to randomized polynomial kernelizations for
modulators to -quasi-bipartite and -quasi-integral graphs. There are
lower bounds of and
for the bit size of such kernels
Approximate Turing Kernelization for Problems Parameterized by Treewidth
We extend the notion of lossy kernelization, introduced by Lokshtanov et al.
[STOC 2017], to approximate Turing kernelization. An -approximate
Turing kernel for a parameterized optimization problem is a polynomial-time
algorithm that, when given access to an oracle that outputs -approximate
solutions in time, obtains an -approximate solution to
the considered problem, using calls to the oracle of size at most for
some function that only depends on the parameter.
Using this definition, we show that Independent Set parameterized by
treewidth has a -approximate Turing kernel with
vertices, answering an open question posed by
Lokshtanov et al. [STOC 2017]. Furthermore, we give
-approximate Turing kernels for the following graph problems
parameterized by treewidth: Vertex Cover, Edge Clique Cover, Edge-Disjoint
Triangle Packing and Connected Vertex Cover.
We generalize the result for Independent Set and Vertex Cover, by showing
that all graph problems that we will call "friendly" admit
-approximate Turing kernels of polynomial size when
parameterized by treewidth. We use this to obtain approximate Turing kernels
for Vertex-Disjoint -packing for connected graphs , Clique Cover,
Feedback Vertex Set and Edge Dominating Set
Constraint satisfaction parameterized by solution size
In the constraint satisfaction problem (CSP) corresponding to a constraint
language (i.e., a set of relations) , the goal is to find an assignment
of values to variables so that a given set of constraints specified by
relations from is satisfied. The complexity of this problem has
received substantial amount of attention in the past decade. In this paper we
study the fixed-parameter tractability of constraint satisfaction problems
parameterized by the size of the solution in the following sense: one of the
possible values, say 0, is "free," and the number of variables allowed to take
other, "expensive," values is restricted. A size constraint requires that
exactly variables take nonzero values. We also study a more refined version
of this restriction: a global cardinality constraint prescribes how many
variables have to be assigned each particular value. We study the parameterized
complexity of these types of CSPs where the parameter is the required number
of nonzero variables. As special cases, we can obtain natural and
well-studied parameterized problems such as Independent Set, Vertex Cover,
d-Hitting Set, Biclique, etc.
In the case of constraint languages closed under substitution of constants,
we give a complete characterization of the fixed-parameter tractable cases of
CSPs with size constraints, and we show that all the remaining problems are
W[1]-hard. For CSPs with cardinality constraints, we obtain a similar
classification, but for some of the problems we are only able to show that they
are Biclique-hard. The exact parameterized complexity of the Biclique problem
is a notorious open problem, although it is believed to be W[1]-hard.Comment: To appear in SICOMP. Conference version in ICALP 201
Preprocessing Under Uncertainty: Matroid Intersection
We continue the study of preprocessing under uncertainty that was initiated independently by Assadi et al. (FSTTCS 2015) and Fafianie et al. (STACS 2016). Here, we are given an instance of a tractable problem with a large static/known part and a small part that is dynamic/uncertain, and ask if there is an efficient algorithm that computes an instance of size polynomial in the uncertain part of the input, from which we can extract an optimal solution to the original instance for all (usually exponentially many) instantiations of the uncertain part.
In the present work, we focus on the Matroid Intersection problem. Amongst others we present a positive preprocessing result for the important case of finding a largest common independent set in two linear matroids. Motivated by an application for intersecting two gammoids we also revisit Maximum Flow. There we tighten a lower bound of Assadi et al. and give an alternative positive result for the case of low uncertain capacity that yields a Maximum Flow instance as output rather than a matrix
Cara a un Patrimonio Público Galego: Vinte apuntamentos
La versión castellana de este texto está accesible en el siguiente enlace: http://hdl.handle.net/10261/32906[GA] Este traballo foi preparado por solicitude do Consello da Cultura Galega para contribuír a un debate sobre o Patrimonio Cultural de Galicia vinculado cas posibilidades de redactar unha nova lei galega do Patrimonio. No presente texto destácanse algúns trazos relativos ao valor actual do Patrimonio Cultural, á situación presente dos estudos de investigación e xestión do Patrimonio Cultural, á necesidade de conxugar múltiples narrativas na interpretación deste, á conveniencia de establecer un sistema ordenado de diálogo entre estas narrativas, á interacción dos proxectos patrimoniais coas comunidades, á construción dun Patrimonio Público e á xeración de novos modelos de relación co medio e produción de valor nas industrias e mercado de traballo en Patrimonio.[ES] Este trabajo fue preparado atendiendo a una solicitud del Consello da Cultura Galega para contribuir a un debate sobre el Patrimonio Cultural de Galicia y sobre las posibilidades de redactar una nueva ley gallega de Patrimonio. Existe una versión gallega y otra castellana del mismo texto. En el presente texto se destacan algunos rasgos sobre el valor actual del Patrimonio Cultural, la situación presente de los estudios de investigación y gestión del Patrimonio Cultural, la necesidad de conjugar múltiples narrativas en la interpretación de éste, la conveniencia de establecer un sistema ordenado de diálogo entre estas narrativas, la interacción de los proyectos patrimoniales con las comunidades, la construcción de un Patrimonio Público y la generación de nuevos modelos de relación con el entorno y producción de valor en las industrias y mercado de trabajo en Patrimonio
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
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