Greedy Approximation Research Articles

Greedy approximation by arbitrary set

Определяются различные алгоритмы жадных приближений элементами произвольного множества $M$ в банаховом пространстве. Исследуется сходимость этих алгоритмов в гильбертовом пространстве при различных геометрических условиях на $M$. Как следствие получаются достаточные условия плотности аддитивной полугруппы, порожденной множеством $M$. Библиография: 12 наименований.

Improved adaptive forward-backward matching pursuit algorithm to compressed sensing signal recovery

As a novel two-stage greedy approximation algorithm, Forward-Backward Pursuit (FBP) algorithm attracts wide attention because of its high reconstruction accuracy and no need for sparsity as a priori information. However, the FBP algorithm has to spend much more time to get a higher accuracy. In view of this, an Improved Adaptive Forward-Backward Matching Pursuit (IAFBP) algorithm is proposed in this paper. In the forward stage, the IAFBP algorithm uses an adaptive threshold to select the appropriate number of atoms into support set, so that the number of selected atoms is more random. In the backward stage, the projection coefficient of the atoms is taken as the basis of rejection, and the deletion threshold is introduced to reject the atoms adaptively, so that more right atoms are retained in each iteration and the reconstruction speed can be accelerated. At the same time, it overcomes the excessive backtracking phenomenon existing in the adaptive process and improves the accuracy of the algorithm. The simulation results of one-dimensional sparse signals and two-dimensional images show that the IAFBP algorithm has more advantages than the FBP algorithm in reconstruction performance and computational time.

Coverage Centrality Maximization in Undirected Networks

Centrality metrics are among the main tools in social network analysis. Being central for a user of a network leads to several benefits to the user: central users are highly influential and play key roles within the network. Therefore, the optimization problem of increasing the centrality of a network user recently received considerable attention. Given a network and a target user v, the centrality maximization problem consists in creating k new links incident to v in such a way that the centrality of v is maximized, according to some centrality metric. Most of the algorithms proposed in the literature are based on showing that a given centrality metric is monotone and submodular with respect to link addition. However, this property does not hold for several shortest-path based centrality metrics if the links are undirected.In this paper we study the centrality maximization problem in undirected networks for one of the most important shortestpath based centrality measures, the coverage centrality. We provide several hardness and approximation results. We first show that the problem cannot be approximated within a factor greater than 1 − 1/e, unless P = NP, and, under the stronger gap-ETH hypothesis, the problem cannot be approximated within a factor better than 1/no(1), where n is the number of users. We then propose two greedy approximation algorithms, and show that, by suitably combining them, we√ can guarantee an approximation factor of Ω(1/ n). We experimentally compare the solutions provided by our approximation algorithm with optimal solutions computed by means of an exact IP formulation. We show that our algorithm produces solutions that are very close to the optimum.

Big Data Acquisition Under Failures in FiWi Enhanced Smart Grid

With the increasing of monitoring devices and advanced measurement infrastructures, smart grids (SGs) collect large amounts of data every moment, which gives rise to the SG big data. In order to fulfill diverse communication requirements of various energy-related data in SG, it is obviously impractical to rely on a single communication technology, and a hybrid communication architecture of low-latency fiber optic and cost-effective wireless technologies could be a promising solution. Note that, low-latency data acquisition under failures is of particular importance to SG reliability, considering that SGs are vulnerable to various failures. Toward this end, we provide in this paper a hybrid SG communication architecture integrating fiber optic and WiFi-based mesh networks, i.e., fiber-wireless (FiWi) enhanced SG, and study the problem of data acquisition under failures in FiWi enhanced SG. The problem is first formulated as a constrained optimization problem, and then three algorithms are proposed as our solutions, i.e., an optimal enumeration routing algorithm (OERA), a greedy approximation routing algorithm (GARA), and a heuristic greedy routing algorithm (HGRA). Numerical results reveal that both GARA and HGRA can achieve near-optimal solutions to the problem of data acquisition under failures, and have higher computational efficiency compared to our benchmark, i.e., OERA.

Spreading social influence with both positive and negative opinions in online networks

Spreading social influence with both positive and negative opinions in online networks Social networks are important media for spreading information, ideas, and influence among individuals. Most existing research focuses on understanding the characteristics of social networks, investigating how information is spread through the word-of-mouth effect of social networks, or exploring social influences among individuals and groups. However, most studies ignore negative influences among individuals and groups. Motivated by the goal of alleviating social problems, such as drinking, smoking, and gambling, and influence-spreading problems, such as promoting new products, we consider positive and negative influences, and propose a new optimization problem called the Minimum-sized Positive Influential Node Set (MPINS) selection problem to identify the minimum set of influential nodes such that every node in the network can be positively influenced by these selected nodes with no less than a threshold of θ. Our contributions are threefold. First, we prove that, under the independent cascade model considering positive and negative influences, MPINS is APX-hard. Subsequently, we present a greedy approximation algorithm to address the MPINS selection problem. Finally, to validate the proposed greedy algorithm, we conduct extensive simulations and experiments on random graphs and seven different realworld data sets that represent small-, medium-, and large-scale networks.

A budgeted maximum multiple coverage model for cybersecurity planning and management

This article studies how to identify strategies for mitigating cyber-infrastructure vulnerabilities. We propose an optimization framework that prioritizes the investment in security mitigations to maximize the coverage of vulnerabilities. We use multiple coverage to reflect the implementation of a layered defense, and we consider the possibility of coverage failure to address the uncertainty in the effectiveness of some mitigations. Budgeted Maximum Multiple Coverage (BMMC) problems are formulated, and we demonstrate that the problems are submodular maximization problems subject to a knapsack constraint. Other variants of the problem are formulated given different possible requirements for selecting mitigations, including unit cost cardinality constraints and group cardinality constraints. We design greedy approximation algorithms for identifying near-optimal solutions to the models. We demonstrate an optimal (1–1/e)-approximation ratio for BMMC and a variation of BMMC that considers the possibility of coverage failure, and a 1/2-approximation ratio for a variation of BMMC that uses a cardinality constraint and group cardinality constraints. The computational study suggests that our models yield robust solutions that use a layered defense and provide an effective mechanism to hedge against the risk of possible coverage failure. We also find that the approximation algorithms efficiently identify near-optimal solutions, and that a Benders branch-and-cut algorithm we propose can find provably optimal solutions to the vast majority of our test instances within an hour for the variations of the proposed models that consider coverage failures.

Sparsity in max-plus algebra and systems

We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the $\ell_{1}$ residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations.

Quality-Assured Synchronized Task Assignment in Crowdsourcing

With the rapid development of crowdsourcing platforms that aggregate the intelligence of Internet workers, crowdsourcing has been widely utilized to address problems that require human cognitive abilities. Considering great dynamics of worker arrival and departure, it is of vital importance to design a task assignment scheme to adaptively select the most beneficial tasks for the available workers. In this paper, in order to make the most efficient utilization of the worker labor and balance the accuracy of answers and the overall latency, we a) develop a parameter estimation model that assists in estimating worker expertise, question easiness, and answer confidence; b) propose a quality-assured synchronized task assignment scheme that executes in batches and maximizes the number of potentially completed questions (MCQ) within each batch. We prove that MCQ problem is NP-hard and present two greedy approximation solutions to address the problem. The effectiveness and efficiency of the approximation solutions are further evaluated through extensive experiments on synthetic and real datasets. The experimental results show that the accuracy and the overall latency of the MCQ approaches outperform the existing online task assignment algorithms in the synchronized task assignment scenario.

Efficient computation of Coulomb and exchange integrals for multi-million atom nanostructures

Atomistic modeling of nanostructures such as quantum dots or nanowires often involves numbers of atoms reaching and even exceeding well beyond 1 million. Such a large quantity of atoms presents a very complex computational challenge especially at a stage of many-body calculation where numerous Coulomb matrix elements need to be calculated. Here we present a practical solution to this problem by performing calculations in the momentum space and utilizing fast Fourier transform combined with a memory-efficient way to compute the convolution that overcomes the problem of spurious interactions with quasi-charges from other super-cells. Finally, calculation of multiple integrals is optimized by reducing the problem to finding a minimum vertex cover of a graph. All these algorithms are implemented and presented here in a self-contained and highly parallelized computer program named Coulombo. Coulombo demonstrates quasi-linear scaling of computational time of Coulomb matrix elements with respect to the number of points in the computational box and, at the same time, significantly reduced memory demand. The proposed solution can have potential applications not only in the realm of nano-physics, but could be applied to other mesoscopic simulations or large-scale quantum chemistry problems. Program summaryProgram Title: CoulomboProgram Files doi:http://dx.doi.org/10.17632/98bhm5zbrd.1Licensing provisions: CC BY 4.0Programming language: C++Nature of problem: Computing the Coulomb matrix elements (Coulomb and exchange integrals), while being a demanding computational task, is a necessary step in a range of quantum mechanical calculations. For example, in the field of nanostructures, such as quantum dots and nanowires, this stage of calculation even after numerous approximations is at least an O(N2) operation (a summation over all pairs of N atoms or grid points in the analyzed system). Moreover, calculating the full Coulomb matrix usually requires computation of thousands of such elements, thus presenting a formidable computational challenge.Solution method: We provide a ready-to-use implementation for calculating Coulomb matrix elements for a given set of input wavefunctions. This implementation is based on the approach introduced in [1], by using a fast Fourier transform to compute the convolution. In this work we further significantly improved the method by eliminating the need to extend the computational domain with padding, thus reducing the memory consumption by a factor of 8. The optimal computational plan for each run is prepared by calculating a minimum vertex cover on a graph representing the subset of requested Coulomb matrix elements.Additional comments including restrictions and unusual features: The implementation is fully parallelized in a distributed-memory model, using MPI and parallel routines from FFTW [2]. A minimum vertex cover is computed by our greedy approximation algorithm, which we found to perform significantly better than the standard text-book heuristic.

The maximum trajectory coverage query in spatial databases

With the widespread use of GPS-enabled mobile devices, an unprecedented amount of trajectory data has become available from various sources such as Bikely, GPS-wayPoints, and Uber. The rise of smart transportation services and recent break-throughs in autonomous vehicles increase our reliance on trajectory data in a wide variety of applications. Supporting these services in emerging platforms requires more efficient query processing in trajectory databases. In this paper, we propose two new coverage queries for trajectory databases: (i) k Best Facility Trajectory Search ( k BFT); and (ii) k Best Coverage Facility Trajectory Search ( k BCovFT). We propose a novel index structure, the Trajectory Quadtree (TQ-tree) that utilizes a quadtree to hierarchically organize trajectories into different nodes, and then applies a z-ordering to further organize the trajectories by spatial locality inside each node. This structure is highly effective in pruning the trajectory search space, which is of independent interest. By exploiting the TQ-tree, we develop a divide-and-conquer approach to efficiently process a k BFT query. To solve the k BCovFT, which is a non-submodular NP-hard problem, we propose a greedy approximation. We evaluate our algorithms through an extensive experimental study on several real datasets, and demonstrate that our algorithms outperform baselines by two to three orders of magnitude.

Improving the Betweenness Centrality of a Node by Adding Links

Betweenness is a well-known centrality measure that ranks the nodes according to their participation in the shortest paths of a network. In several scenarios, having a high betweenness can have a positive impact on the node itself. Hence, in this article, we consider the problem of determining how much a vertex can increase its centrality by creating a limited amount of new edges incident to it. In particular, we study the problem of maximizing the betweenness score of a given node—Maximum Betweenness Improvement (MBI)—and that of maximizing the ranking of a given node—Maximum Ranking Improvement (MRI). We show that MBI cannot be approximated in polynomial-time within a factor (1−1/2e) and that MRI does not admit any polynomial-time constant factor approximation algorithm, both unless P = NP . We then propose a simple greedy approximation algorithm for MBI with an almost tight approximation ratio and we test its performance on several real-world networks. We experimentally show that our algorithm highly increases both the betweenness score and the ranking of a given node and that it outperforms several competitive baselines. To speed up the computation of our greedy algorithm, we also propose a new dynamic algorithm for updating the betweenness of one node after an edge insertion, which might be of independent interest. Using the dynamic algorithm, we are now able to compute an approximation of MBI on networks with up to 10 5 edges in most cases in a matter of seconds or a few minutes.

Approximate and incremental network function placement

The virtualization of modern computer networks introduces interesting new opportunities for a more flexible placement of network functions and middleboxes (firewalls, proxies, traffic optimizers, virtual switches, etc.). This paper studies approximation algorithms for the incremental deployment of a minimum number of middleboxes, such that capacity constraints at the middleboxes and length constraints on the communication routes are respected.Based on a new, purely combinatorial and rigorous proof of submodularity, we obtain our main result: a deterministic greedy approximation algorithm which only employs augmenting paths to serve future communication pairs. Hence, our algorithm does not require any changes to the locations of existing middleboxes or the preemption of previously served communication pairs when additional middleboxes are deployed. It is hence particularly attractive for incremental deployments. We prove that the achieved polynomial-time approximation bound is optimal, unless P=NP holds.This paper also initiates the study of a weighted problem variant, in which entire groups of nodes need to communicate via a middlebox, possibly at different rates. We present an LP relaxation and randomized rounding algorithm for this problem, leveraging an interesting connection to scheduling.We complement our formal results with a simulation study of a large set of synthetically generated instances. Our results indicate that the presented algorithms yield near-optimal solutions in practice.

On Non-Preemptive VM Scheduling in the Cloud

We study the problem of scheduling VMs (Virtual Machines) in a distributed server platform, motivated by cloud computing applications. The VMs arrive dynamically over time to the system, and require a certain amount of resources (e.g. memory, CPU, etc) for the duration of their service. To avoid costly preemptions, we consider non-preemptive scheduling: Each VM has to be assigned to a server which has enough residual capacity to accommodate it, and once a VM is assigned to a server, its service cannot be disrupted (preempted). Prior approaches to this problem either have high complexity, require synchronization among the servers, or yield queue sizes/delays which are excessively large. We propose a non-preemptive scheduling algorithm that resolves these issues. In general, given an approximation algorithm to Knapsack with approximation ratio r , our scheduling algorithm can provide rβ fraction of the throughput region for β < r. In the special case of a greedy approximation algorithm to Knapsack, we further show that this condition can be relaxed to β<1. The parameters β and r can be tuned to provide a tradeoff between achievable throughput, delay, and computational complexity of the scheduling algorithm. Finally extensive simulation results using both synthetic and real traffic traces are presented to verify the performance of our algorithm.

Computing and Updating Hypervolume Contributions in Up to Four Dimensions

Arising in connection with multiobjective selection and archiving, the hypervolume subset selection problem (HSSP) consists in finding a subset of size k ≤ n of a set X ⊂ ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> of n nondominated points in d-dimensional space that maximizes the hypervolume indicator. The incremental greedy approximation to the HSSP has an approximation guarantee of 1 - 1/e, and is polynomial in n and k, while no polynomial exact algorithms are known for d ≥ 3. The decremental greedy counterpart has no known approximation guarantee, but is potentially faster for large k, and still leads to good approximations in practice. The computation and update of individual hypervolume contributions are at the core of the implementation of this greedy strategy. In this paper, new algorithms for the computation and update of hypervolume contributions are developed. In three dimensions, updating the total hypervolume and all individual contributions under single-point changes is performed in linear time, while in the 4-D case all contributions are computed in O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) time. As a consequence, the decremental greedy approximation to the HSSP can now be obtained in O(n(n - k) +n log n) and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (n - k)) time for d = 3 and d = 4, respectively. Experimental results show that the proposed algorithms significantly outperform existing ones.

1-greedy renormings of Garling sequence spaces

We show that all Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In particular, our approach to the study of the superreflexivity of Garling sequence spaces provides an example of how essentially non-linear tools from greedy approximation can be used to shed light on the linear structure of these spaces.

Robust sensor placement for pipeline monitoring: Mixed integer and greedy optimization

Intelligent water systems – aided by sensing technologies – have been identified as an important mechanism towards ensuring the resilience of urban systems. In this work, we study the problem of sensor placement that is robust to intermittent failures of sensors, i.e. sensor interruptions. We propose robust mixed integer optimization (RMIO) and robust greedy approximation (RGA) solution approaches. The underlying idea of both approaches is to promote solutions that achieve multiple detectability of events, such that these events remain detectable even when some sensors are interrupted. Additionally, we apply a previously proposed greedy approximation approach for solving the robust submodular function optimization (RSFO) problem. We compare scalability of these approaches and the quality of the solutions using a set of real water-networks. Our results demonstrate that considering sensor interruptions in the design stage improves sensor network performance. Importantly, we find that although the detection performances of RMIO and RGA approaches are comparable, RMIO generally has better performance than RGA, and is scalable to large-scale networks. Furthermore, the results demonstrate that RMIO and RGA approaches tend to outperform the RSFO approach.

Structural Target Controllability of Linear Networks.

Computational analysis of the structure of intra-cellular molecular interaction networks can suggest novel therapeutic approaches for systemic diseases like cancer. Recent research in the area of network science has shown that network control theory can be a powerful tool in the understanding and manipulation of such bio-medical networks. In 2011, Liu etal. developed a polynomial time algorithm computing the size of the minimal set of nodes controlling a linear network. In 2014, Gao etal. generalized the problem for target control, minimizing the set of nodes controlling a target within a linear network. The authors developed a Greedy approximation algorithm while leaving open the complexity of the optimization problem. We prove here that the target controllability problem is NP-hard in all practical setups, i.e., when the control power of any individual input is bounded by some constant. We also show that the algorithm provided by Gao etal. fails to provide a valid solution in some special cases, and an additional validation step is required. We fix and improve their algorithm using several heuristics, obtaining in the end an up to 10-fold decrease in running time and also a decrease in the size of solutions.

Data Quality Aware Task Allocation With Budget Constraint in Mobile Crowdsensing

With the pervasiveness of mobile devices, satisfying spatial-temporal coverage requirements in interested regions while considering the quality of the sensing data and the budget constraint is a major research challenge in mobile crowdsensing (MCS). In this paper, we define a novel coverage metric, quality coverage, which considers both the fraction of subareas covered by sensor readings and the quality of sensing data in each covered subarea, for diversified location-based MCS tasks. In our system, as participants with high reliability level can contribute a lot to sensing tasks, we design an incentive model that provides higher bonus to more reliable participants. After knowing the reward for the per amount of sensing data, each worker can submit an interested task set that contains several unit tasks. The platform assigns MCS tasks to these mobile workers by solving the task allocation optimization problems, which select the optimal mobile worker set under the budget of data requesters to maximize diverse and spatial coverage level in the whole sensing area. As the task allocation problems are proven to be NP-hard, we propose a heuristic and a polynomial-time greedy approximation algorithm to solve optimization problems in two different application scenarios. Simulations using both real and synthetic data sets show that our algorithms outperform existing approaches.

The Team Surviving Orienteers problem: routing teams of robots in uncertain environments with survival constraints

We study the following multi-robot coordination problem: given a graph, where each edge is weighted by the probability of surviving while traversing it, find a set of paths for K robots that maximizes the expected number of nodes collectively visited, subject to constraints on the probabilities that each robot survives to its destination. We call this the Team Surviving Orienteers (TSO) problem, which is motivated by scenarios where a team of robots must traverse a dangerous environment, such as aid delivery after disasters. We present the TSO problem formally along with several variants, which represent "survivability-aware" counterparts for a wide range of multi-robot coordination problems such as vehicle routing, patrolling, and informative path planning. We propose an approximate greedy approach for selecting paths, and prove that the value of its output is within a factor $$1-e^{-p_s/\lambda }$$1-e-ps/ź of the optimum where $$p_s$$ps is the per-robot survival probability threshold, and $$1/\lambda \le 1$$1/ź≤1 is the approximation factor of an oracle routine for the well-known orienteering problem. We also formalize an on-line update version of the TSO problem, and a generalization to heterogeneous teams where both robot types and paths are selected. We provide numerical simulations which verify our theoretical findings, apply our approach to real-world scenarios, and demonstrate its effectiveness in large-scale problems with the aid of a heuristic for the orienteering problem.

On Non-Preemptive VM Scheduling in the Cloud

We study the problem of scheduling VMs (Virtual Machines) in a distributed server platform, motivated by cloud computing applications. The VMs arrive dynamically over time to the system, and require a certain amount of resources (e.g. memory, CPU, etc) for the duration of their service. To avoid costly preemptions, we consider non-preemptive scheduling: Each VM has to be assigned to a server which has enough residual capacity to accommodate it, and once a VM is assigned to a server, its service cannot be disrupted (preempted). Prior approaches to this problem either have high complexity, require synchronization among the servers, or yield queue sizes/delays which are excessively large. We propose a non-preemptive scheduling algorithm that resolves these issues. In general, given an approximation algorithm to Knapsack with approximation ratio r , our scheduling algorithm can provide rβ fraction of the throughput region for β &lt; r . In the special case of a greedy approximation algorithm to Knapsack, we further show that this condition can be relaxed to β&lt;1 . The parameters β and r can be tuned to provide a trade-off between achievable throughput, delay, and computational complexity of the scheduling algorithm. Finally extensive simulation results using both synthetic and real traffic traces are presented to verify the performance of our algorithm.