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A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k. An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q=k=2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [47] and [41], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [56], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes. [ABSTRACT FROM AUTHOR]

We consider the numerical taxonomy problem of fitting a positive distance function \({\mathcal {D}:{S\choose 2}\rightarrow \mathbb {R}_{\gt 0}}\) by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in \(\mathcal {D}\). A nice application is in evolutionary biology where the tree T aims to approximate thebranching process leading to the observed distances in \(\mathcal {D}\) [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is, the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees and for the special case of ultrametrics with a root having the same distance to all vertices in S. The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O((log n)(log log n)) by Ailon and Charikar [2005], who wrote "determining whether an O(1) approximation can be obtained is a fascinating question." [ABSTRACT FROM AUTHOR]

We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree. [ABSTRACT FROM AUTHOR]

The article focuses on studying randomized greedy matching algorithms, particularly the Modified Randomized Greedy (MRG) algorithm and its weaker version named Random Decision Order (RDO). It mentions the RDO algorithm is shown to provide a 0.639-approximation for bipartite graphs and a 0.531-approximation for general graphs, leading to a substantial improvement in the approximation ratio of MRG.

Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in n. Given the sketch and a query item y, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to y. Most works to date focused on additive εn error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This article investigates multiplicative (1±ε )-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either O(log(ε²n)/ε²) or O(log³ (εn)/ε ) universe items. We present a randomized sketch storing O(log1.5 (εn)/ε ) items that can (1 ± ε )-approximate the rank of each universe item with high constant probability; this space bound is within an O(√log(εn)) factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments. [ABSTRACT FROM AUTHOR]

The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, we still have a limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximalmatching either take polylogarithmic time which is considered inefficient, or require a strictly superlinear space of n1+Ω(1) per machine. In this work, we close this gap by providing a novel analysis of an extremely simple algorithm, which is a variant of an algorithm conjectured to work by Czumaj, Lacki, Madry, Mitrovic, Onak, and Sankowski [15]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition.We show that this algorithm drastically reduces the vertex degrees. This, among other results, leads to an O(log log Δ) round algorithm for maximal matching with O(n) space (or even mildly sublinear in n using standard techniques). As an immediate corollary, we get a 2 approximate minimum vertex cover in essentially the same rounds and space, which is the optimal approximation factor under standard assumptions. We also get an improved O(log log Δ) round algorithm for 1 + ε approximate matching. All these results can also be implemented in the congested clique model in the same number of rounds. [ABSTRACT FROM AUTHOR]

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multiplayer model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two-player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the connections mentioned previously, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets (i.e., sets respecting the cardinality constraint) are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the link to the (non-streaming) robust setting mentioned previously, both of these algorithms improve on the current state of the art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2 + e hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best-known approximation algorithm. [ABSTRACT FROM AUTHOR]

Transportation networks frequently employ hub-and-spoke network architectures to route flows between many origin and destination pairs. Hub facilities work as switching points for flows in large networks. In this study, we focus on a problem, called the single allocation hub-and-spoke network design problem. In the problem, the goal is to allocate each non-hub node to exactly one of the given hub nodes so as to minimize the total transportation cost. The problem is essentially equivalent to another combinatorial optimization problem, called the metric labeling problem. The metric labeling problem was first introduced by Kleinberg and Tardos in 2002, motivated by application to segmentation problems in computer vision and related areas. In this study, we deal with a case where the set of hubs forms a star, which is encountered especially in telecommunication networks. We propose a polynomial-time randomized approximation algorithm for the problem, whose approximation ratio is less than 5.281. Our algorithms solve a linear relaxation problem and apply dependent rounding procedures. [ABSTRACT FROM AUTHOR]

We call a (not necessarily properly) edge-colored graph edge-color-avoiding connected if after the removal of edges of any single color, the graph remains connected. For vertex-colored graphs, similar definitions of color-avoiding connectivity can be given. In this article, we investigate the problem of determining the maximum number of edges that can be removed from either an edge- or a vertex-colored, color-avoiding connected graph so that it remains color-avoiding connected. First, we prove that this problem is NP-hard, and then, we give a polynomial-time approximation algorithm for it. To analyze the approximation factor of this algorithm, we determine the minimum number of edges of color-avoiding connected graphs on a given number of vertices and with a given number of colors. Furthermore, we also consider a generalization of edge-color-avoiding connectivity to matroids. [ABSTRACT FROM AUTHOR]

