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Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other "path-related" temporal notions, only few attempts have been made to investigate "non-path" temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions. [ABSTRACT FROM AUTHOR]

In an offload-capable Long Term Evolution (LTE)- Wireless Fidelity (WiFi) Heterogeneous Network (HetNet), we consider the problem of maximization of the total system throughput under voice user blocking probability constraint. The optimal policy is threshold in nature. However, computation of optimal policy requires the knowledge of the statistics of system dynamics, viz., arrival processes of voice and data users, which may be difficult to obtain in reality. Motivated by the Post-Decision State (PDS) framework to learn the optimal policy under unknown statistics of system dynamics, we propose, in this paper, an online Radio Access Technology (RAT) selection algorithm using Relative Value Iteration Algorithm (RVIA). However, the convergence speed of this algorithm can be further improved if the underlying threshold structure of the optimal policy can be exploited. To this end, we propose a novel structure-aware online RAT selection algorithm which reduces the feasible policy space, thereby offering lesser storage and computational complexity and faster convergence. This algorithm provides a novel framework for designing online learning algorithms for other problems and hence is of independent interest. We prove that both the algorithms converge to the optimal policy. Simulation results demonstrate that the proposed algorithms converge faster than a traditional scheme. Also, the proposed schemes perform better than other benchmark algorithms under realistic network scenarios. [ABSTRACT FROM AUTHOR]

A mixed shop is a manufacturing infrastructure designed to process a mixture of a set of flow-shop jobs and a set of open-shop jobs. Mixed shops are in general much more complex to schedule than flow-shops and open-shops, and have been studied since the 1980's. We consider the three machine proportionate mixed shop problem denoted as M 3 | p r p t | C max , in which by "proportionate" each job has equal processing times on all three machines. Koulamas and Kyparisis (2015) [6] showed that the problem is solvable in polynomial time in some very special cases; for the non-solvable case, they proposed a 5/3-approximation algorithm. In this paper, we first present an improved 4/3-approximation algorithm and show that this ratio of 4/3 is asymptotically tight; when the largest job is a flow-shop job, we then present a fully polynomial-time approximation scheme (FPTAS). On the negative side, while the F 3 | p r p t | C max problem is polynomial-time solvable, we show an interesting hardness result that adding one open-shop job to the job set makes the problem NP-hard if this open-shop job is larger than any flow-shop job. We are able to design an FPTAS for this special case too. [ABSTRACT FROM AUTHOR]

Motivated by emerging resource allocation and data placement problems such as web caches and peer-to-peer systems, we consider and study a class of resource allocation problems over a network of agents (nodes). In this model, which can be viewed as a homogeneous data placement problem, nodes can store only a limited number of resources while accessing the remaining ones through their closest neighbors. We consider this problem under both optimization and gametheoretic frameworks. In the case of optimal resource allocation, we will first show that when there are only k = 2 resources, the optimal allocation can be found efficiently in O(n ² log n) steps, where n denotes the total number of nodes. However, for k \geq 3 this problem becomes NP-hard with no polynomial-time approximation algorithm with a performance guarantee better than 1 + 1 102k², even under metric access costs. We then provide a 3-approximation algorithm for the optimal resource allocation which runs only in O(kn² ). Subsequently, we look at this problem under a selfish setting formulated as a noncooperative game and provide a 3-approximation algorithm for obtaining its pure Nash equilibria under metric access costs. We then establish an equivalence between the set of pure Nash equilibria and flip-optimal solutions of the Max-k-Cut problem over a specific weighted complete graph. While this reduction suggests that finding a pure Nash equilibrium using best response dynamics might be PLS-hard, it allows us to use tools from complementary slackness and quadratic programming to devise systematic and more efficient algorithms towards obtaining Nash equilibrium points. [ABSTRACT FROM AUTHOR]

