The prize-collecting call control problem

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

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.

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.

Sums of squares based approximation algorithms for MAX-SAT

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

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.

Load-balanced routing in SONET rings has attracted much attention recently. Most prior works modeled the SONET rings as undirected rings and the traffic as undirected chords. While this model fits well to the traditional telephony applications, it is inefficient for the explosive Internet traffic and multimedia data communications, which exhibit an unidirectional and asymmetric nature. For these applications, it is proper to model the SONET rings as a pair of counter rotated rings and the traffic as directed chords. In this paper, we first explore general flow properties in counter rotated rings and then introduce flow rounding and unsplitting techniques. Afterward, an optimal integral routing algorithm is provided. Finally, we show the NP-completeness of optimal unsplit routing and present several polynomial-time approximation algorithms. © 2000 John Wiley & Sons, Inc.

In the universal facility location problem, we are given a set of clients and facilities. Our goal is to find an assignment such that the total connection and facility cost is minimized. The connection cost is proportional to the distance between each client and its assigned facility, whereas the facility cost is a nondecreasing function with respect to the total number of clients assigned to the facility. The universal facility location problem generalizes several classical facility location problems, including the uncapacitated facility location problem and the capacitated facility location problem (both hard and soft capacities). This work considers the universal facility location problem with linear penalties, where each client can be rejected for service with a penalty. The objective is to minimize the total connection, facility and penalty cost. We present a $$(5.83+\epsilon )$$ -approximation local search algorithm for this problem.

Modeling Pooled Purchasing Strategy in Purchasing Consortium to Optimize Total Purchasing Cost

Inner approximation algorithms for optimization over the weakly efficient set

In this paper, we propose a general extension of constraint propagation for constraint optimization based on cooperative computation. It is similar both in principle and operations to constraint propagation. In principle, it eliminates variable values by checking the feasibility that they can be in any global optimum. In operations, it is based on parallel, local iterative computations. The proposed algorithm returns both a solution and a global lower bound at each iteration. As an approximation algorithm for optimization, it significantly outperform classical optimization methods, such as simulated annealing and local search with multi-restarts in practice.

In recent years, there have been several attempts to use machine learning techniques to improve the performance of exact and approximate optimization algorithms. Along this line of research, the present paper shows how supervised and unsupervised techniques can be used to improve the quality of the solutions generated by a heuristic for the Time-Dependent Travelling Salesman Problem with no increased computing time. This can be useful in a real-time setting where a speed update (or the arrival of a new customer request) may lead to the reoptimization of the planned route. The main contribution of this work is to show how to reuse the information gained in those settings in which instances with similar features have to be solved over and over again, as it is customary in distribution management. We use a method based on the nearest neighbor procedure (supervised learning) and the K-means algorithm with the Euclidean distance (unsupervised learning). In order to show the effectiveness of this approach, the computational experiments have been carried out for the dataset generated based on the real travel time functions of two European cities: Paris and London. The overall average improvement of our heuristic over the classical nearest neighbor procedure is about 5% for London, and about 4% for Paris.

The flying sidekick traveling salesman problem with stochastic travel time: A reinforcement learning approach

This article presents PackageCargo. A modular open-source application developed using the Unity game engine to calculate, visualize, and save efficient packing patterns to instances of the Container Loading Problem (CLP). The packing patterns are obtained through approximate optimization algorithms (metaheuristics). Additionally, the proposed tool allows us to estimate cargo stability metrics through the implementation of mathematical models and verify the results of said models using a simulation environment built with the PhysX library. The goal of this application was to create a usable decision support system suitable for industrial purposes as well as a platform for academic research. It is offering a modifiable framework that can adapt to the necessities of its users, saving them software development time while continuing to extend PackageCargo through community contributions. The resulting application was compared with commercial software solutions. Furthermore, each module was tested using the most successful approaches found in literature as benchmarks. The packing module was compared against the top-performing algorithm published to date, obtaining similar results in similar computational times. The simulation module for cargo stability was benchmarked against high-performance simulation software, validating its accuracy and performance. Accordingly, PackageCargo was found to have a competitive feature set useful in both academic and commercial settings. As future work, it is proposed to combine the different modules to solve more sophisticated variants of the CLP, like the container loading problem constrained to weight distribution profiles.