117 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs
We investigate a fundamental vertex-deletion problem called (Induced)
Subgraph Hitting: given a graph and a set of forbidden
graphs, the goal is to compute a minimum-sized set of vertices of such
that does not contain any graph in as an (induced)
subgraph. This is a generic problem that encompasses many well-known problems
that were extensively studied on their own, particularly (but not only) from
the perspectives of both approximation and parameterization. We focus on the
design of efficient approximation schemes, i.e., with running time
, which are also of significant
interest to both communities. Technically, our main contribution is a
linear-time approximation-preserving reduction from (Induced) Subgraph Hitting
on any graph class of bounded expansion to the same problem on
bounded degree graphs within . This yields a novel algorithmic
technique to design (efficient) approximation schemes for the problem on very
broad graph classes, well beyond the state-of-the-art. Specifically, applying
this reduction, we derive approximation schemes with (almost) linear running
time for the problem on any graph classes that have strongly sublinear
separators and many important classes of geometric intersection graphs (such as
fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel
concepts and combinatorial observations that may be of independent interest
(and, which we believe, will find other uses) for studies of approximation
algorithms, parameterized complexity, sparse graph classes, and geometric
intersection graphs. As a byproduct, we also obtain the first robust algorithm
for -Subgraph Isomorphism on intersection graphs of fat objects and
pseudo-disks, with running time .Comment: 60 pages, abstract shortened to fulfill the length limi
Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU
The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set
intersecting each of the sets in a given input collection a given number of
times. Generalizing a well-known data reduction algorithm due to Weihe, we show
a problem kernel for Multiple Hitting Set parameterized by the Dilworth number,
a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far
unexplored in the context of parameterized complexity theory. Using matrix
multiplication, we speed up the algorithm to quadratic sequential time and
logarithmic parallel time. We experimentally evaluate our algorithms. By
implementing our algorithm on GPUs, we show the feasability of realizing
kernelization algorithms on SIMD (Single Instruction, Multiple Date)
architectures.Comment: Added experiments on one more data se
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Lossy Kernelization for (Implicit) Hitting Set Problems
We re-visit the complexity of polynomial time pre-processing (kernelization) for the d-Hitting Set problem. This is one of the most classic problems in Parameterized Complexity by itself, and, furthermore, it encompasses several other of the most well-studied problems in this field, such as Vertex Cover, Feedback Vertex Set in Tournaments (FVST) and Cluster Vertex Deletion (CVD). In fact, d-Hitting Set encompasses any deletion problem to a hereditary property that can be characterized by a finite set of forbidden induced subgraphs. With respect to bit size, the kernelization complexity of d-Hitting Set is essentially settled: there exists a kernel with ?(k^d) bits (?(k^d) sets and ?(k^{d-1}) elements) and this it tight by the result of Dell and van Melkebeek [STOC 2010, JACM 2014]. Still, the question of whether there exists a kernel for d-Hitting Set with fewer elements has remained one of the most major open problems in Kernelization.
In this paper, we first show that if we allow the kernelization to be lossy with a qualitatively better loss than the best possible approximation ratio of polynomial time approximation algorithms, then one can obtain kernels where the number of elements is linear for every fixed d. Further, based on this, we present our main result: we show that there exist approximate Turing kernelizations for d-Hitting Set that even beat the established bit-size lower bounds for exact kernelizations - in fact, we use a constant number of oracle calls, each with "near linear" (?(k^{1+?})) bit size, that is, almost the best one could hope for. Lastly, for two special cases of implicit 3-Hitting set, namely, FVST and CVD, we obtain the "best of both worlds" type of results - (1+?)-approximate kernelizations with a linear number of vertices. In terms of size, this substantially improves the exact kernels of Fomin et al. [SODA 2018, TALG 2019], with simpler arguments
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Preprocessing Complexity for Some Graph Problems Parameterized by Structural Parameters
Structural graph parameters play an important role in parameterized
complexity, including in kernelization. Notably, vertex cover, neighborhood
diversity, twin-cover, and modular-width have been studied extensively in the
last few years. However, there are many fundamental problems whose
preprocessing complexity is not fully understood under these parameters.
Indeed, the existence of polynomial kernels or polynomial Turing kernels for
famous problems such as Clique, Chromatic Number, and Steiner Tree has only
been established for a subset of structural parameters. In this work, we use
several techniques to obtain a complete preprocessing complexity landscape for
over a dozen of fundamental algorithmic problems.Comment: 24 pages, 1 table, 1 figur
Representative set statements for delta-matroids and the Mader delta-matroid
We present representative sets-style statements for linear delta-matroids,
which are set systems that generalize matroids, with important connections to
matching theory and graph embeddings. Furthermore, our proof uses a new
approach of sieving polynomial families, which generalizes the linear algebra
approach of the representative sets lemma to a setting of bounded-degree
polynomials. The representative sets statements for linear delta-matroids then
follow by analyzing the Pfaffian of the skew-symmetric matrix representing the
delta-matroid. Applying the same framework to the determinant instead of the
Pfaffian recovers the representative sets lemma for linear matroids.
Altogether, this significantly extends the toolbox available for kernelization.
As an application, we show an exact sparsification result for Mader networks:
Let be a graph and a partition of a set of terminals , . A -path in is a path with endpoints
in distinct parts of and internal vertices disjoint from . In
polynomial time, we can derive a graph with ,
such that for every subset there is a packing of
-paths with endpoints in if and only if there is one in
, and . This generalizes the (undirected version of the)
cut-covering lemma, which corresponds to the case that contains
only two blocks.
To prove the Mader network sparsification result, we furthermore define the
class of Mader delta-matroids, and show that they have linear representations.
This should be of independent interest
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