• Models the scheduling problem as an optimization problem with congruence constraints. • Proposes a new graph representation of the problem. • Provides optimal inapproximability results for the optimization problem. • Exposes the close bounds that relate this problem with the MAX DICUT problem. • Customizes and applies approximation algorithms for MAX DICUT in real case studies. This paper aims to synchronize timetables in a transit network so as to minimize the total passenger transfer waiting time. Assuming a fixed headway for each line, we first formulate the problem as an optimization problem with congruence constraints. We show that the problem is NP-hard, and investigate several special cases of the problem that are solvable in polynomial time. Furthermore, we show that the local search for the general problem is equivalent to the well-studied maximum directed cut problem. As such, we use an approximation algorithm for the maximum directed cut problem to solve the timetable synchronization problem. We assess the quality of the algorithm on a real-world case study, and show that our algorithm significantly outperforms the state-of-the-art in the literature. Lastly, we relax the fixed headway assumption, and propose an efficient recursive quasi-linear time algorithm to minimize the total transfer waiting time in this general setting. [ABSTRACT FROM AUTHOR]

We study Parallel Task Scheduling Pm|sizej|Cmax with a constant number of machines. This problem is known to be strongly NP-complete for each m ≥ 5, while it is solvable in pseudo-polynomial time for each m ≤ 3. We give a positive answer to the long-standing open question whether this problem is strongly NP-complete for m = 4. As a second result, we improve the lower bound of 12 11 for approximating pseudo-polynomial Strip Packing to 5 4 . Since the best known approximation algorithm for this problem has a ratio of 5 4 + ε , this result almost closes the gap between approximation ratio and inapproximability result. Both results are proved by a reduction from the strongly NP-complete problem 3-Partition. [ABSTRACT FROM AUTHOR]

In recent years the t-distributed Stochastic Neighbor Embedding (t-SNE) algorithm has become one of the most used and insightful techniques for exploratory data analysis of high-dimensional data. It reveals clusters of high-dimensional data points at different scales while only requiring minimal tuning of its parameters. However, the computational complexity of the algorithm limits its application to relatively small datasets. To address this problem, several evolutions of t-SNE have been developed in recent years, mainly focusing on the scalability of the similarity computations between data points. However, these contributions are insufficient to achieve interactive rates when visualizing the evolution of the t-SNE embedding for large datasets. In this work, we present a novel approach to the minimization of the t-SNE objective function that heavily relies on graphics hardware and has linear computational complexity. Our technique decreases the computational cost of running t-SNE on datasets by orders of magnitude and retains or improves on the accuracy of past approximated techniques. We propose to approximate the repulsive forces between data points by splatting kernel textures for each data point. This approximation allows us to reformulate the t-SNE minimization problem as a series of tensor operations that can be efficiently executed on the graphics card. An efficient implementation of our technique is integrated and available for use in the widely used Google TensorFlow.js, and an open-source C++ library. [ABSTRACT FROM AUTHOR]

Inspired by scaffold filling, a recent approach for genome reconstruction from incomplete data, we consider a variant of the well-known longest common subsequence problem for the comparison of two sequences. The new problem, called Longest Filled Common Subsequence, aims to compare a complete sequence with an incomplete one, i.e. with some missing elements. Longest Filled Common Subsequence (LFCS), given a complete sequence A , an incomplete sequence B , and a multiset M of symbols missing in B , asks for a sequence B ⁎ obtained by inserting the symbols of M into B so that B ⁎ induces a common subsequence with A of maximum length. We investigate the computational and approximation complexity of the problem and we show that it is NP-hard and APX-hard when A contains at most two occurrences of each symbol, and we give a polynomial time algorithm when the input sequences are over a constant-size alphabet. We give a 3 5 − approximation algorithm for the Longest Filled Common Subsequence problem. Finally, we present a fixed-parameter algorithm for the problem, when it is parameterized by the number of symbols inserted in B that "match" symbols of A. [ABSTRACT FROM AUTHOR]

Min-Hash is a popular technique for efficiently estimating the Jaccard similarity of binary sets. Consistent Weighted Sampling (CWS) generalizes the Min-Hash scheme to sketch weighted sets and has drawn increasing interest from the community. Due to its constant-time complexity independent of the values of the weights, Improved CWS (ICWS) is considered as the state-of-the-art CWS algorithm. In this paper, we revisit ICWS and analyze its underlying mechanism to show that there actually exists dependence between the two components of the hash-code produced by ICWS, which violates the condition of independence. To remedy the problem, we propose an Improved ICWS (I $^2$ 2 CWS) algorithm which not only shares the same theoretical computational complexity as ICWS but also abides by the required conditions of the CWS scheme. The experimental results on a number of synthetic data sets and real-world text data sets demonstrate that our I $^2$ 2 CWS algorithm can estimate the Jaccard similarity more accurately, and also competes with or outperforms the compared methods, including ICWS, in classification and top- $K$ K retrieval, after relieving the underlying dependence. [ABSTRACT FROM AUTHOR]

