Optimal Piecewise Polynomial Approximation for Minimum Computing Cost by Using Constrained Least Squares.

A rapid method for calculating maximal and minimal inter-satellite distances

In this paper, we consider two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club for positive integers d. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$ for any real $$\varepsilon > 0$$ unless $${{{\mathcal {P}}}} = {{\mathcal {NP}}}$$ , since they are identical to Max Clique (Hastad in Acta Math 182(1):105–142, 1999; Zuckerman in Theory Comput 3:103–128, 2007). In addition, it is $${{\mathcal {NP}}}$$ -hard to approximate Maxd-Clique and Maxd-Club to within a factor of $$n^{1/2 - \varepsilon }$$ for any fixed integer $$d\ge 2$$ and any real $$\varepsilon > 0$$ (Asahiro et al. in Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010, Springer, pp 615–626, 2010; Asahiro et al. in Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA, Springer, pp 586–600, 2015). As for approximability of Maxd-Clique and Maxd-Club, a polynomial-time algorithm, called ReFindStar $$_d$$ , that achieves an optimal approximation ratio of $$O(n^{1/2})$$ for Maxd-Clique and Maxd-Club was designed for any integer $$d\ge 2$$ in Asahiro et al. (2015, Algorithmica 80(6):1834–1856, 2018). Moreover, a simpler algorithm, called ByFindStar $$_d$$ , was proposed and it was shown in Asahiro et al. (2010, 2018) that although the approximation ratio of ByFindStar $$_d$$ is much worse for any odd $$d\ge 3$$ , its time complexity is better than ReFindStar $$_d$$ . In this paper, we implement those approximation algorithms and evaluate their quality empirically for random graphs. The experimental results show that (1) ReFindStar $$_d$$ can find larger d-clubs (d-cliques) than ByFindStar $$_d$$ for odd d, (2) the size of d-clubs (d-cliques) output by ByFindStar $$_d$$ is the same as ones by ReFindStar $$_d$$ for even d, and (3) ByFindStar $$_d$$ can find the same size of d-clubs (d-cliques) much faster than ReFindStar $$_d$$ . Furthermore, we propose and implement two new heuristics, Hclub $$_d$$ for Maxd-Club and Hclique $$_d$$ for Maxd-Clique. Then, we present the experimental evaluation of the solution size of ReFindStar $$_d$$ , Hclub $$_d$$ , Hclique $$_d$$ and previously known heuristic algorithms for random graphs and Erdős collaboration graphs.

Piecewise Polynomial Model Identification using Constrained Least Squares for UAS Stall

We consider a game of persuasion with evidence between a sender and a receiver. The sender has private information. By presenting evidence on the information, the sender wishes to persuade the receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether or not the receiver takes the action. The receiver’s utility depends on both the action as well as the sender’s private information. We study three natural variations. First, we consider sequential equilibria of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and then the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and then the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, as well as polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm for a natural problem in algorithmic economics.

Optimal recovery of multivariate contractions

Optimal router node placement (RNP) is an effective method for improving the performance of wireless mesh networks (WMN). However, solving the RNP problem in WMN is difficult because it is NP-hard. As a result, this problem can only be solved using approximate optimization algorithms such as heuristics and meta-heuristics. In this study, we propose a new and effective method for solving the RNP problem. The idea behind this method is to solve the RNP problem in two stages using an optimal algorithm with fewer variables than the original RNP problem. In stage 1, we build an RNP sub problem using 15% to 20% of the number of routers, with the objective function of minimizing coverage overlap between routers to form a core network. Stage 2 is built into another RNP sub problem with the remaining number of routers, and the objective function is to maximize the network connectivity. Each sub problem was solved using an approximate optimal algorithm. The experimental results demonstrate that, in terms of client coverage and network connectivity, our proposed method outperforms widely used RNP problem-solving methods.

From a visual point of view, the shape of an image is mainly determined by the edges. Conventional polynomial interpolation of image enlarging methods would produce blurred edges, while edge-directed interpolation based methods would cause distortion in the non-edge areas. A new method for image enlarging is presented. The image is enlarged in two steps. In the first step, a fitting surface is constructed to interpolate the image data. To remove the zigzagging artifact, for each pixel, a fitting patch is constructed using edge information as constraints. The combination of all the patches forms the fitting surface which has the shape suggested by image data. Each point on the fitting surface can be regarded as a sampling point taken from a unit square domain, which means that when the fitting surface is used to enlarge the image, each sampling domain of the enlarged pixels is also a unit square, causing the enlarged image to lose some details. To make the enlarged image keep the details as many as possible, the sampling domain of the enlarged pixels should be less than a unit square. Then, in the second step, using the points taken from the fitting surface, new pixels are computed using constrained optimization technique to form the enlarged image, and the size of the sampling domain of the enlarged pixels is inversely proportional to the size of the enlarged image. The image enlarged by the new method has a quadratic polynomial precision. Comparison results show that the new method produces resized image with better quality.

