10,637 research outputs found

    The covert set-cover problem with application to Network Discovery

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    We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OPTOPT within O(logN)O(\log N) factor with high probability using O(OPTlog2N)O(OPT \cdot \log^2 N) queries where NN is the input size. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown nn-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log2n)O(\log^2 n)-competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of Ω(nlogn)\Omega (\sqrt{n\log n}) and therefore our result achieves an exponential improvement

    The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs

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    Loebl, Koml\'os and S\'os conjectured that every nn-vertex graph GG with at least n/2n/2 vertices of degree at least kk contains each tree TT of order k+1k+1 as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of kk. For our proof, we use a structural decomposition which can be seen as an analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of GG to embed a given tree TT. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050]

    Strictly convex drawings of planar graphs

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    Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma

    The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs

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    This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every α>0\alpha>0 there exists a number k0k_0 such that for every k>k0k>k_0 every nn-vertex graph GG with at least (12+α)n(\frac12+\alpha)n vertices of degree at least (1+α)k(1+\alpha)k contains each tree TT of order kk as a subgraph. In the first paper of the series, we gave a decomposition of the graph GG into several parts of different characteristics; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree TT.Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDM
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