3 research outputs found

    Efficient Second-Order Shape-Constrained Function Fitting

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    We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-L∞L_{\infty} norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error Ρ\varepsilon in O(nlog⁑UΡ)O\left(n \log \frac{U}{\varepsilon} \right) time, where UU captures the range of input values. We also give a simple greedy algorithm that runs in O(n)O(n) time for the special case of unweighted L∞L_{\infty} convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming.Comment: accepted for WADS 2019; (v2 fixes various typos

    Efficient Second-Order Shape-Constrained Function Fitting

    Get PDF
    We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-L∞L_{\infty} norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error Ρ\varepsilon in O(nlog⁑UΡ)O\left(n \log \frac{U}{\varepsilon} \right) time, where UU captures the range of input values. We also give a simple greedy algorithm that runs in O(n)O(n) time for the special case of unweighted L∞L_{\infty} convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming

    A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path

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    Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis. Many NP-hard problems are known to be solvable in O~(nβ‹…2O(tw))\widetilde{O}(n \cdot 2^{O(\mathrm{tw})}) time, where tw\mathrm{tw} is the treewidth of the input graph. Analogously, many problems in P should be solvable in O~(nβ‹…twO(1))\widetilde{O}(n \cdot \mathrm{tw}^{O(1)}) time; however, due to the lack of appropriate tools, only a few such results are currently known. [Fom+18] conjectured this to hold as broadly as all linear programs; in our paper, we show this is true: Given a linear program of the form min⁑Ax=b,ℓ≀x≀uc⊀x\min_{Ax=b,\ell \leq x\leq u} c^{\top} x, and a width-Ο„\tau tree decomposition of a graph GAG_A related to AA, we show how to solve it in time O~(nβ‹…Ο„2log⁑(1/Ξ΅)),\widetilde{O}(n \cdot \tau^2 \log (1/\varepsilon)), where nn is the number of variables and Ξ΅\varepsilon is the relative accuracy. Combined with recent techniques in vertex-capacitated flow [BGS21], this leads to an algorithm with O~(nβ‹…tw2log⁑(1/Ξ΅))\widetilde{O}(n \cdot \mathrm{tw}^2 \log (1/\varepsilon)) run-time. Besides being the first of its kind, our algorithm has run-time nearly matching the fastest run-time for solving the sub-problem Ax=bAx=b (under the assumption that no fast matrix multiplication is used). We obtain these results by combining recent techniques in interior-point methods (IPMs), sketching, and a novel representation of the solution under a multiscale basis similar to the wavelet basis
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