Self-reconfigurable modular robots (SRM-robots) can autonomously change their shape according to different tasks and work environments, and have received considerable attention recently. Many reshaping/reconfiguration algorithms have been proposed. In this paper, we present a theoretical analysis of computational complexity on a reshape planning for a kind of lattice-type 3D SRM-robots, whose modules are of cubic shape and can move by rotating on the surfaces of other modules. Different from previous NP-completeness study on general chain-type robots (i.e. the motion of any chains and the location of modules can be arbitrary), we consider more practical constraints on modules' shape (i.e. cubic shape), position (lying in 2D/3D grids) and motion (using orthogonal rotations) in this paper. We formulate the reshape planning problem of SRM-robots with these practical constraints by a (p, q) optimization problem, where p and q characterize two widely used metrics, i.e. the number of disconnecting/reconnecting operations and the number of reshaping steps. Proofs are presented, showing that this optimization problem is NP-complete. Therefore, instead of finding global optimization results, most likely approximation solution can be obtained for the problem instead of seeking polynomial algorithm. We also present the upper and lower bounds for the 2-tuple (p, q), which is useful for evaluating the approximation algorithms in future research. [ABSTRACT FROM AUTHOR]

Orthogonal greedy algorithms are popular sparse signal reconstruction algorithms. Their principle is to select atoms one by one. A series of unconstrained least-square subproblems of gradually increasing size is solved to compute the approximation coefficients, which is efficiently performed using a fast recursive update scheme. When dealing with non-negative sparse signal reconstruction, a series of non-negative least-squares subproblems have to be solved. Fast implementation becomes tricky since each subproblem does not have a closed-form solution anymore. Recently, non-negative extensions of the classical orthogonal matching pursuit and orthogonal least squares algorithms were proposed, using slow (i.e., non-recursive) or recursive but inexact implementations. In this paper, we revisit these algorithms in a unified way. We define a class of non-negative orthogonal greedy algorithms and exhibit their structural properties. We propose a fast and exact implementation based on the active-set resolution of non-negative least-squares and exploiting warm start initializations. The algorithms are assessed in terms of accuracy and computational complexity for a sparse spike deconvolution problem. We also present an application to near-infrared spectra decomposition. [ABSTRACT FROM AUTHOR]

Many resource allocation tasks are challenging global (i.e., non-convex) optimization problems. The main issue is that the computational complexity of these problems grows exponentially in the number of variables instead of polynomially as for many convex optimization problems. However, often the non-convexity stems only from a subset of variables. Conventional global optimization frameworks like monotonic optimization or DC programming treat all variables as global variables and require complicated, problem specific decomposition approaches to exploit the convexity in some variables. To overcome this challenge, we develop an easy-to-use algorithm that inherently differentiates between convex and non-convex variables, preserving the low computational complexity in the number of convex variables. Another issue with these widely used frameworks is that they may suffer from severe numerical problems. We discuss this issue in detail and provide a clear motivating example. The solution to this problem is to replace the traditional approach of finding an $\varepsilon $ -approximate solution by the novel concept of $\varepsilon $ -essential feasibility. The underlying algorithmic approach is called successive incumbent transcending (SIT) algorithm and builds the foundation of our developed algorithm. A further highlight of this algorithm is that it inherently treats fractional objectives making the use of Dinkelbach's iterative algorithm obsolete. Numerical experiments show a speed-up of four orders of magnitude over state-of-the-art algorithms and almost three orders of magnitude of additional speed-up over Dinkelbach's algorithm for fractional programs. [ABSTRACT FROM AUTHOR]

