Covering Pairs in Directed Acyclic Graphs

The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this paper, we study the computational complexity of two constrained variants of Minimum Path Cover motivated by the recent introduction of next-generation sequencing technologies in bioinformatics. The first problem (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum cardinality set of paths "covering" all the vertices such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The second problem (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W[1]-hardness when parametrized by the number of covered pairs. On the positive side, we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree, a natural parameter in the bioinformatics applications of the problem

Bounded Representations of Interval and Proper Interval Graphs

Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals L_v and R_v called bounds. We ask whether there exists a bounded representation in which each interval I_v has its left endpoint in L_v and its right endpoint in R_v. We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently. The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems

Algorithmic aspects of disjunctive domination in graphs

For a graph $G=(V,E)$, a set $D\subseteq V$ is called a \emph{disjunctive dominating set} of $G$ if for every vertex $v\in V\setminus D$, $v$ is either adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it. The cardinality of a minimum disjunctive dominating set of $G$ is called the \emph{disjunctive domination number} of graph $G$, and is denoted by $\gamma_{2}^{d}(G)$. The \textsc{Minimum Disjunctive Domination Problem} (MDDP) is to find a disjunctive dominating set of cardinality $\gamma_{2}^{d}(G)$. Given a positive integer $k$ and a graph $G$, the \textsc{Disjunctive Domination Decision Problem} (DDDP) is to decide whether $G$ has a disjunctive dominating set of cardinality at most $k$. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a $(\ln(\Delta^{2}+\Delta+2)+1)$-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within $(1-\epsilon) \ln(|V|)$ for any $\epsilon>0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log \log |V|)})$. Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree $3$

Minimal Obstructions for Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension

Identification of a negative regulatory role for Spi-C in the murine B cell lineage

Spi-C is an E26 transformation-specific family transcription factor that is highly related to PU.1 and Spi-B. Spi-C is expressed in developing B cells, but its function in B cell development and function is not well characterized. To determine whether Spi-C functions as a negative regulator of Spi-B (encoded by Spib), mice were generated that were germline knockout for Spib and heterozygous for Spic (Spib-/-Spic+/-). Interestingly, loss of one Spic allele substantially rescued B cell frequencies and absolute numbers in Spib-/- mouse spleens. Spib-/-Spic+/- B cells had restored proliferation compared with Spib-/- B cells in response to anti-IgM or LPS stimulation. Investigation of a potential mechanism for the Spib-/-Spic+/- phenotype revealed that steady-state levels of Nfkb1, encoding p50, were elevated in Spib-/-Spic+/- B cells compared with Spib-/- B cells. Spi-B was shown to directly activate the Nfkb1 gene, whereas Spi-C was shown to repress this gene. These results indicate a novel role for Spi-C as a negative regulator of B cell development and function

Smoothed Analysis of the Minimum-Mean Cycle Canceling Algorithm and the Network Simplex Algorithm

The minimum-cost flow (MCF) problem is a fundamental optimization problem with many applications and seems to be well understood. Over the last half century many algorithms have been developed to solve the MCF problem and these algorithms have varying worst-case bounds on their running time. However, these worst-case bounds are not always a good indication of the algorithms' performance in practice. The Network Simplex (NS) algorithm needs an exponential number of iterations for some instances, but it is considered the best algorithm in practice and performs best in experimental studies. On the other hand, the Minimum-Mean Cycle Canceling (MMCC) algorithm is strongly polynomial, but performs badly in experimental studies. To explain these differences in performance in practice we apply the framework of smoothed analysis. We show an upper bound of $O(mn^2\log(n)\log(\phi))$ for the number of iterations of the MMCC algorithm. Here $n$ is the number of nodes, $m$ is the number of edges, and $\phi$ is a parameter limiting the degree to which the edge costs are perturbed. We also show a lower bound of $\Omega(m\log(\phi))$ for the number of iterations of the MMCC algorithm, which can be strengthened to $\Omega(mn)$ when $\phi=\Theta(n^2)$. For the number of iterations of the NS algorithm we show a smoothed lower bound of $\Omega(m \cdot \min \{ n, \phi \} \cdot \phi)$.Comment: Extended abstract to appear in the proceedings of COCOON 201

Reducing Occupational Distress in Veterinary Medicine Personnel with Acceptance and Commitment Training: A Pilot Study

Aims To determine whether an educational programme targeting the reaction of veterinary personnel to difficult client interactions reduced burden transfer, stress and burnout in veterinary staff. Methods Employees of three small-animal veterinary hospitals in the south-western United States of America were recruited and randomised to intervention (educational programme; n = 16) or control (no intervention; n = 18) groups. Participants of this randomised, parallel arms trial completed pre-programme assessment including the Burden Transfer Inventory (BTI), Perceived Stress Scale, and Copenhagen Burnout Inventory. Assessment was followed by two, group-format educational sessions, based on acceptance and commitment training, tailored to reducing reactivity to difficult veterinary client interactions (intervention group only). After training was completed, both groups were assessed using the same measures and the intervention participants provided use and acceptability ratings. Results Intervention participants rated the programme as useful and appropriate, and reported that programme techniques were used a median of 43 (min 9, max 68) times during the 2 weeks prior to retesting. Relative to pre-programme scores, median post-programme scores for reaction (subscore of BTI) to difficult client interactions decreased in the intervention group (33 vs. 54; p = 0.047), but not in the control group (51 vs. 59; p = 0.210). Changes in median scores for stress and burnout from pre- to post-programme were non-significant for both groups. Conclusions This pilot and feasibility trial showed high rates of acceptability and use by participants, as well as promising reductions in burden transfer. A larger scale clinical trial with follow-up at extended time points is needed to more fully examine the efficacy of this novel programme

Labels direct infants’ attention to commonalities during novel category learning

Recent studies have provided evidence that labeling can influence the outcome of infants’ visual categorization. However, what exactly happens during learning remains unclear. Using eye-tracking, we examined infants’ attention to object parts during learning. Our analysis of looking behaviors during learning provide insights going beyond merely observing the learning outcome. Both labeling and non-labeling phrases facilitated category formation in 12-month-olds but not 8-month-olds (Experiment 1). Non-linguistic sounds did not produce this effect (Experiment 2). Detailed analyses of infants’ looking patterns during learning revealed that only infants who heard labels exhibited a rapid focus on the object part successive exemplars had in common. Although other linguistic stimuli may also be beneficial for learning, it is therefore concluded that labels have a unique impact on categorization

Reconfiguration of Cliques in a Graph

We study reconfiguration problems for cliques in a graph, which determine whether there exists a sequence of cliques that transforms a given clique into another one in a step-by-step fashion. As one step of a transformation, we consider three different types of rules, which are defined and studied in reconfiguration problems for independent sets. We first prove that all the three rules are equivalent in cliques. We then show that the problems are PSPACE-complete for perfect graphs, while we give polynomial-time algorithms for several classes of graphs, such as even-hole-free graphs and cographs. In particular, the shortest variant, which computes the shortest length of a desired sequence, can be solved in polynomial time for chordal graphs, bipartite graphs, planar graphs, and bounded treewidth graphs