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Spectral clustering has been one of the widely used methods for community detection in networks. However, large-scale networks bring computational challenges to the eigenvalue decomposition therein. In this paper, we study the spectral clustering using randomized sketching algorithms from a statistical perspective, where we typically assume the network data are generated from a stochastic block model that is not necessarily of full rank. To do this, we first use the recently developed sketching algorithms to obtain two randomized spectral clustering algorithms, namely, the random projection-based and the random sampling-based spectral clustering. Then we study the theoretical bounds of the resulting algorithms in terms of the approximation error for the population adjacency matrix, the misclassification error, and the estimation error for the link probability matrix. It turns out that, under mild conditions, the randomized spectral clustering algorithms lead to the same theoretical bounds as those of the original spectral clustering algorithm. We also extend the results to degree-corrected stochastic block models. Numerical experiments support our theoretical findings and show the efficiency of randomized methods. A new R package called Rclust is developed and made available to the public. for this article are available online. [ABSTRACT FROM AUTHOR]

In order to enhance the calculation efficiency of steady-state incompressible fluid flow, based on the existing IDEAL (i.e. the inner doubly iterative efficient algorithm for linked equations) algorithm, a novel segregated pressure-based algorithm (named as IDEALC) has been developed in this paper. The novel algorithm inherits the superiorities of the original IDEAL algorithm and at the same time improves its potential shortcomings. The specific promotions of the IDEALC algorithm consist of three aspects, including the first approximation of the SIMPLE algorithm, the approach to solving the momentum equation and the second approximation of the SIMPLE algorithm. Detailed comparisons between the IDEAL and the IDEALC algorithms are carried out by three numerical experiments on structured collocated grids. The numerical results demonstrate that the IDEALC algorithm can accurately predict the information of the flow field and has substantial advantages over the IDEAL algorithm in terms of the efficiency because of the consistent property of the former. The superiority of the IDEALC algorithm over the IDEAL algorithm is especially remarkable when the time step multiplier gets a moderate value. [ABSTRACT FROM AUTHOR]

In this paper, the application of randomized projection functional algorithms for numerical approximation of solutions of Fredholm equations of the second kind is discussed. Special attention is paid to the problems of numerical stability of the used orthonormal functional bases. The numerical instability of the randomized projection functional algorithm is noticed for the simple test one-dimensional equation and for the Hermite orthonormal basis. [ABSTRACT FROM AUTHOR]

A rough volatility model contains a stochastic Volterra integral with a weakly singular kernel. The classical Euler-Maruyama algorithm is not very efficient for simulating this kind of model because one needs to keep records of all the past path-values and thus the computational complexity is too large. This paper develops a fast two-step iteration algorithm using an approximation of the weakly singular kernel with a sum of exponential functions. Compared to the Euler-Maruyama algorithm, the complexity of the fast algorithm is reduced from O (N 2) to O (N log ⁡ N) or O (N log 2 ⁡ N) for simulating one path, where N is the number of time steps. Further, the fast algorithm is developed to simulate rough Heston models with (or without) regime switching, and multi-factor approximation algorithms are also studied and compared. The convergence rates of the Euler-Maruyama algorithm and the fast algorithm are proved. A number of numerical examples are carried out to confirm the high efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this article, we aim to solve Bayesian Risk Optimization (BRO), which is a recently proposed framework that formulates simulation optimization under input uncertainty. In order to efficiently solve the BRO problem, we derive nested stochastic gradient estimators and propose corresponding stochastic approximation algorithms. We show that our gradient estimators are asymptotically unbiased and consistent, and that the algorithms converge asymptotically. We demonstrate the empirical performance of the algorithms on a two-sided market model. Our estimators are of independent interest in extending the literature of stochastic gradient estimation to the case of nested risk measures. [ABSTRACT FROM AUTHOR]

The problem of minimizing the makespan on parallel identical machines is considered in the presence of additional resources, provided that some jobs at any time of their processing require one unit of a particular resource. We establish a lower bound on the worst-case performance of any group technology algorithm, which schedules the composite jobs formed of the original jobs that require the same resource. A simple group technology algorithm is given such that in the worst case no group technology algorithm performs better. An algorithm for the two-machine case is presented which guarantees a tight worst-case performance ratio of 6/5. [ABSTRACT FROM AUTHOR]

Integration of production and distribution operations at the scheduling level is one of the most important issues concerned in recent years in supply chain management. We investigate the scheduling problem with job delivery coordination. Concretely, we are given a set of jobs with non-identical processing times and non-identical sizes to be first processed by two parallel-identical machines and then delivered to a common customer by a single vehicle, which has a limited capacity that can carry up to a fixed total size of jobs in one batch. The objective is to minimize the makespan, ie, the time when all the jobs have reached the customer and the delivery vehicle has returned to the machine. The problem does not admit a polynomial time approximation algorithm with worst-case ratio smaller than 3 2 unless P = NP. By introducing some novel ideas and new methods, including algorithms solving multiple knapsack problem as subprocedures and the rediscovery of the profound nature of the bin-packing algorithm, we propose a polynomial time approximation algorithm with worst-case ratio arbitrary close to 3 2 . The algorithm, acquired after long-term research containing many earlier related works, is the first algorithm for this kind of problems and its theoretical performance is nearly best. [ABSTRACT FROM AUTHOR]

