A Shortcut to (Sun)Flowers: Kernels in Logarithmic Space or Linear Time

Abstract

We investigate whether kernelization results can be obtained if we restrict kernelization algorithms to run in logarithmic space. This restriction for kernelization is motivated by the question of what results are attainable for preprocessing via simple and/or local reduction rules. We find kernelizations for d-hitting set(k), d-set packing(k), edge dominating set(k) and a number of hitting and packing problems in graphs, each running in logspace. Additionally, we return to the question of linear-time kernelization. For d-hitting set(k) a linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We give a simpler procedure and save a large constant factor in the size bound. Furthermore, we show that we can obtain a linear-time kernel for d-set packing(k) as well.

The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

“…Our proof of Theorem 5.2 exploits expressive kernelization algorithms for d-Hitting Set [2,3,19], which preserve inclusion-minimal solutions and that return subgraphs of the input hypergraph as kernels: Herein, given a hypergraph H = (U, C) with |C| ≤ d for each C ∈ C, and an integer k, d-Hitting Set asks whether there is a hitting set S ⊆ U with |S| ≤ k, that is, C ∩ S = ∅ for each C ∈ C. Our kernelization for Secluded F -free Vertex Deletion is based on transforming the input instance (G, k) to a d-Hitting Set instance (H, k), computing an expressive d-Hitting Set problem kernel (H ′ , k), and outputting a Secluded F -free Vertex Deletion instance (G ′ , k), where G ′ is the graph induced by the vertices remaining in H ′ together with at most k + 1 additional neighbors for each vertex in G.…”

“…To prove Theorem 5.2, we show that (G ′ , k) is a problem kernel for the input instance (G, k). The subgraph H ′ exists and is computable in linear time from H [3,19]. Moreover, for constant c, one can compute H from G and G ′ from H ′ in polynomial time.…”

Fixed-parameter algorithms for DAG Partitioning

“…Since Suen et al [33] point out that the input instances of DAG PARTITIONING can be so large that even running times quadratic in the input size are prohibitively large, we focus on finding algorithms that run in linear time if the parameter k is a constant. An important ingredient of our algorithms is linear-time data reduction, which recently received increased interest since data reduction is potentially applied to large input data [32,4,18,3,22,13].…”

“…From a practical point of view, problem kernelization is potentially applicable to speed up exact and heuristic algorithms to solve a problem. Since kernelization is applied to shrink potentially large input instances, recently the running time of kernelization has stepped into the focus and linear-time kernelization algorithms have been developed for various NP-hard problems [32,4,18,3,22,13].…”

Space-Efficient FPT Algorithms

“…In contrast, Directed Feedback Vertex Set restricted to tournaments can be shown to be in Para-L (Corollary 2). Later work in this setting includes the paper of Chen and Müller [7], who studied additional complexity-theoretic aspects of restrictedspace classes and showed that Longest Path is in Para-L. d-Hitting Set, a generalization of Vertex Cover to hypergraphs was shown by Fafanie and Kratsch [13] to be kernelizable in logarithmic space, which puts it in Para-L (see Lemma 2). As a consequence they showed that various graph deletion problems where the target classes are characterized by finite forbidden sets are also in Para-L. For related results, see [3,2] which give constant-time parallel kernelization algorithms for Hitting Set.…”