Given a set of $n$ input integers, the Equal Subset Sum problem asks us to find two distinct subsets with the same sum. In this paper we present an algorithm that runs in time $O^*(3^{0.387n})$ in the~average case, significantly improving over the $O^*(3^{0.488n})$ running time of the best known worst-case algorithm and the Meet-in-the-Middle benchmark of $O^*(3^{0.5n})$. Our algorithm generalizes to a number of related problems, such as the ``Generalized Equal Subset Sum'' problem, which asks us to assign a coefficient $c_i$ from a set $C$ to each input number $x_i$ such that $\sum_{i} c_i x_i = 0$. Our algorithm for the average-case version of this problem runs in~time $|C|^{(0.5-c_0/|C|)n}$ for some positive constant $c_0$, whenever $C=\{0, \pm 1, \dots, \pm d\}$ or $\{\pm 1, \dots, \pm d\}$ for some positive integer $d$ (with $O^*(|C|^{0.45n})$ when $|C|<10$). Our results extend to the~problem of finding ``nearly balanced'' solutions in which the target is a not-too-large nonzero offset $\tau$. Our approach relies on new structural results that characterize the probability that $\sum_{i} c_i x_i$ $=\tau$ has a solution $c \in C^n$ when $x_i$'s are chosen randomly; these results may be of independent interest. Our algorithm is inspired by the ``representation technique'' introduced by Howgrave-Graham and Joux. This requires several new ideas to overcome preprocessing hurdles that arise in the representation framework, as well as a novel application of dynamic programming in the solution recovery phase of the algorithm.
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We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. The boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. Optimal convergence rates under the $L^2$ norm and the energy norm are derived. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.
Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m>>n collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit "m=infinity" of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to some related literature.
Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires $\Theta(2^n)$ queries to an $n$-bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted $\lambda$, with query complexities that depend polynomially on $\lambda$. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in $n$, already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of $n$. Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.
The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.
Under-approximations of reachable sets and tubes have received recent research attention due to their important roles in control synthesis and verification. Available under-approximation methods designed for continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general. In this note, we attempt to overcome this drawback for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes utilizing approximations of the matrix exponential. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and uncertain initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes with first order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, the proposed method is implemented in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.
Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.
The Discrete Periodic Radon Transform (DPRT) has been extensively used in applications that involve image reconstructions from projections. This manuscript introduces a fast and scalable approach for computing the forward and inverse DPRT that is based on the use of: (i) a parallel array of fixed-point adder trees, (ii) circular shift registers to remove the need for accessing external memory components when selecting the input data for the adder trees, (iii) an image block-based approach to DPRT computation that can fit the proposed architecture to available resources, and (iv) fast transpositions that are computed in one or a few clock cycles that do not depend on the size of the input image. As a result, for an $N\times N$ image ($N$ prime), the proposed approach can compute up to $N^{2}$ additions per clock cycle. Compared to previous approaches, the scalable approach provides the fastest known implementations for different amounts of computational resources. For example, for a $251\times 251$ image, for approximately $25\%$ fewer flip-flops than required for a systolic implementation, we have that the scalable DPRT is computed 36 times faster. For the fastest case, we introduce optimized architectures that can compute the DPRT and its inverse in just $2N+\left\lceil \log_{2}N\right\rceil+1$ and $2N+3\left\lceil \log_{2}N\right\rceil+B+2$ cycles respectively, where $B$ is the number of bits used to represent each input pixel. On the other hand, the scalable DPRT approach requires more 1-bit additions than for the systolic implementation and provides a trade-off between speed and additional 1-bit additions. All of the proposed DPRT architectures were implemented in VHDL and validated using an FPGA implementation.
The problem of aligning Erd\"os-R\'enyi random graphs is a noisy, average-case version of the graph isomorphism problem, in which a pair of correlated random graphs is observed through a random permutation of their vertices. We study a polynomial time message-passing algorithm devised to solve the inference problem of partially recovering the hidden permutation, in the sparse regime with constant average degrees. We perform extensive numerical simulations to determine the range of parameters in which this algorithm achieves partial recovery. We also introduce a generalized ensemble of correlated random graphs with prescribed degree distributions, and extend the algorithm to this case.
The design of methods for inference from time sequences has traditionally relied on statistical models that describe the relation between a latent desired sequence and the observed one. A broad family of model-based algorithms have been derived to carry out inference at controllable complexity using recursive computations over the factor graph representing the underlying distribution. An alternative model-agnostic approach utilizes machine learning (ML) methods. Here we propose a framework that combines model-based algorithms and data-driven ML tools for stationary time sequences. In the proposed approach, neural networks are developed to separately learn specific components of a factor graph describing the distribution of the time sequence, rather than the complete inference task. By exploiting stationary properties of this distribution, the resulting approach can be applied to sequences of varying temporal duration. Learned factor graph can be realized using compact neural networks that are trainable using small training sets, or alternatively, be used to improve upon existing deep inference systems. We present an inference algorithm based on learned stationary factor graphs, which learns to implement the sum-product scheme from labeled data, and can be applied to sequences of different lengths. Our experimental results demonstrate the ability of the proposed learned factor graphs to learn to carry out accurate inference from small training sets for sleep stage detection using the Sleep-EDF dataset, as well as for symbol detection in digital communications with unknown channels.
We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in $[0,1]^d$ of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most $B$ in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to "learn" characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints.
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