Benson's outer approximation algorithm and its variants are the most frequently used methods for solving linear multiobjective optimization problems. These algorithms have two intertwined parts: single-objective linear optimization on one hand, and a combinatorial part closely related to vertex enumeration on the other. Their separation provides a deeper insight into Benson's algorithm, and points toward a dual approach. Two skeletal algorithms are defined which focus on the combinatorial part. Using different single-objective optimization problems yields different algorithms, such as a sequential convex hull algorithm, another version of Benson's algorithm with the theoretically best possible iteration count, and the dual algorithm of Ehrgott et al. [A dual variant of Benson's 'outer approximation algorithm' for multiple objective linear programming. J Glob Optim. 2012;52:757–778]. The implemented version is well suited to handle highly degenerate problems where there are many linear dependencies among the constraints. On problems with 10 or more objectives, it shows a significant increase in efficiency compared to Bensolve – due to the reduced number of iterations and the improved combinatorial handling. [ABSTRACT FROM AUTHOR]

In an application of map labelling to air-traffic control, labels should be placed with as few overlaps as possible since labels include important information about airplanes. Motivated by this application, de Berg and Gerrits (Comput. Geom. 2012) proposed a problem of maximizing the number of free labels (i.e. labels not intersecting with any other label) and developed approximation algorithms for their problem under various label-placement models. In this paper, we propose an alternative problem of minimizing a degree of overlap at a point. Specifically, the objective of this problem is to minimize the maximum of λ (p) over p ∈ R 2 , where λ (p) is defined as the sum of weights of labels that overlap with a point p. We develop a 4-approximation algorithm by LP-rounding under the 4-position model. We also investigate the case when labels are rectangles with bounded height/length ratios. [ABSTRACT FROM AUTHOR]

This work deals with the robust maximum weighted independent set problem, i.e. finding a subset of graph vertices that are not adjacent to each other and whose sum of weights is as large as possible. Uncertainty in problem formulation is restricted to vertex weights and expressed explicitly by a finite set of scenarios. Three criteria of robustness are considered: absolute robustness (max-min), robust deviation (min-max regret), and relative robustness (relative min-max regret). Since the conventional maximum weighted independent set problem is already NP-hard, finding the exact solution of its robust counterpart should obviously have a prohibitive computational complexity. Therefore, we propose an approximate algorithm for solving the considered robust problem, which is based on evolutionary computing and on various crossover and mutation operators. The algorithm is experimentally evaluated on appropriate problem instances. It is shown that satisfactory solutions can be obtained for any of the three robustness criteria in reasonable time. [ABSTRACT FROM AUTHOR]

A randomised algorithm is proposed for computing globally optimal static-output-feedbacks for large-scale systems. The algorithm is based on the Ray-Shooting Method and involves some heuristics to accelerate the search. We also improve the basic Ray-Shooting Algorithm and make the search in the controller parameter-space (which generally, is much more smaller than the certificate parameter-space), thus enabling its efficient use for large-scale systems. [ABSTRACT FROM AUTHOR]

We present a method to solve constrained convex stochastic optimization problems when the objective is a finite sum of convex functions. Our method is based on Incremental Stochastic Subgradient Algorithms and String-Averaging techniques, with an assumption that the subgradient directions are affected by random errors in each iteration. Our analysis allows the method to perform approximate projections onto the feasible set in each iteration. We provide convergence results for the case where a diminishing step-size rule is used. We test our method in a large set of random instances of a stochastic convex programming problem and we compare its performance with the robust mirror descent stochastic approximation algorithm proposed in Nemirovski et al. (Robust stochastic approximation approach to stochastic programming, SIAM J Optim 19 (2009), pp. 15741609). [ABSTRACT FROM AUTHOR]

The classical pursuit problem considers the path traced by a point in space as it charges directly toward a moving target. But what if the target has more lofty goals than mere escape? To what degree can it control the path taken by its pursuer? We prove that the pursuer can be led to any point in without being allowed to close more than an arbitrarily small distance en route. [ABSTRACT FROM AUTHOR]

In allocation problems with indivisible goods, money compensation is used to distribute worth in a fair way. Coalitional games provide a formal mathematical framework to model such problems, and the Shapley value is a solution concept widely used to realise a fair distribution. To overcome its intractability, we describe how to simplify allocation problems and we propose algorithms for computing lower bounds and upper bounds of the Shapley value that can be combined with approximation algorithms. The proposed techniques have been implemented and tested on a real-world application of allocation problems, namely, the Italian research assessment program known as VQR. [ABSTRACT FROM AUTHOR]

This article studies computation problem in the context of estimating parameters of linear mixed model for massive data. Our algorithms combine the factored spectrally transformed linear mixed model method with a sequential singular value decomposition calculation algorithm. This combination solves the operation limitation of the method and also makes this algorithm feasible to big dataset, especially when the data has a tall and thin design matrix. Our simulation studies show that our algorithms make the calculation of linear mixed model feasible for massive data on ordinary desktop and have same estimating accuracy with the method based on the whole data. [ABSTRACT FROM AUTHOR]

One of the most important issues in the design of distributed systems is process scheduling, which maps applications to resources in an attempt to reduce the application execution time or maximise resource utilisation. The complexity involved in finding good scheduling solutions has motivated the design of several heuristics and approximation algorithms. Although heuristics aim at finding good solutions within acceptable time constraints, they do not guarantee solution quality. On the other hand, approximation-based algorithms provide optimal solution bounds, but they are usually designed to address simple scenarios. In an attempt to combine the best of both algorithmic approaches, this paper proposes the min-heap-based scheduling algorithm (MHSA), which finds solutions for homogeneous and heterogeneous distributed environments within acceptable time constraints and optimal solution bounds. MHSA considers applications that can be bag-of-tasks or which communicate among each other. Such solutions minimise the application execution time and maximise resource utilisation; therefore, MHSA is a system-oriented scheduling algorithm. Results confirm that MHSA provides better solutions than schedulers such as Random, Round-Robin, Workload-Queue and List-Scheduling. In addition, MHSA provides an-approximation ratio for reaching optimal solution bounds. [ABSTRACT FROM AUTHOR]

Common deficiencies found in previous studies that modelled quay crane scheduling with non-crossing constraints (QCSNC) in port container terminals are identified. Remedies for the problems are proposed to prevent the occurrence of unrealistic optimal solutions for QCSNC. Additionally, two approximation schemes for QCSNC are studied. The first one is called the best partition (BP) method, which can be solved by dynamic programming (DP) with time complexity of O(mn2). A more efficient algorithm is designed to improve the original DP by adopting binary search. Next, based on BP and an exact algorithm for QCSNC (with pseudo-polynomial computational time), the second approximation scheme, called the enhanced best partition (EBP) method, is proposed to achieve a better estimation of the optimal makespan for QCSNC. The results of numerical experiments show that EBP is an effective approximation scheme for the problem of QCSNC. [ABSTRACT FROM AUTHOR]

This paper develops an iterative algorithm for the solution to a variable-coefficient semilinear heat equation with nonlocal boundary conditions in the reproducing space. It is proved that the approximate sequence un(x, t) converges to the exact solution u(x, t). Moreover, the partial derivatives of un(x, t) are also convergent to the partial derivatives of u(x, t). And the approximate sequence un(x, t) is the best approximation under a complete normal orthogonal system. [ABSTRACT FROM AUTHOR]

To improve reachability of a snake-like robot depending on the problem, the links of the robot need to be resizable. In such a case folding links of the robot help to better plan obstacle avoidance. Optimum folding of the robot is the aim of our paper. We introduce a practical idea to construct reconfigurable and resizable snake robots which can be folded and an approximation algorithm to find near-optimum folding for the robot. Since an open chain is an abstract model of a snake-like robot, folding algorithms for the proposed robot are given in terms of an open chain. Ruler folding is a well-known NP-Complete problem. It considers folding of an n-link open chain linkage to the minimum length. The best previously known approximation algorithm for this problem has been developed by Hopcroft et al. They achieved the upper bound of 2m1 for the length of the folded chain in all cases, where m1 is the length of the longest link of the given chain. Already there are no any algorithms for open chain folding which guarante that the folded length of the open chain is less than 2m1. In this paper, we introduce an approximation algorithm which runs in O (n log n) using O (n) space. We introduce a function for the upper bound of the folded chain which depends to the lengths of all links in the given chain. Our experimental results show that for more than 95% of the problem instances we can achieve the same results in O (n) time. Using our folding algorithm, we can design the length of each link in an open chain to get x · m1 folded length where 1 < x < 2 is given and m1 is the length of the longest link of the chain. We introduce how to design a snake-like robot for which it can be folded in the given interval. [ABSTRACT FROM AUTHOR]

This paper considers the problem of scheduling operations in bufferless robotic cells that produce identical parts. Finding a cyclic solution that minimizes the long-run average time to produce a part has long been a fundamental open problem. Most research has been focused on finding an optimal 1-unit cyclic solution. However, it is known that an optimal multi-unit cyclic solution can be better than an optimal 1-unit cyclic solution for cells with four or more machines. For additive travel-time cells, we present a linear-time algorithm that produces a cyclic solution whose per unit cycle time is within a factor of 10/7 of the optimum. This result improves the current best 3/2-approximation guarantee. [ABSTRACT FROM AUTHOR]

Given a translating and rotating rod robot in a plane in the presence of polygonal obstacles with the initial and final placements of the rod known, the d 1 -optimal motion planning problem is defined as finding a collision-free motion of the rod such that the orbit length of a fixed but arbitrary point F on the rod is minimized. In this paper we study a special case of this problem in which the rod can translate freely, but can only rotate by some pre-specified given angles around F . We first characterize the d 1 -optimal motion of the robot under the given conditions and then present a (1 + ε)–approximation algorithm for finding the optimal path. The running time of the algorithm is bounded by a polynomial in terms of some parameters related to the problem input. [ABSTRACT FROM AUTHOR]