This paper presents a new algorithm for computing the maximal turning state of an arbitrary mesh. This is accomplished by intersecting the mesh with a plane at even intervals along the mesh, and computing the farthest point in that plane from an axis of rotation. The algorithm will run in O(|E|*n) time and θ(n) space, where E is the list of the edges on the mesh and n is the number of intervals used. Because of this, a very high degree of accuracy can be obtained with very little computational cost. The developed algorithm has been implemented and tested on some meshed models.

The article examines the approximation error of the chi-square distribution using a normal distribution, which is relevant when applying statistical tests to evaluate the quality of random number generators. The aim of the study is to determine under what conditions it is acceptable to replace chi-square statistics with their approximations (normal distributions) in order to simplify the calculation of p-values using the complementary error function. To achieve this goal, the research employed mathematical analysis methods, including analysis of the gamma distribution, which includes the chi-square distribution as a special case, and the application of the Berry — Esseen inequality for evaluating the accuracy of approximation. An analytical expression for the third absolute central moment of the distribution was obtained, allowing for an analytical estimation of the minimum length of a bit sequence necessary to achieve a specified level of approximation accuracy. The results showed that, in order to achieve the high level of accuracy required in cryptographic applications, there are significant practical limitations due to the required sample size. These limitations are related to computational complexity, memory requirements, and time costs. The issue of determining the optimal number of intervals in a test using chi-square statistics is considered to optimize the balance between the desired sensitivity with resistance to random fluctuations. The scientific novelty of this work lies in formalizing the conditions for using normal approximation to calculate p-values and developing recommendations for selecting the number of intervals and estimating the minimum sample size. The results obtained contribute to increasing the statistical significance and validity of statistical tests for verifying pseudo-random number generators. They also reduce the influence of heuristic considerations in determining the sample size, which increases the reliability of evaluating random number generator characteristics. In practical terms, this work aims to unify statistical analysis procedures in cryptography by formalizing conditions under which it is correct to replace the chi-square distribution with the normal distribution.

This paper addresses the problem of efficient coding of an important class of signals, namely piecewise polynomials. For this signal class, we develop a coding algorithm, which achieves oracle like rate-distortion (R-D) behavior in the high bit rate regime and with a reasonable computational complexity. For the 1-D case, our scheme is based on the binary tree segmentation of the signal and an optimal bit allocation strategy among the different signal segments. The scheme further encodes the similar neighbors jointly to achieve the right exponentially decaying R-D behavior (D(R) /spl sim/ c/sub 0/2/sup -c1R/). We have also shown that the computational cost of the scheme is of the order O(N log N). We then show that the scheme can be easily extended to the 2-D case, as the quadtree based coding scheme, with the similar R-D behavior and computational cost. Finally, we conclude with some numerical results.

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form [Formula: see text]. Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order [Formula: see text] in the [Formula: see text]-norm for the solution, when piecewise polynomials of degree at most [Formula: see text] are used. We use these results to prove that the UWDG solution is superconvergent with order [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] towards a special projection of the exact solution. We further prove that the [Formula: see text]-degree UWDG solution and its derivative are [Formula: see text] superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree [Formula: see text]. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.

In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the order algorithms produces polytopes for which the accuracy in Hausdorff metric is inversely proportional to the number of vertices (faces) in the degree of 2/(d-1). Numerical approximation algorithms can be adaptive (active) when the vertices or faces are constructed successively, depending on the information obtained in the process of approximation, and non-adaptive (passive) when parameters of algorithms are defined on the basis of a priory information available. Approximation algorithms differ in the use of operations applied to the approximated body. Most common are indicator, support and distance (Minkowski) functions calculations. Some optimal active algorithms for arbitrary bodies approximation are known using support or distance function calculation operation. Optimal passive algorithms for smooth bodies approximation are known using support function calculation operation and extremal curvature information. It is known that there are no optimal non-adaptive algorithms for arbitrary bodies approximation using support function calculation operation. We consider optimal non-adaptive algorithms for arbitrary bodies approximation using projection function calculation operation.

Quasi-interpolation by quadratic piecewise polynomials in three variables

The prize-collecting call control problem is to minimize the sum of the maximum load on the edges and the total penalty costs of the rejected calls. In this paper, we design a 2-approximation algorithm using linear programming rounding technique, and a 1.58-approximation algorithm using randomized rounding technique for this problem. We also present some optimal algorithms and approximation algorithms for some special graphs including rings, lines and stars.

The problem of recursive robust identification of linear discrete-time single-input single-output dynamic systems with correlated disturbances is considered. Problems related to the construction of optimal robust stochastic approximation algorithms in the min-max sense are demonstrated. Since the optimal solution cannot be achieved in practice, several robustified stochastic approximation algorithms are derived on the basis of a suitable non-linear transformation of normalized residuals, as well as step-by-step optimization with respect to the weighting matrix of the algorithm. The convergence of the developed algorithms is established theoretically using the ordinary differential equation approach. Monte Carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms. The results indicate the most suitable algorithms for applications in engineering practice.