174 research outputs found

    Intersection Graphs of L-Shapes and Segments in the Plane

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    An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: ‚Ćä,‚Ćą,‚Ćč and ‚ĆČ. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an ‚Ćä, an ‚Ćä or ‚Ćą, a k-bend path, or a segment, then this graph is called an ‚Ćä-graph, ‚Ćä,‚Ćą-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365‚Äď372, 1992], stating that every ‚Ćä,‚Ćą-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are ‚Ćä-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are ‚Ćä-graphs. Furthermore we show that all complements of planar graphs are B 19-VPG-graphs and all complements of full subdivisions are B 2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once

    On the Threshold of Intractability

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    We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in 2O(‚ąöklog‚Ā°k)+poly(n)2^{O(\surd k \log k)} + \text{poly}(n) time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems

    Separating Hierarchical and General Hub Labelings

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    In the context of distance oracles, a labeling algorithm computes vertex labels during preprocessing. An s,ts,t query computes the corresponding distance from the labels of ss and tt only, without looking at the input graph. Hub labels is a class of labels that has been extensively studied. Performance of the hub label query depends on the label size. Hierarchical labels are a natural special kind of hub labels. These labels are related to other problems and can be computed more efficiently. This brings up a natural question of the quality of hierarchical labels. We show that there is a gap: optimal hierarchical labels can be polynomially bigger than the general hub labels. To prove this result, we give tight upper and lower bounds on the size of hierarchical and general labels for hypercubes.Comment: 11 pages, minor corrections, MFCS 201

    Approximation Algorithms for Capacitated Rectangle Stabbing

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    Sviluppo pre-clinico e clinico di inibitori della cellula staminale leucemica nelle leucemie acute

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    In Leukemias, recent developments have demonstrated that the Hedgehog pathway plays a key-role in the peculiar ability of self renewal of leukemia stem cells. The aim of this research activity was to investigate, through a first in man, Phase I, open label, clinical trial, the role and the impact, mainly in terms of safety profile, adverse events and pharmacokinetics, of a Sonic Hedgehog inhibitor compound on a population of heavely pretreated patients affected by AML, CML, MF, or MDS, resistant or refractory to standard chemotherapy. Thirty-five patients have been enrolled. The drug was administered orally, in 28 days cycles, without rest periods. The compound showed a good safety profile. The half life was of 17-35 hours, justifying the daily administration. Significant signs of activity, in terms of reduction of bone marrow blast cell amount were seen in most of the patients enrolled. Interestingly, correlative biological studies demonstrated that, comparing the gene expression profyiling signature of separated CD34+ cells before and after one cycle of treatment, the most variably expressed genes were involved in the Hh pathway. Moreover, we observed that many genes involved in MDR (multidrug resistance)were significantly down regulated after treatment. These data might lead to future clinical trials based on combinatory approaches, including, for instance, Hh inhibitors and conventional chemotherapy

    A SAT Approach to Clique-Width

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    Clique-width is a graph invariant that has been widely studied in combinatorics and computer science. However, computing the clique-width of a graph is an intricate problem, the exact clique-width is not known even for very small graphs. We present a new method for computing the clique-width of graphs based on an encoding to propositional satisfiability (SAT) which is then evaluated by a SAT solver. Our encoding is based on a reformulation of clique-width in terms of partitions that utilizes an efficient encoding of cardinality constraints. Our SAT-based method is the first to discover the exact clique-width of various small graphs, including famous graphs from the literature as well as random graphs of various density. With our method we determined the smallest graphs that require a small pre-described clique-width.Comment: proofs in section 3 updated, results remain unchange
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