Maximal induced matchings in triangle-free graphs

An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} \approx 1.5849^n$ maximal induced matchings, and this bound is best possible. We prove that any $n$-vertex triangle-free graph has at most $3^{n/3} \approx 1.4423^n$ maximal induced matchings, and this bound is attained by any disjoint union of copies of the complete bipartite graph $K_{3,3}$. Our result implies that all maximal induced matchings in an $n$-vertex triangle-free graph can be listed in time $O(1.4423^n)$, yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.Comment: 17 page

Network conduciveness with application to the graph-coloring and independent-set optimization transitions

We introduce the notion of a network's conduciveness, a probabilistically interpretable measure of how the network's structure allows it to be conducive to roaming agents, in certain conditions, from one portion of the network to another. We exemplify its use through an application to the two problems in combinatorial optimization that, given an undirected graph, ask that its so-called chromatic and independence numbers be found. Though NP-hard, when solved on sequences of expanding random graphs there appear marked transitions at which optimal solutions can be obtained substantially more easily than right before them. We demonstrate that these phenomena can be understood by resorting to the network that represents the solution space of the problems for each graph and examining its conduciveness between the non-optimal solutions and the optimal ones. At the said transitions, this network becomes strikingly more conducive in the direction of the optimal solutions than it was just before them, while at the same time becoming less conducive in the opposite direction. We believe that, besides becoming useful also in other areas in which network theory has a role to play, network conduciveness may become instrumental in helping clarify further issues related to NP-hardness that remain poorly understood

Lower bounds for several online variants of bin packing

We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201

Enumerating Cyclic Orientations of a Graph

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the problem of efficiently enumerating all the \emph{cyclic} orientations of an undirected connected graph with $n$ vertices and $m$ edges, observing that it cannot be solved using algorithmic techniques previously employed for enumerating acyclic orientations.We show that the problem is of independent interest from both combinatorial and algorithmic points of view, and that each cyclic orientation can be listed with $\tilde{O}(m)$ delay time. Space usage is $O(m)$ with an additional setup cost of $O(n^2)$ time before the enumeration begins, or $O(mn)$ with a setup cost of $\tilde{O}(m)$ time

It’s about time: A synthesis of changing phenology in the Gulf of Maine ecosystem

The timing of recurring biological and seasonal environmental events is changing on a global scale relative to temperature and other climate drivers. This study considers the Gulf of Maine ecosystem, a region of high social and ecological importance in the Northwest Atlantic Ocean and synthesizes current knowledge of (a) key seasonal processes, patterns, and events; (b) direct evidence for shifts in timing; (c) implications of phenological responses for linked ecological-human systems; and (d) potential phenology-focused adaptation strategies and actions. Twenty studies demonstrated shifts in timing of regional marine organisms and seasonal environmental events. The most common response was earlier timing, observed in spring onset, spring and winter hydrology, zooplankton abundance, occurrence of several larval fishes, and diadromous fish migrations. Later timing was documented for fall onset, reproduction and fledging in Atlantic puffins, spring and fall phytoplankton blooms, and occurrence of additional larval fishes. Changes in event duration generally increased and were detected in zooplankton peak abundance, early life history periods of macro-invertebrates, and lobster fishery landings. Reduced duration was observed in winter-spring ice-affected stream flows. Two studies projected phenological changes, both finding diapause duration would decrease in zooplankton under future climate scenarios. Phenological responses were species-specific and varied depending on the environmental driver, spatial, and temporal scales evaluated. Overall, a wide range of baseline phenology and relevant modeling studies exist, yet surprisingly few document long-term shifts. Results reveal a need for increased emphasis on phenological shifts in the Gulf of Maine and identify opportunities for future research and consideration of phenological changes in adaptation efforts

Local Guarantees in Graph Cuts and Clustering

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled $+$ or $-$ and the goal is to produce a clustering that agrees with the labels as much as possible: $+$ edges within clusters and $-$ edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max $s-t$ Cut: find an $s-t$ cut minimizing the largest number of cut edges incident on any node. We present the following results: $(1)$ an $O(\sqrt{n})$-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems), $(2)$ a remarkably simple $7$-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of $48$), and $(3)$ a $1/(2+\varepsilon)$-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the $1/(4+\varepsilon)$-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games

