International audienceWe present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X⊆V(G), called a treewidth-modulator, such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has a finite integer index and such that Yes-instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar-F-Deletion asks whether G has a set X⊆V(G) such that |X| ⩽ k and G − X is H-minor-free for every H ε F. As our second application, we present the first single-exponential algorithm to solve Planar-F-Deletion. Namely, our algorithm runs in time 2O(k) · n2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F

Genomes computationally inferred from large metagenomic data sets are often incomplete and may be missing functionally important content and strain variation. We introduce an information retrieval system for large metagenomic data sets that exploits the sparsity of DNA assembly graphs to efficiently extract subgraphs surrounding an inferred genome. We apply this system to recover missing content from genome bins and show that substantial genomic sequence variation is present in a real metagenome. Our software implementation is available at https://github.com/ spacegraphcats/spacegraphcats under the 3-Clause BSD License.

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International audienceWe present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V(G), called a treewidth-modulator, such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and such that positive instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs.Let Fbe a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar F- Deletion asks whether G has a set X ⊆ V(G) such that |X|⩽k and G − X is H-minor-free for every H∈F. As our second application, we present the first single-exponential algorithm to solve Planar F- Deletion. Namely, our algorithm runs in time 2 O(k)·n 2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F

The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which-given as input an n-vertex graph, a tree decomposition of the graph of width w, and an integer t-decides Treedepth, i.e. whether the treedepth of the graph is at most t, in time 2 O(wt) · n. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time 2 2 O(t) · n and 2 O(t 2 ) · n, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nešetřil as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed t) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of 2 O(t log t) · n for the same algorithm. * Research funded by DFG-Project RO 927/13-1 "Pragmatic Parameterized Algorithms".

We prove that graph problems with nite integer index have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. We also argue that such a linear kernelization result with a weaker parameter would fail to include some of the problems covered by our framework. We only require the problems to have FII on graphs of constant treedepth. This allows to prove linear kernels also for problems such as LongestPath/Cycle, Exact-s, t-Path, Treewidth, and Pathwidth, which do not have FII on general graphs.

The Flow Decomposition problem, which asks for the smallest set of weighted paths that "covers" a flow on a DAG, has recently been used as an important computational step in transcript assembly. We prove the problem is in FPT when parameterized by the number of paths by giving a practical linear fpt algorithm. Further, we implement and engineer a Flow Decomposition solver based on this algorithm, and evaluate its performance on RNA-sequence data. Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic. Finally, we contextualize our design choices with two hardness results related to preprocessing and weight recovery. Specifically, k-Flow Decomposition does not admit polynomial kernels under standard complexity assumptions, and the related problem of assigning (known) weights to a given set of paths is NP-hard.

This research establishes that many real-world networks exhibit bounded expansion 2 , a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis.We analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed sparse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabási-Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks.