3,706 research outputs found

    Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

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    A hypergraph is said to be χ\chi-colorable if its vertices can be colored with χ\chi colors so that no hyperedge is monochromatic. 22-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 22-colorable kk-uniform hypergraph, it is NP-hard to find a 22-coloring miscoloring fewer than a fraction 2k+12^{-k+1} of hyperedges (which is achieved by a random 22-coloring), and the best algorithms to color the hypergraph properly require n11/k\approx n^{1-1/k} colors, approaching the trivial bound of nn as kk increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 22-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 22-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy k\ell \ll \sqrt{k}, we give an algorithm to color the it with nO(2/k)\approx n^{O(\ell^2/k)} colors. However, for the maximization version, we prove NP-hardness of finding a 22-coloring miscoloring a smaller than 2O(k)2^{-O(k)} (resp. kO(k)k^{-O(k)}) fraction of the hyperedges when =O(logk)\ell = O(\log k) (resp. =2\ell=2). Assuming the UGC, we improve the latter hardness factor to 2O(k)2^{-O(k)} for almost discrepancy-11 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (k)(k-\ell)-coloring such that each hyperedge is polychromatic with all these colors, we give a 22-coloring algorithm that miscolors at most kΩ(k)k^{-\Omega(k)} of the hyperedges when k\ell \ll \sqrt{k}, and complement this with a matching UG hardness result showing that when =k\ell =\sqrt{k}, it is hard to even beat the 2k+12^{-k+1} bound achieved by a random coloring.Comment: Approx 201

    Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

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    We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

    Discrete Convex Functions on Graphs and Their Algorithmic Applications

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    The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

    Precise Coulomb wave functions for a wide range of complex l, eta and z

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    A new algorithm to calculate Coulomb wave functions with all of its arguments complex is proposed. For that purpose, standard methods such as continued fractions and power/asymptotic series are combined with direct integrations of the Schrodinger equation in order to provide very stable calculations, even for large values of |eta| or |Im(l)|. Moreover, a simple analytic continuation for Re(z) < 0 is introduced, so that this zone of the complex z-plane does not pose any problem. This code is particularly well suited for low-energy calculations and the calculation of resonances with extremely small widths. Numerical instabilities appear, however, when both |eta| and |Im(l)| are large and |Re(l)| comparable or smaller than |Im(l)|

    Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor

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    We present a mathematica package that performs the symbolic calculation of integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_{\nu} (x) j_{\mu} (x) dx where jν(x)j_{\nu} (x) and jμ(x)j_{\mu} (x) denote spherical Bessel functions of integer orders, with ν0\nu \ge 0 and μ0\mu \ge 0. With the real parameter u>0u>0 and the integer nn, convergence of the integral requires that n+ν+μ0n+\nu +\mu \ge 0. The package provides analytical result for the integral in its most simplified form. The novel symbolic method employed enables the calculation of a large number of integrals of the above form in a fraction of the time required for conventional numerical and Mathematica based brute-force methods. We test the accuracy of such analytical expressions by comparing the results with their numerical counterparts.Comment: 17 pages; updated references for the introductio
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