20,815 research outputs found

    Frame Permutation Quantization

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    Frame permutation quantization (FPQ) is a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented, and reconstruction algorithms based on linear programming, quadratic programming, and recursive orthogonal projection are derived. Implementations of the linear and quadratic programming algorithms for uniform and Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate. Monte Carlo evaluation of the recursive algorithm shows that mean-squared error (MSE) decays as 1/M^4 for an M-element frame, which is consistent with previous results on optimal decay of MSE. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions.Comment: 29 pages, 5 figures; detailed added to proof of Theorem 4.3 and a few minor correction

    Deciding first-order properties of nowhere dense graphs

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    Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page

    Hitting forbidden minors: Approximation and Kernelization

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    We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most kk vertices can be deleted from a graph GG such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding tt-claw K1,tK_{1,t}, the star with tt leves, as an induced subgraph, where tt is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of O(log3/2OPT)O(\log^{3/2} OPT), where OPTOPT is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph θc\theta_c as a minor for a fixed integer cc. The graph θc\theta_c consists of two vertices connected by cc parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes

    Metric Embedding via Shortest Path Decompositions

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    We study the problem of embedding shortest-path metrics of weighted graphs into p\ell_p spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth 11. General graph has an SPD of depth kk if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most k1k-1. In this paper we give an O(kmin{1p,12})O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})-distortion embedding for graphs of SPD depth at most kk. This result is asymptotically tight for any fixed p>1p>1, while for p=1p=1 it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth kk embed into p\ell_p with distortion O(kmin{1p,12})O(k^{\min\{\frac{1}{p},\frac{1}{2}\}}). For p=1p=1, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in kk; moreover, for other values of pp it gives the first embeddings whose distortion is independent of the graph size nn. Furthermore, we use the fact that planar graphs have SPD depth O(logn)O(\log n) to give a new proof that any planar graph embeds into 1\ell_1 with distortion O(logn)O(\sqrt{\log n}). Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor
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