In data centers, up to dozens of tasks are colocated on a single physical machine. Machines are used more efficiently, but the performance of the tasks deteriorates, as the colocated tasks compete for shared resources. Since the tasks are heterogeneous, the resulting performance dependencies are complex. In our previous work [1] , [2] we proposed a new combinatorial optimization model that uses two parameters of a task — its size and its type — to characterize how a task influences the performance of other tasks allocated to the same machine. In this paper, we study the egalitarian optimization goal: the aim is to optimize the performance of the worst-off task. This problem generalizes the classic makespan minimization on multiple processors ($P||C_{\max }$P||Cmax). We prove that polynomially-solvable variants of $P||C_{\max }$P||Cmax are NP-hard for this generalization, and that the problem is hard to approximate when the number of types is not constant. For a constant number of types, we propose a PTAS, a fast approximation algorithm, and a series of heuristics. We simulate the algorithms on instances derived from a trace of one of Google clusters. Compared with baseline algorithms solving $P||C_{\max }$P||Cmax, our proposed algorithms aware of the types of the jobs lead to significantly better tasks' performance. The notion of type enables us to extend standard combinatorial optimization methods to handle degradation of performance caused by colocation. Types add a layer of additional complexity. However, our results — approximation algorithms and good average-case performance — show that types can be handled efficiently. [ABSTRACT FROM AUTHOR]

We introduce a variant of the well-known minimum-cost circulation problem in directed networks, where vertex demand values are taken from the integers modulo λ , for some integer λ ≥ 2 . More formally, given a directed network G = (V , E) , each of whose edges is associated with a weight and each of whose vertices is associated with a demand taken over the integers modulo λ , the objective is to compute a flow of minimum weight that satisfies all the vertex demands modulo λ . This problem is motivated by a problem of computing a periodic schedule for traffic lights in an urban transportation network that minimizes the total delay time of vehicles. We show that this modular circulation problem is solvable in polynomial time when λ = 2 and that the problem is NP-hard when λ = 3 . We also present a polynomial time algorithm that achieves a 4 (λ - 1) -approximation. [ABSTRACT FROM AUTHOR]

In this paper, a class of single machine scheduling problems is considered. It is assumed that job processing times and due dates can be uncertain and they are specified in the form of discrete scenario set. A probability distribution in the scenario set is known. In order to choose a schedule, some risk criteria such as the value at risk and conditional value at risk are used. Various positive and negative complexity results are provided for basic single machine scheduling problems. In this paper, new complexity results are shown and some known complexity results are strengthened. [ABSTRACT FROM AUTHOR]

Ordered statistics-based decoding (OSD), which exhibits a near maximum likelihood decoding performance, suffers from huge computational complexity as the order increases. In this letter, we propose a fast and scalable OSD by considering the OSD as a fast searching problem. In the searching process, if the up-to-date minimum cost value is less than a predicted threshold value, then we can safely skip the search for remaining higher orders. The computational complexity of the proposed algorithm converges quickly to that of order one, regardless of the maximum order. Compared with the probabilistic necessary conditions-based OSD, the proposed algorithm exhibits speed-up gains of a factor of approximately 2,740 (at 3.0dB) for (127,64) BCH codes, with an indistinguishable decoding performance. [ABSTRACT FROM AUTHOR]

In sketched clustering, a dataset of $T$ samples is first sketched down to a vector of modest size, from which the centroids are subsequently extracted. Its advantages include 1) reduced storage complexity and 2) centroid extraction complexity independent of $T$. For the sketching methodology recently proposed by Keriven et al., which can be interpreted as a random sampling of the empirical characteristic function, we propose a sketched clustering algorithm based on approximate message passing. Numerical experiments suggest that our approach is more efficient than the state-of-the-art sketched clustering algorithm “CL-OMPR” (in both computational and sample complexity) and more efficient than k-means++ when $T$ is large. [ABSTRACT FROM AUTHOR]

We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of jobs in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. (ACM Trans Algorithms 3(4), 2007), its exact computational complexity has been repeatedly posed as an open question (see e.g. Albers and Antoniadis in ACM Trans Algorithms 10(2):9, 2014; Baptiste et al. in ACM Trans Algorithms 8(3):26, 2012; Irani and Pruhs in SIGACT News 36(2):63–76, 2005). The currently best known upper and lower bounds are a 4 / 3-approximation algorithm and NP-hardness due to Albers and Antoniadis (2014) and Kumar and Shannigrahi (CoRR, 2013. arXiv:1304.7373), respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules. [ABSTRACT FROM AUTHOR]

In this letter, we consider the problem of detecting finite-alphabet sparse signals from noisy and coarsely quantized measurements. To solve this problem, we propose a greedy sparse signal detection algorithm referred to as Bayesian matching pursuit (BMP). The key idea of BMP is to identify the non-zero elements of a sparse signal that produce the largest a posteriori probabilities in an iterative fashion. Our simulation results show that the BMP algorithm outperforms the existing sparse signal reconstruction algorithms in terms of frame error rates even with a significantly reduced computational complexity. [ABSTRACT FROM AUTHOR]

The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of "bad" events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is non-constructively guaranteed by the LLL, culminating in the recent breakthrough of Moser and Tardos [2010] for the full asymmetric LLL. We show that the output distribution of the Moser-Tardos algorithm well-approximates the conditional LLL-distribution, the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results. [ABSTRACT FROM AUTHOR]

k -means and k -median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependencies on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k -means and k -median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the "best explanation" by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k -means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering. [ABSTRACT FROM AUTHOR]

The problem of phase-noise compensation for correlated phase noise in coded multichannel optical transmission is investigated. To that end, a simple multichannel phase-noise model is considered and the maximum a posteriori detector for this model is approximated using two frameworks, namely factor graphs (FGs) combined with the sum–product algorithm (SPA) and a variational Bayesian (VB) inference method. The resulting pilot-aided algorithms perform iterative phase-noise compensation in cooperation with a decoder, using extended Kalman smoothing to estimate the a posteriori phase-noise distribution jointly for all channels. The system model and the proposed algorithms are verified using experimental data obtained from space-division multiplexed multicore-fiber transmission. Through Monte Carlo simulations, the algorithms are further evaluated in terms of phase-noise tolerance for coded transmission. It is observed that they significantly outperform the conventional approach to phase-noise compensation in the optical literature. Moreover, the FG/SPA framework performs similarly or better than the VB framework in terms of phase-noise tolerance of the resulting algorithms, for a slightly higher computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we study unbounded parallel-batch scheduling with drop-line tasks to minimize a regular objective function, where by "drop-line tasks" we mean that the completion time of each task (job) is equal to the sum of the starting time of the batch containing the task and the processing time of the task. In the problems considered in this paper, we assume that the tasks have individual release dates and the general regular objective function to be minimized is either of the sum-form or of the max-form. We then study the computational complexity of these problems on an unbounded parallel-batch processor. We show that (i) the problems are binary NP-hard and are solvable in pseudo-polynomial times, and (ii) when the number of processing times or release dates is a constant, the problems are solvable in polynomial times. We also point out some consequences of approximation algorithms. [ABSTRACT FROM AUTHOR]

A solution extension problem consists in an instance and a partial feasible solution which is given in advance and the goal is to extend this partial solution to a feasible one. Many well-known problems like Coloring Extension Problems, Traveling Salesman Problem and Routing problems, have been studied in this framework. In this paper, we focus on the edge-weighted spanning star forest problem for both minimization and maximization versions. The goal here is to find a vertex partition of an edge-weighted graph into disjoint non-trivial stars and the value of a solution is given by the sum of the edge-weights of the stars. We propose NP -hardness, parameterized complexity, positive and negative approximation results. • An extension variant of the weighted edge cover problem is studied. • The problem is NP -complete in complete graphs while polynomial solvable in bounded tree-width graphs. • For non-negative weights, the minimization version is in-approximable at all while the maximization version is in RAPX. • If weight function satisfies (extended) c -relaxed triangle inequalities, the minimization version is in RAPX. [ABSTRACT FROM AUTHOR]

We research a scheduling algorithm for a feasible schedule construction on one machine by the total earliness of jobs minimization criterion. We propose an improvement to this algorithm. During the research, we have developed a software system to verify reliability of the algorithms. We compared the schedules built by the algorithm with those constructed by an exhaustive enumeration. We calculated the percentage of a non-exact schedule construction by the investigated algorithm for different characteristics of the schedule. The object of research is the process of schedules' optimization and their investigation. The purpose of the paper is to increase the efficiency of feasible schedules construction process for one machine in the presence of penalties for jobs completion prior to their due dates. We need to use appropriate economic and mathematical methods, in particular those from scheduling theory, to organize the production and planning on an enterprise efficiently. Scheduling problems have a great application value today. Rapid development of communication and electronics technologies increasingly requires the construction of schedules. The schedules are related to functioning of industrial enterprises and the service sector, education, transport and many other areas. Tasks of scheduling theory include study of the computational complexity of the problems and development of exact, approximation and heuristic algorithms for their solution. Most papers are devoted to combinatorial approaches' development. However, as practice shows, possibilities of combinatorial algorithms are substantially limited by the dimension of the problems they solve. This is why we believe in importance of the research and improvement of the scheduling algorithm to minimize the total earliness of jobs on one machine. We use combinatorial optimization methods, scheduling theory, heuristic algorithms and statistical analysis. The scientific novelty of the results consists in modification to the existing scheduling algorithm that minimizes the total earliness on one machine. New algorithm has better efficiency than the existing ones. [ABSTRACT FROM AUTHOR]

