On the complexity of computing the kk-restricted edge-connectivity of a graph

The \emph{$k$-restricted edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least $k$ vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing $\lambda_k(G)$. Very recently, in the parameterized complexity community the notion of \emph{good edge separation} of a graph has been defined, which happens to be essentially the same as the $k$-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.Comment: 16 pages, 4 figure

Structurally Parameterized d-Scattered Set

In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $d\ge2$, an $O^*(d^{\textrm{tw}})$-time algorithm, where $\textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(\textrm{td}^2)}$-time algorithm parameterized by tree-depth ($\textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $\epsilon > 0$, runs in time $O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})})$ and returns a $d/(1+\epsilon)$-scattered set of size $k$, if a $d$-scattered set of the same size exists

Total knee arthroplasty: good agreement of clinical severity scores between patients and consultants

BACKGROUND: Nearly 20,000 patients per year in the UK receive total knee arthroplasty (TKA). One of the problems faced by the health services of many developed countries is the length of time patients spend waiting for elective treatment. We therefore report the results of a study in which the Salisbury Priority Scoring System (SPSS) was used by both the surgeon and their patients to ascertain whether there were differences between the surgeon generated and patient generated Salisbury Priority Scores. METHODS: The Salisbury Priority Scoring System (SPSS) was used to assign relative priority to patients with knee osteoarthritis as part of a randomised controlled trial comparing the standard medial parapatellar approach versus the sub-vastus approach in TKA. The operating surgeons and each patient completed the SPSS at the same pre-assessment clinic. The SPSS assesses four criteria, namely progression of disease, pain or distress, disability or dependence on others, and loss of usual occupation. Crosstabs and agreement measures (Cohen's kappa) were performed. RESULTS: Overall, the four SPSS criteria showed a kappa value of 0.526, 0.796, 0.813, and 0.820, respectively, showing moderate to very good agreement between the patient and the operating consultant. Male patients showed better agreement than female patients. CONCLUSION: The Salisbury Priority Scoring System is a good means of assessing patients' needs in relation to elective surgery, with high agreement between the patient and the operating surgeon

The Power of Cut-Based Parameters for Computing Edge Disjoint Paths

This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter treecut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded treecut width. Our second result shows that EDP parameterized by treecut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph

Fixed-Parameter Tractable Distances to Sparse Graph Classes

We show that for various classes $\mathcal{C}$ of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph $\small{G}$ to $\mathcal{C}$ is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph $\small{G}$ to a minor-closed class $\mathcal{C}$ is FPT. We demonstrate that several prior results (of Golovach, Moser and Thilikos and Mathieson) on the fixed-parameter tractability of distance measures are special cases of our first method

Calcium Sulfate and Platelet-Rich Plasma make a novel osteoinductive biomaterial for bone regeneration

BACKGROUND: With the present study we introduce a novel and simple biomaterial able to induce regeneration of bone. We theorized that nourishing a bone defect with calcium and with a large amount of activated platelets may initiate a series of biological processes that culminate in bone regeneration. Thus, we engineered CS-Platelet, a biomaterial based on the combination of Calcium Sulfate and Platelet-Rich Plasma in which Calcium Sulfate also acts as an activator of the platelets, therefore avoiding the need to activate the platelets with an agonist. METHODS: First, we tested CS-Platelet in heterotopic (muscle) and orthotopic (bone) bone regeneration bioassays. We then utilized CS-Platelet in a variety of dental and craniofacial clinical cases, where regeneration of bone was needed. RESULTS: The heterotopic bioassay showed formation of bone within the muscular tissue at the site of the implantation of CS-Platelet. Results of a quantitative orthotopic bioassay based on the rat calvaria critical size defect showed that only CS-Platelet and recombinant human BMP2 were able to induce a significant regeneration of bone. A non-human primate orthotopic bioassay also showed that CS-Platelet is completely resorbable. In all human clinical cases where CS-Platelet was used, a complete bone repair was achieved. CONCLUSION: This study showed that CS-Platelet is a novel biomaterial able to induce formation of bone in heterotopic and orthotopic sites, in orthotopic critical size bone defects, and in various clinical situations. The discovery of CS-Platelet may represent a cost-effective breakthrough in bone regenerative therapy and an alternative or an adjuvant to the current treatments

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP