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Router nodes placement (RNP) is an important issue in the design and implementation of wireless mesh networks (WMN). This is known as an P-hard problem, which cannot be solved using conventional algorithms. Consequently, approximate optimization strategies are commonly used to solve this problem. With heavy node density and wide-area WMNs, solving the RNP problem using approximation algorithms often faces many difficulties, therefore, a more effective solution is necessary. This motivated us to conduct this work. We propose a new method for solving the RNP problem using reinforcement learning (RL). The RNP problem is modeled as an RL model with environment, agent, action, and reward are equivalent to the network system, routers, coordinate adjustment, and connectivity of the RNP problem, respectively. To the best of our knowledge, this is the first study that applies RL to solve the RNP problem. The experimental results showed that the proposed method increased the network connectivity by up to 22.73% compared to the most recent methods. [ABSTRACT FROM AUTHOR]

Quantum approximate optimization algorithms are hybrid quantum-classical variational algorithms designed to approximately solve combinatorial optimization problems such as the MAX-CUT problem. In spite of its potential for near-term quantum applications, it has been known that quantum approximate optimization algorithms have limitations for certain instances to solve the MAX-CUT problem, at any constant level p. Recently, the recursive quantum approximate optimization algorithm, which is a non-local version of quantum approximate optimization algorithm, has been proposed to overcome these limitations. However, it has been shown by mostly numerical evidences that the recursive quantum approximate optimization algorithm outperforms the original quantum approximate optimization algorithm for specific instances. In this paper, we analytically prove that the recursive quantum approximate optimization algorithm is more competitive than the original one to solve the MAX-CUT problem for complete graphs with respect to the approximation ratio. [ABSTRACT FROM AUTHOR]

Solving combinatorial problems on complex networks represents a primary issue which, on a large scale, requires the use of heuristics and approximate algorithms. Recently, neural methods have been proposed in this context to find feasible solutions for relevant computational problems over graphs. However, such methods have some drawbacks: (1) they use the same neural architecture for different combinatorial problems without introducing customizations that reflects the specificity of each problem; (2) they only use a nodes local information to compute the solution; (3) they do not take advantage of common heuristics or exact algorithms. Following this interest, in this research we address these three main points by designing a customized attention-based mechanism that uses both local and global information from the adjacency matrix to find approximate solutions for the Minimum Vertex Cover Problem. We evaluate our proposal with respect to a fast two-factor approximation algorithm and a widely adopted state-of-the-art heuristic both on synthetically generated instances and on benchmark graphs with different scales. Experimental results demonstrate that, on the one hand, the proposed methodology is able to outperform both the two-factor approximation algorithm and the heuristic on the test datasets, scaling even better than the heuristic with harder instances and, on the other hand, is able to provide a representation of the nodes which reflects the combinatorial structure of the problem. [ABSTRACT FROM AUTHOR]

Bayesian optimization algorithms are widely used for solving problems with a high computational complexity in terms of objective function evaluation. The efficiency of Bayesian optimization is strongly dependent on the quality of the surrogate models of an objective function, which are built and refined at each iteration. The quality of surrogate models, and hence the performance of an optimization algorithm, can be greatly improved by selecting the appropriate hyperparameter values of the approximation algorithm. The common approach to finding good hyperparameter values for each iteration of Bayesian optimization is to build surrogate models with different hyperparameter values and choose the best one based on some estimation of the approximation error, for example, a cross-validation score. Building multiple surrogate models for each iteration of Bayesian optimization is computationally demanding and significantly increases the time required to solve an optimization problem. This paper suggests a new approach, called exploratory landscape validation, to find good hyperparameter values with less computational effort. Exploratory landscape validation metrics can be used to predict the best hyperparameter values, which can improve both the quality of the solutions found by Bayesian optimization and the time needed to solve problems. [ABSTRACT FROM AUTHOR]

We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P = NP. Therefore, to solve the DDF problem effectively, we propose two convex integer-programming formulations and investigate their corresponding complementary and Lagrangian-dual problems. Leveraging the concavity of the objective functions in the two proposed convex integer-programming formulations, we design an exact algorithm, aimed at solving the DDF problem to optimality. We further derive a family of submodular valid inequalities and optimality cuts, which can significantly enhance the algorithm performance. We also develop scalable randomized-sampling and local-search algorithms with provable performance guarantees. Finally, we test our algorithms using real-world data on the new phasor-measurement-units placement problem for modern power grids, considering the existing conventional sensors. Our numerical study demonstrates the efficiency of our exact algorithm and the scalability and high-quality outputs of our approximation algorithms. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: Y. Li and W. Xie were supported in part by Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046414] and Division of Computing and Communication Foundations [Grant 2246417]. J. Lee was supported in part by Air Force Office of Scientific Research [Grants FA9550-19-1-0175 and FA9550-22-1-0172]. M. Fampa was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 305444/2019-0 and 434683/2018-3]. F. Qiu and R. Yao were supported in part by the U.S. Department of Energy Advanced Grid Modeling Program under [Grant DE-OE0000875]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.0235. [ABSTRACT FROM AUTHOR]

Let P be a set of points in Rd, where each point p ∈ P has an associated transmission range, denoted (p). The range assignment induces a directed communication graph Gp (P) on P, which contains an edge (p, q) iff | pq| (p). In the broadcast range-assignment problem, the goal is to assign the ranges such that (P) contains an arborescence rooted at a designated root node and the costΣp∈ P(p)² of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution-the number of ranges that are modified when a point is inserted into or deleted from P-and its approximation ratio. To this end we study k-stable algorithms, which are algorithms that modify the range of at most k points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm \mathrm{A}\mathrm{L}\mathrm{G} that, for any given fixed parameter ε >0, is kε-stable and that maintains a solution with approximation ratio 1 + \ε, where the stability parameter k(\varepsilon) only depends on ε and not on the size of P. We study such trade-offs in three settings. (1) For the problem in R¹, we present a SAS with kε = O(1/ε. Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have kε =ε (1ε). We also present 1-, 2-, and 3-stable algorithms with constant approximation ratio. (2) For the problem in S1 (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in S1, we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in R¹. (3) For the problem in R2, we also prove that no SAS exists, and we present a O(1)-stable O(1)-approximation algorithm. Most results γ generalize to the setting where, for any given constant α > 1, the range-assignment cost is pΣ (p)∈α. [ABSTRACT FROM AUTHOR]

Discrete random variables are essential ingredients in various artificial intelligence problems. These include the estimation of the probability of missing the deadline in a series-parallel schedule and the assignment of suppliers to tasks in a project in a manner that maximizes the probability of meeting the overall project deadline. The solving of such problems involves repetitive operations, such as summation, over random variables. However, these computations are NP-hard. Therefore, we explore techniques and methods for approximating random variables with a given support size and minimal Kolmogorov distance. We examine both the general problem of approximating a random variable and a one-sided version in which over-approximation is allowed but not under-approximation. We propose several algorithms and evaluate their performance through computational complexity analysis and empirical evaluation. All the presented algorithms are optimal in the sense that given an input random variable and a requested support size, they return a new approximated random variable with the requested support size and minimal Kolmogorov distance from the input random variable. Our approximation algorithms offer useful estimations of probabilities in situations where exact computations are not feasible due to NP-hardness complexity. [ABSTRACT FROM AUTHOR]

Surrogate-assisted evolutionary algorithms (SAEAs) are a promising approach for solving expensive multiobjective optimization problems, but they often cannot address high-dimensional problems. Although one common approach to overcoming this challenge is to construct reliable surrogates, their accuracy inevitably deteriorates in a high-dimensional search space. Thus, this paper presents a novel SAEA based on scalarization function approximation, which is designed to strengthen its robustness against this deterioration. The proposed algorithm constructs an approximation model for each scalarization function defined in a decomposition-based framework. Each decomposed problem is then solved using multiple independent models trained for its neighbor problems. The intent is to decrease the risk of search performance degradations caused by unreliable approximations and retain the redundancy of the surrogate-assisted search to hedge the risk of over-fitting. Furthermore, each approximation model is adapted to a promising region of its corresponding decomposed problem to reduce the complexity of model fitting given a limited number of training samples. Experimental results show that the proposed algorithm is competitive with state-of-the-art SAEAs adapted for high-dimensional problems. [ABSTRACT FROM AUTHOR]

The fair k -median problem is one of the important clustering problems. The current best approximation ratio is 4.675 for this problem with 1-fairness violation, which was proposed by Bercea et al. [APPROX-RANDOM'2019]. To our best knowledge, there is no available approximation algorithm for this problem without any fairness violation in doubling metrics. In this paper, we consider the fair k -median problem in doubling metrics and general metrics. We provide the first QPTAS for the fair k -median problem based on the hierarchical decomposition and dynamic programming process in doubling metrics. For applying a dynamic programming process to solve this problem, the distances from portals to facilities cannot be directly enumerated since each client may not be assigned to its closest open facility. To overcome the difficulties caused by the fairness constraints, we construct an auxiliary graph and use minimum weighted perfect matching to get the cost between the portals of each block and the ones in its children. In order to satisfy the fairness constraints, we bound the fairness constraints of each open facility in the leaves of the split-tree based on the relation between the subproblem and the subproblems of its children. To obtain the assignment of the given instance and remove the fairness violation, we construct a new b -value min-cost max-flow model based on the set of open facilities. For the fair k -median problem in general metrics, we present a polynomial-time approximation algorithm with ratio O (log ⁡ k). Our approximation algorithm for the fair k -median problem in doubling metrics is the first result for the corresponding problem without any fairness violation in doubling metrics. [ABSTRACT FROM AUTHOR]