This paper investigates the hybrid precoding design for millimeter wave (mmWave) multiple-input-multiple-output (MIMO) systems with finite-alphabet inputs. The mmWave MIMO system employs partially-connected hybrid precoding architecture with dynamic subarrays, where each radio frequency (RF) chain is connected to a dynamic subset of antennas. We consider the design of analog and digital precoders utilizing statistical and/or mixed channel state information (CSI), which involve solving an extremely difficult problem in theory: First, designing the optimal partition of antennas over RF chains is a combinatorial optimization problem, whose optimal solution requires an exhaustive search over all antenna partitioning solutions; Second, the average mutual information under mmWave MIMO channels lacks closed-form expression and involves prohibitive computational burden; and Third, the hybrid precoding problem with given partition of antennas is nonconvex with respect to the analog and digital precoders. To address these issues, this paper first presents a simple criterion and the corresponding low complexity algorithm to design the optimal partition of antennas using statistical CSI. Then, it derives the lower bound and its approximation for the average mutual information, in which the computational complexity is greatly reduced compared to calculating the average mutual information directly. In addition, it also shows that the lower bound with a constant shift offers a very accurate approximation to the average mutual information. This paper further proposes utilizing the lower bound approximation as a low-complexity and accurate alternative for developing a manifold-based gradient ascent algorithm to find near-optimal analog and digital precoders. Several numerical results are provided to show that our proposed algorithm outperforms the existing hybrid precoding algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

Multi-access edge computing (MEC) has formed a major improvement in the existing mobile cloud computing paradigm, due to its ability in addressing the rising number of latency-sensitive services. However, bearing in mind the limited capacity that edge servers possess which offsets their benefits in the periods of high load, a hierarchical arrangement of the edge cloudlets has been studied and has shown to be successful in expanding their capabilities. Yet, considering the emerging business models in 5G networks, the cost disparity between the edge tiers has been until now ignored, leading to cost-inefficient solutions with respect to the network operators (NOs). In this paper, we consider an NO that is leasing resources of a high-tier central cloudlet for task offload, where we jointly minimize the NO’s computational cost and devices’ energy consumption in a multi-tier MEC system, by optimizing the offloading decision, the allocated transmission power and radio resources on the uplink channel, and the assigned servers’ computation, while respecting the devices’ latency requirement. We mathematically formulate our mixed-integer non-convex program and propose a Branch-and-Bound (BnB) algorithm for obtaining the optimal solution. Due to the BnB complexity, we propose a low-complexity algorithm based on the successive convex approximation method to solve and obtain a high-quality solution and also present an inflation-based algorithm for obtaining a polynomial-time and efficient solution. The numerical results show the performance and scalability of the algorithms, demonstrate their efficiency, and uncover insights for helping NOs to better manage their resources following various configurations. [ABSTRACT FROM AUTHOR]

Hypergraphs are generalizations of graphs where the (hyper)edges can connect any number of vertices. They are powerful tools for representing complex and non-pairwise relationships. However, existing graph computation frameworks cannot accommodate hypergraphs without converting them into graphs, because they do not offer APIs that support (hyper)edges directly. This graph conversion may create excessive replicas and result in very large graphs, causing difficulties in workload balancing. A few tools have been developed for hypergraph partitioning, but they are not general-purpose frameworks for hypergraph processing. In this paper, we propose HyperX, a general-purpose distributed hypergraph processing framework built on top of Spark. HyperX is based on the computation paradigm "Pregel", which is user-friendly and has been widely adopted by popular graph computation frameworks. To help create balanced workloads for distributed hypergraph processing, we further investigate the hypergraph partitioning problem and propose a novel label propagation partitioning (LPP) algorithm. We conduct extensive experiments using both real and synthetic data. The result shows that HyperX achieves an order of magnitude improvement for running hypergraph learning algorithms compared with graph conversion based approaches in terms of running time, network communication costs, and memory consumption. For hypergraph partitioning, LPP outperforms the baseline algorithms significantly in these measures as well. [ABSTRACT FROM AUTHOR]

We study two variants of the Balanced Tree Partition (BTP for short) problem, whose goal is to find the minimum number of edges such that a partition of vertices into sets of equal size can be obtained after deleting those edges. We consider the BTP problem over trees with virtual nodes, which is motivated by the real applications in storing tree structured data distributively. Different from the traditional BTP problem, after deleting an edge from the tree, two virtual nodes must be added to provide the ability to recover the edge when computing, which will increase the size of some set in the partition and the set may become too large and need further splitting. Depending on whether or not to consider the partition set number k to be a parameter, to investigate the balanced tree partition methods over trees with virtual nodes, we formally defined two specific problems, k -VBTP and VBTP. The computational complexity and inapproximability of k -VBTP are analyzed, for the VBTP problem, a polynomial time algorithm is designed to find the optimal solution based on dynamic programming. Finally, the connections between k -VBTP and VBTP are built, and based the algorithm for VBTP, we design a heuristic algorithm for k -VBTP with performance guarantee. [ABSTRACT FROM AUTHOR]

We study the complexity of motivating time-inconsistent agents to complete long term projects in a graph-based planning model proposed by Kleinberg and Oren (2014). Given a task graph G with n nodes, our objective is to guide an agent towards a target node t under certain budget constraints. The crux is that the agent may change its strategy over time due to its present-bias. We consider two strategies to guide the agent. First, a single reward is placed at t and arbitrary edges can be removed from G. Secondly, rewards can be placed at arbitrary nodes of G but no edges must be deleted. In both cases we show that it is NP-complete to decide if a given budget is sufficient to keep the agent motivated. For the first setting, we give complementing upper and lower bounds on the approximability of the minimum required budget. In particular, we devise a (1 + n) -approximation algorithm and prove NP-hardness for ratios greater than n / 3 . We also argue that the second setting does not permit any efficient approximation unless P = NP. [ABSTRACT FROM AUTHOR]

In this paper, we show that every (2P2, K4)-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon [J. Combin. Theory Ser. B, 29 (1980), pp. 345-346] from the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for (4P1,C4)-free graphs. Lozin and Malyshev [Discrete Appl. Math., 216 (2017), pp. 273-280] conjecture that the answer is yes. As an application of our main result, we provide the first positive evidence to the conjecture by giving a 2-approximation algorithm for coloring (4P1,C4)-free graphs. [ABSTRACT FROM AUTHOR]

In solving many real-world optimization problems, neither mathematical functions nor numerical simulations are available for evaluating the quality of candidate solutions. Instead, surrogate models must be built based on historical data to approximate the objective functions and no new data will be available during the optimization process. Such problems are known as offline data-driven optimization problems. Since the surrogate models solely depend on the given historical data, the optimization algorithm is able to search only in a very limited decision space during offline data-driven optimization. This paper proposes a new offline data-driven evolutionary algorithm to make the full use of the offline data to guide the search. To this end, a surrogate management strategy based on ensemble learning techniques developed in machine learning is adopted, which builds a large number of surrogate models before optimization and adaptively selects a small yet diverse subset of them during the optimization to achieve the best local approximation accuracy and reduce the computational complexity. Our experimental results on the benchmark problems and a transonic airfoil design example show that the proposed algorithm is able to handle offline data-driven optimization problems with up to 100 decision variables. [ABSTRACT FROM AUTHOR]

We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor α = 2 poly (m , n) for the volume of a tropical polytope given by n for the volume of a tropical polytope given by n vertices in a space of dimension m , unless P = NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. It follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank. [ABSTRACT FROM AUTHOR]

