44 research outputs found
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
On the Descriptive Complexity of Color Coding
Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing - purely in terms of the syntactic structure of describing formulas - when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in fpt just because of the way they are commonly described using logical formulas
Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph
We present fixed-parameter tractable (FPT) algorithms for two problems,
Maximum Happy Set (MaxHS) and Maximum Edge Happy Set (MaxEHS)--also known as
Densest k-Subgraph. Given a graph and an integer , MaxHS asks for a set
of vertices such that the number of with
respect to is maximized, where a vertex is happy if and all its
neighbors are in . We show that MaxHS can be solved in time
and , where and denote the
and the of , respectively.
This resolves the open questions posed in literature. The MaxEHS problem is an
edge-variant of MaxHS, where we maximize the number of ,
the edges whose endpoints are in . In this paper we show that MaxEHS can be
solved in time and
, where
and denote the
and the of , respectively, and is
some computable function. This result implies that MaxEHS is also
fixed-parameter tractable by