4,528 research outputs found

    Better Sparsifiers for Directed Eulerian Graphs

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    Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distribution, hitting and commute time, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph G,G, (1+Īµ)(1+\varepsilon)-sparsifiers with O(nĪµāˆ’2)O(n\varepsilon^{-2}) edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require Ī©(nĪµāˆ’2logā”4n)\Omega(n\varepsilon^{-2}\log^4 n) edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1. We show that for every directed Eulerian graph Gāƒ—,\vec{G}, there exist an Eulerian sparsifier with O(nĪµāˆ’2logā”2nlogā”2logā”n+nĪµāˆ’4/3logā”8/3n)O(n\varepsilon^{-2} \log^2 n \log^2\log n + n\varepsilon^{-4/3}\log^{8/3} n) edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2. We give an improved analysis of the constructions based on short-cycle decompositions, giving an m1+Ī“m^{1+\delta}-time algorithm for any constant Ī“>0\delta > 0 for constructing Eulerian sparsifiers with O(nĪµāˆ’2logā”3n)O(n\varepsilon^{-2}\log^3 n) edges

    Graph Kernels

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    We present a unified framework to study graph kernels, special cases of which include the random walk (GƤrtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; MahƩ et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Frƶhlich et al. (2006) yet provably positive semi-definite

    The Interlace Polynomial

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    In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials, edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL

    Embedded graph invariants in Chern-Simons theory

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    Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines - an embedded graph invariant. Using a slight generalization of the variational method, lowest-order results for invariants for arbitrary valence graphs are derived; gauge invariant operators are introduced; and some higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. Though, without a global projection of the graph, there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity - as a way of relating frames at distinct vertices.Comment: 20 pages; RevTex; with approx 50 ps figures; References added, introduction rewritten, version to be published in Nuc. Phys.

    Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes

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    Bipartite on-shell diagrams are the latest tool in constructing scattering amplitudes. In this paper we prove that a Britto-Cachazo-Feng-Witten (BCFW)-decomposable on-shell diagram process a rational top-form if and only if the algebraic ideal comprised of the geometrical constraints is shifted linearly during successive BCFW integrations. With a proper geometric interpretation of the constraints in the Grassmannian manifold, the rational top-form integration contours can thus be obtained, and understood, in a straightforward way. All rational top-form integrands of arbitrary higher loops leading singularities can therefore be derived recursively, as long as the corresponding on-shell diagram is BCFW-decomposable.Comment: 13 pages with 12 figures; final version appeared in Eur.Phys.J. C77 (2017) no.2, 8
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