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In Layout Synthesis, the logical qubits of a quantum circuit are mapped to the physical qubits of a given quantum hardware platform, taking into account the connectivity of physical qubits. This involves inserting SWAP gates before an operation is applied on distant qubits. Optimal Layout Synthesis is crucial for practical Quantum Computing on current error-prone hardware: Minimizing the number of SWAP gates directly mitigates the error rates when running quantum circuits. In recent years, several approaches have been proposed for minimizing the required SWAP insertions. The proposed exact approaches can only scale to a small number of qubits. Proving that a number of swap insertions is optimal is much harder than producing near optimal mappings. In this paper, we provide two encodings for Optimal Layout Synthesis as a classical planning problem. We use optimal classical planners to synthesize the optimal layout for a standard set of benchmarks. Our results show the scalability of our approach compared to previous leading approaches. We can optimally map circuits with 9 qubits onto a 14 qubit platform, which could not be handled before by exact methods.

Computational aspects of solution notions such as Nash equilibrium have been extensively studied, including settings where the ultimate goal is to find an equilibrium that possesses some additional properties. Furthermore, in order to address issues of tractability, attention has been given to approximate versions of these problems. Our work extends this direction by considering games with constraints in which players are subject to some form of restrictions on their strategic choices. We also consider the relationship between Nash equilibria and so-called constrained or social equilibria in this context, with particular attention to how they are related with respect to totality and complexity. Our results demonstrate that the computational complexity of finding an equilibrium varies significantly between games with slightly different strategic constraints. In addition to examining the computational aspects of such strategic constraints, we also demonstrate that these constraints are useful for modeling problems involving strategic resource allocation and also are of interest from the perspective of behavioral game theory.

Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-$k$AND problem. This generalizes the definition by Feige and Jozeph (Algorithmica '15) of oblivious algorithms for Max-DICUT, a special case of Max-$2$AND. Oblivious algorithms round each variable with probability depending only on a quantity called the variable's bias. For each oblivious algorithm, we design a so-called "factor-revealing linear program" (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all $k$, oblivious algorithms for Max-$k$AND provably outperform a special subclass of algorithms we call "superoblivious" algorithms. Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every $k$, $O(\log n)$-space sketching algorithms for Max-$k$AND known to be optimal in $o(\sqrt n)$-space can be beaten in (a) $O(\log n)$-space under a random-ordering assumption, and (b) $O(n^{1-1/k} D^{1/k})$ space under a maximum-degree-$D$ assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.

This paper concerns the numerical solution of the two-dimensional time-dependent partial integro-differential equation (PIDE) that holds for the values of European-style options under the two-asset Kou jump-diffusion model. A main feature of this equation is the presence of a nonlocal double integral term. For its numerical evaluation, we extend a highly efficient algorithm derived by Toivanen (2008) in the case of the one-dimensional Kou integral. The acquired algorithm for the two-dimensional Kou integral has optimal computational cost: the number of basic arithmetic operations is directly proportional to the number of spatial grid points in the semidiscretization. For the effective discretization in time, we study seven contemporary operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind. All these schemes allow for a convenient, explicit treatment of the integral term. We analyze their (von Neumann) stability. By ample numerical experiments for put-on-the-average option values, the actual convergence behavior as well as the mutual performance of the seven operator splitting schemes are investigated. Moreover, the Greeks Delta and Gamma are considered.

Profile likelihoods are rarely used in geostatistical models due to the computational burden imposed by repeated decompositions of large variance matrices. Accounting for uncertainty in covariance parameters can be highly consequential in geostatistical models as some covariance parameters are poorly identified, the problem is severe enough that the differentiability parameter of the Matern correlation function is typically treated as fixed. The problem is compounded with anisotropic spatial models as there are two additional parameters to consider. In this paper, we make the following contributions: 1, A methodology is created for profile likelihoods for Gaussian spatial models with Mat\'ern family of correlation functions, including anisotropic models. This methodology adopts a novel reparametrization for generation of representative points, and uses GPUs for parallel profile likelihoods computation in software implementation. 2, We show the profile likelihood of the Mat\'ern shape parameter is often quite flat but still identifiable, it can usually rule out very small values. 3, Simulation studies and applications on real data examples show that profile-based confidence intervals of covariance parameters and regression parameters have superior coverage to the traditional standard Wald type confidence intervals.

We develop the notion of discrete degrees of freedom of a log-concave sequence and use it to prove that geometric distribution minimises R\'enyi entropy of order infinity under fixed variance, among all discrete log-concave random variables in $\mathbb{Z}$. We also show that the quantity $\mathbb{P}(X=\mathbb{E} X)$ is maximised, among all ultra-log-concave random variables with fixed integral mean, for a Poisson distribution.

We study the connections between sorting and the binary search tree (BST) model, with an aim towards showing that the fields are connected more deeply than is currently appreciated. While any BST can be used to sort by inserting the keys one-by-one, this is a very limited relationship and importantly says nothing about parallel sorting. We show what we believe to be the first formal relationship between the BST model and sorting. Namely, we show that a large class of sorting algorithms, which includes mergesort, quicksort, insertion sort, and almost every instance-optimal sorting algorithm, are equivalent in cost to offline BST algorithms. Our main theoretical tool is the geometric interpretation of the BST model introduced by Demaine et al., which finds an equivalence between searches on a BST and point sets in the plane satisfying a certain property. To give an example of the utility of our approach, we introduce the log-interleave bound, a measure of the information-theoretic complexity of a permutation $\pi$, which is within a $\lg \lg n$ multiplicative factor of a known lower bound in the BST model; we also devise a parallel sorting algorithm with polylogarithmic span that sorts a permutation $\pi$ using comparisons proportional to its log-interleave bound. Our aforementioned result on sorting and offline BST algorithms can be used to show existence of an offline BST algorithm whose cost is within a constant factor of the log-interleave bound of any permutation $\pi$.

The question of whether $Y$ can be predicted based on $X$ often arises and while a well adjusted model may perform well on observed data, the risk of overfitting always exists, leading to poor generalization error on unseen data. This paper proposes a rigorous permutation test to assess the credibility of high $R^2$ values in regression models, which can also be applied to any measure of goodness of fit, without the need for sample splitting, by generating new pairings of $(X_i, Y_j)$ and providing an overall interpretation of the model's accuracy. It introduces a new formulation of the null hypothesis and justification for the test, which distinguishes it from previous literature. The theoretical findings are applied to both simulated data and sensor data of tennis serves in an experimental context. The simulation study underscores how the available information affects the test, showing that the less informative the predictors, the lower the probability of rejecting the null hypothesis, and emphasizing that detecting weaker dependence between variables requires a sufficient sample size.

Although data augmentation is a powerful technique for improving the performance of image classification tasks, it is difficult to identify the best augmentation policy. The optimal augmentation policy, which is the latent variable, cannot be directly observed. To address this problem, this study proposes $\textit{LatentAugment}$, which estimates the latent probability of optimal augmentation. The proposed method is appealing in that it can dynamically optimize the augmentation strategies for each input and model parameter in learning iterations. Theoretical analysis shows that LatentAugment is a general model that includes other augmentation methods as special cases, and it is simple and computationally efficient in comparison with existing augmentation methods. Experimental results show that the proposed LatentAugment has higher test accuracy than previous augmentation methods on the CIFAR-10, CIFAR-100, SVHN, and ImageNet datasets.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.