Constructive Relationships Between Algebraic Thickness and Normality

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < \epsilon<2$, any function with algebraic thickness $n^{3-\epsilon}$ is constant on some affine subspace of dimension $\Omega\left(n^{\frac{\epsilon}{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $\Theta(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $\Theta(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $\Omega\left(2^{n^{1/6}}\right)$.Comment: Final version published in FCT'201

Exploring Graphs with Time Constraints by Unreliable Collections of Mobile Robots

A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored. We concentrate on line graphs and rings, for which we give positive results. We start with the case of the collections involving only reliable robots. We give algorithms finding optimal times needed for exploration when the robots are assigned to fixed initial positions as well as when such starting positions may be determined by the algorithm. We extend our consideration to the case when some number of robots may be unreliable. Our most surprising result is that solving the line exploration problem with robots at given positions, which may involve crash-faulty ones, is NP-hard. The same problem has polynomial solutions for a ring and for the case when the initial robots' positions on the line are arbitrary. The exploration problem is shown to be NP-hard for star graphs, even when the team consists of only two reliable robots

Nurses' workarounds in acute healthcare settings: A scoping review

Background: Workarounds circumvent or temporarily 'fix' perceived workflow hindrances to meet a goal or to achieve it more readily. Behaviours fitting the definition of workarounds often include violations, deviations, problem solving, improvisations, procedural failures and shortcuts. Clinicians implement workarounds in response to the complexity of delivering patient care. One imperative to understand workarounds lies in their influence on patient safety. This paper assesses the peer reviewed empirical evidence available on the use, proliferation, conceptualisation, rationalisation and perceived impact of nurses' use of workarounds in acute care settings. Methods. A literature assessment was undertaken in 2011-2012. Snowballing technique, reference tracking, and a systematic search of twelve academic databases were conducted to identify peer reviewed published studies in acute care settings examining nurses' workarounds. Selection criteria were applied across three phases. 58 studies were included in the final analysis and synthesis. Using an analytic frame, these studies were interrogated for: workarounds implemented in acute care settings by nurses; factors contributing to the development and proliferation of workarounds; the perceived impact of workarounds; and empirical evidence of nurses' conceptualisation and rationalisation of workarounds. Results: The majority of studies examining nurses' workarounds have been published since 2008, predominantly in the United States. Studies conducted across a variety of acute care settings use diverse data collection methods. Nurses' workarounds, primarily perceived negatively, are both individually and collectively enacted. Organisational, work process, patient-related, individual, social and professional factors contribute to the proliferation of workarounds. Group norms, local and organisational culture, 'being competent', and collegiality influence the implementation of workarounds. Conclusion: Workarounds enable, yet potentially compromise, the execution of patient care. In some contexts such improvisations may be deemed necessary to the successful implementation of quality care, in others they are counterproductive. Workarounds have individual and cooperative characteristics. Few studies examine nurses' individual and collective conceptualisation and rationalisation of workarounds or measure their impact. The importance of displaying competency (image management), collegiality and organisational and cultural norms play a role in nurses' use of workarounds. © 2013 Debono et al.; licensee BioMed Central Ltd

Sensitivity of an Ultrasonic Technique for Axial Stress Determination

In machine assembly it is often required that bolts used to fasten machine parts be installed with specific design preloads. Because it is inconvenient to measure preload directly, preload specifications are usually based on some more easily measured quantity with which the level of preload may be correlated. Most often this quantity is the torque to be applied to the bolt at installation. Studies by Blake and Kurtz [1] and Heyman [2] have shown that when bolts are torqued into place, the fraction of applied torque which translates into useful preload is small and widely variable. This is so because the large majority of applied torque is absorbed in overcoming friction in the bolt’s threads and at the underside of the bolt’s head. Consequently, even though the torque to install different bolts may be identical, small variations in frictional conditions from one installation to the next can result in large variations in preload. The unreliability of torque as an indicator of preload has been the motivating factor behind the development of a number of alternate methods of measurement [2–5]