1,135 research outputs found

    Constructive Discrepancy Minimization by Walking on the Edges

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    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

    Generalised Pattern Matching Revisited

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    In the problem of Generalised Pattern Matching (GPM)\texttt{Generalised Pattern Matching}\ (\texttt{GPM}) [STOC'94, Muthukrishnan and Palem], we are given a text TT of length nn over an alphabet ΣT\Sigma_T, a pattern PP of length mm over an alphabet ΣP\Sigma_P, and a matching relationship ΣT×ΣP\subseteq \Sigma_T \times \Sigma_P, and must return all substrings of TT that match PP (reporting) or the number of mismatches between each substring of TT of length mm and PP (counting). In this work, we improve over all previously known algorithms for this problem for various parameters describing the input instance: * D\mathcal{D}\, being the maximum number of characters that match a fixed character, * S\mathcal{S}\, being the number of pairs of matching characters, * I\mathcal{I}\, being the total number of disjoint intervals of characters that match the mm characters of the pattern PP. At the heart of our new deterministic upper bounds for D\mathcal{D}\, and S\mathcal{S}\, lies a faster construction of superimposed codes, which solves an open problem posed in [FOCS'97, Indyk] and can be of independent interest. To conclude, we demonstrate first lower bounds for GPM\texttt{GPM}. We start by showing that any deterministic or Monte Carlo algorithm for GPM\texttt{GPM} must use Ω(S)\Omega(\mathcal{S}) time, and then proceed to show higher lower bounds for combinatorial algorithms. These bounds show that our algorithms are almost optimal, unless a radically new approach is developed

    On a generalization of iterated and randomized rounding

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    We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems where iterated rounding has been useful

    An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound

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    We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an efficient algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t^{1/2} log n) bound. The result also extends to the more general Koml\'{o}s setting and gives an algorithmic O(log^{1/2} n) bound

    Algorithms to Approximate Column-Sparse Packing Problems

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    Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

    A Spectral Bound on Hypergraph Discrepancy

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    Let H\mathcal{H} be a tt-regular hypergraph on nn vertices and mm edges. Let MM be the m×nm \times n incidence matrix of H\mathcal{H} and let us denote λ=maxv1,v=1Mv\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|. We show that the discrepancy of H\mathcal{H} is O(t+λ)O(\sqrt{t} + \lambda). As a corollary, this gives us that for every tt, the discrepancy of a random tt-regular hypergraph with nn vertices and mnm \geq n edges is almost surely O(t)O(\sqrt{t}) as nn grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1811.01491, several changes to the presentatio

    Discrepancy Without Partial Colorings

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    Spencer\u27s theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be "colored" by the values +1 or -1 such that the sum of every set is O(sqrt(n)) in absolute value. All existing proofs of this result recursively construct "partial colorings", which assign +1 or -1 values to half of the ground set. We devise the first algorithm for Spencer\u27s theorem that directly computes a coloring, without recursively computing partial colorings
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