Fast Biclustering by Dual Parameterization

We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails. Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution. However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time $2^{5 \sqrt{pk}} + O(n+m)$ for p-Starforest Editing and $2^{O(p \sqrt{k} \log(pk))} + O(n+m)$ for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for fixed number of clusters, since p is then considered a constant. Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.Comment: Accepted for presentation at IPEC 201

A practical fpt algorithm for Flow Decomposition and transcript assembly

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.Comment: Introduces software package Toboggan: Version 1.0. http://dx.doi.org/10.5281/zenodo.82163

Kernelization and Sparseness: the case of Dominating Set

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded expansion graph

Zig-Zag Numberlink is NP-Complete

When can $t$ terminal pairs in an $m \times n$ grid be connected by $t$ vertex-disjoint paths that cover all vertices of the grid? We prove that this problem is NP-complete. Our hardness result can be compared to two previous NP-hardness proofs: Lynch's 1975 proof without the ``cover all vertices'' constraint, and Kotsuma and Takenaga's 2010 proof when the paths are restricted to have the fewest possible corners within their homotopy class. The latter restriction is a common form of the famous Nikoli puzzle \emph{Numberlink}; our problem is another common form of Numberlink, sometimes called \emph{Zig-Zag Numberlink} and popularized by the smartphone app \emph{Flow Free}

La segregación étnica en la educación secundaria de la ciudad de Madrid : un mapa y una lectura crítica

En esta comunicación analizamos los procesos de distribución del alumnado inmigrante en educación secundaria de la ciudad de Madrid en la red de centros públicos, privados y concertados. En el análisis se ponen en relación tres fuentes de datos: datos globalessobre la educación secundaria en la Comunidad Autónoma de Madrid y la ciudad de Madrid a lo largo de los últimos años, la población estudiantil inmigrante en los diferentes distritos de la ciudad de Madrid y los datos obtenidos a través de una serie de cuestionarios propios que administramos a una muestra de centros de la ciudad de Madrid durante el curso 2005-06. Nuestros datos confirman lo que diferentes informes han señalado: el alumnado inmigrante se concentra en la red pública mientras, en los centros concertados hay una proporción de alumnos inmigrantes menor que la que sería predecible en función de las características del entorno en el que se ubican y en los centros privados la presencia de alumnado extranjero extracomunitario es insignificante. No obstante, nuestros datos sugieren que dentro de cada una de las redes de centros hay cierta diversidad interna en cuanto al papel que desempeñan diferentes centros en la educación del alumnado inmigrante y que en la ciudad de Madrid es posible encontrar un conjunto de centros concertados con una proporción significativa de alumnos de origen inmigrante. Nuestros datos se discuten en relación con la política educativa actual de la Comunidad de Madrid que ha favorecido el crecimiento de la educación privada- concertada y, además, restringe el acceso a los datos estadísticos sobre la distribución de alumnado inmigrante en los centros bajo su responsabilidad

Characterising bounded expansion by neighbourhood complexity

We show that a graph class $\cal G$ has \emph{bounded expansion} if and only if it has bounded \emph{$r$-neighbourhood complexity}, \ie for any vertex set $X$ of any subgraph~$H$ of any $G\in\cal G$, the number of subsets of $X$ which are exact $r$-neighbourhoods of vertices of $H$ on $X$ is linear in the size of $X$. This is established by bounding the $r$-neighbourhood complexity of a graph in terms of both its \emph{$r$-centred colouring number} and its \emph{weak $r$-colouring number}, which provide known characterisations to the property of bounded expansion