Unmanned Aerial Vehicle (UAV) networks have emerged as a promising technique to rapidly provide wireless coverage to a geographical area, where a flying UAV can be fast deployed to serve as cell site. Existing work on UAV-enabled wireless networks overlook the fast UAV deployment for wireless coverage, and such deployment problems have only been studied recently in sensor networks. Unlike sensors, UAVs should be deployed to the air and they are generally different in flying speed, operating altitude and wireless coverage radius. By considering such UAV heterogeneity to cover the whole target area, this paper studies two fast UAV deployment problems: one is to minimize the maximum deployment delay among all UAVs (min-max) for fairness consideration, and the other is to minimize the total deployment delay (min-sum) for efficiency consideration. We prove both min-max and min-sum problems are NP-complete in general. When dispatching UAVs from the same location, we present an optimal algorithm of low computational complexity $O(n^2)$ for the min-max problem. When UAVs are dispatched from different locations, we propose to preserve their location order during deployment and successfully design a fully polynomial time approximation scheme (FPTAS) of computation complexity $O(n^2 \log \frac{1}{\epsilon })$ to arbitrarily approach the global optimum with relative error $\epsilon$ . The min-sum problem is more challenging. When UAVs are dispatched from the same initial location, we present an approximation algorithm of linear time. As for the general case, we further reformulate it as a dynamic program and propose a pseudo polynomial-time algorithm to solve it optimally. [ABSTRACT FROM AUTHOR]

Approximation algorithms with linear complexities are required in the treatments of big data, however, present algorithms cannot output the diameter of a set of points with arbitrary accuracy and near-linear complexity. By introducing the partition technique, we introduce a very simple approximation algorithm with arbitrary accuracy ε and a complexity of O(N + ε−1 log ε−1) for the cases that all points are located in an Euclidean plane. The error bounds are proved strictly, and are verified by numerical tests. This complexity is better than existing algorithms, and the present algorithm is also very simple to be implemented in applications. [ABSTRACT FROM AUTHOR]

In this paper, a fast adaptive cross approximation (FACA) algorithm and the characteristic basis function method (CBFM) are presented to accelerate the computing processing in using the multiscale compressed block decomposition (MS-CBD) method in direct solutions of electrically large problems. The impedance matrix or the reduced matrix can be compressed into block matrices of the MS-CBD structure using the FACA. With the recursive inversion processes on the reduced matrix, it is easy to obtain the solutions. Combined with the fast adaptive cross sampling and the matrix decomposition algorithm method, the speed of the calculation of FACA is proved much faster than that of the conventional adaptive cross approximation. The numerical results of several perfect electric conducting objects are validated to demonstrate the performance of this method. The simulation results show that the computational time of the proposed method is much less than the conventional MS-CBD method with the CBFM, while their computational storages are similar. [ABSTRACT FROM AUTHOR]

Abstract An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing , that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a vertical segment. We prove that the HV-planarity testing problem is NP-complete even for graphs with vertex-degree at most three, which answers an open question posed by both Manuch et al. [30] and Durocher et al. [17]. On the positive side, we prove that the HV-planarity testing problem can be solved in polynomial-time for series-parallel graphs. This result significantly extends a previous result by Durocher et al. about HV-planarity testing of biconnected outerplanar graphs of maximum vertex-degree three. Highlights • HV-planarity testing is NP-complete, even for graphs with vertex-degree at most 3. • Polynomial-time HV-planarity testing for 2-connected series-parallel graphs. • Polynomial-time HV-planarity testing for partial 2-trees. [ABSTRACT FROM AUTHOR]

In this paper, a computationally efficient algorithm for Bayesian joint change point detection (CPD) in multiple time series is presented. The data generation model includes a number of change configurations (CC), each affecting a unique subset of the time series, which introduces correlation between the positions of change points (CPs) in the monitored time series. The inference objective is to identify joint changes and the associated CC. The algorithm consists of two stages: First, a univariate CPD algorithm is applied separately to each of the involved time series. The outcomes of this step are maximum a posteriori (MAP) detected CPs and posterior distributions of CPs conditioned on the MAP CPs. These outcomes are used in combination to approximate the posterior for the CCs. In the second algorithm stage, dynamic programming is used to find the maxima of this approximate CC posterior. The algorithm is applied to synthetic data, and it is shown to be both significantly faster and more accurate compared to a previously proposed algorithm designed to solve similar problems. Also, the initial algorithm is extended with steps from the maximization–maximization algorithm, which allows the hyperparameters of the data generation model to be estimated jointly with the CCs, and we show that these estimates coincide with estimates obtained from a Markov chain Monte Carlo algorithm. [ABSTRACT FROM AUTHOR]