This work studies solutions to determine the macro-elastic moduli of three-phase composite materials with imperfect interfaces in 2D. Which is based on the coated circular assemblage model of Hashin-Strikman and the polarization approximation method (PA) to develop the formulae for elastic moduli of circle inclusions with spring-layer and surface-stress imperfect interfaces. From that, explicit algebraic expressions were obtained to estimate the elastic moduli of three-phase composites with imperfect interfaces, in which two phases are different with circular inclusions distributed randomly in the matrix. Besides, the FFT algorithm and the differential approximation (DA) are also developed to determine the elastic moduli of the three-phase composite with imperfect interfaces. The results of the FFT numerical methods will be compared with the DA and PA results with different material cases to show the effectiveness of the applied methods. [ABSTRACT FROM AUTHOR]

In the past decades, explosive numbers of Internet of Things (IoT) devices (objects) have been connected to the Internet, which enable users to access, control, and monitor their surrounding phenomenons at anytime and anywhere. To provide seamless interactions between the cyber world and the real world, Digital twins (DTs) of objects (IoT devices) are key enablers for real time monitoring, behaviour simulations, and predictive decisions on objects. Compared to centralized cloud computing, mobile edge computing (MEC) has been envisioning as a promising paradigm for low latency IoT applications. Accelerating the usage of DTs in MEC networks will bring unprecedented benefits to diverse services, through the co-evolution between physical objects and their virtual DTs, and DT-assisted service provisioning has attracted increasing attention recently. In this article, we consider novel DT placement and migration problems in an MEC network with the mobility assumption of objects and users, by jointly considering the freshness of DT data and the service cost of users requesting for DT data. To this end, we first propose an algorithm for the DT placement problem with the aim to minimize the sum of the DT update cost of objects and the total service cost of users requesting for DT data, through efficient DT placements and resource allocation to process user requests. We then devise an approximation algorithm with a provable approximation ratio for a special case of the DT placement problem when each user requests the DT data of only one object. Meanwhile, considering the mobility of users and objects, we devise an online, two-layer scheduling algorithm for DT migrations to further reduce the total service cost of users within a given finite time horizon. We finally evaluate the performance of the proposed algorithms through experimental simulations. The simulation results show that the proposed algorithms are promising. [ABSTRACT FROM AUTHOR]

The use of drones for collecting information and detecting bugs in orchards covered by nets is a challenging problem. The nets help in reducing pest damage, but they also constrain the drone's flight path, making it longer and more complex. To address this issue, we model the orchard as an aisle-graph, a regular data structure that represents consecutive aisles where trees are arranged in straight lines. The drone flies close to the trees and takes pictures at specific positions for monitoring the presence of bugs, but its energy is limited, so it can only visit a subset of positions. To tackle this challenge, we introduce the Single-drone Orienteering Aisle-graph Problem (SOAP), a variant of the orienteering problem, where likely infested locations are prioritized by assigning them a larger profit. Additionally, the drone's movements have a cost in terms of energy, and the objective is to plan a drone's route in the most profitable locations under a given drone's battery. We show that SOAP can be optimally solved in polynomial time, but for larger orchards/instances, we propose faster approximation and heuristic algorithms. Finally, we evaluate the algorithms on synthetic and real datasets to demonstrate their effectiveness and efficiency. [ABSTRACT FROM AUTHOR]

We study linear Fisher markets with satiation. In these markets, sellers have earning limits, and buyers have utility limits. Beyond applications in economics, they arise in the context of maximizing Nash social welfare when allocating indivisible items to agents. In contrast to markets with either earning or utility limits, markets with both limits have not been studied before. They turn out to have fundamentally different properties. In general, the existence of competitive equilibria is not guaranteed. We identify a natural property of markets (termed money clearing) that implies existence. We show that the set of equilibria is not always convex, answering a question posed in the literature. We design an FPTAS to compute an approximate equilibrium and prove that the problem of computing an exact equilibrium lies in the complexity class continuous local search (CLS ; i.e., the intersection of polynomial local search (PLS) and polynomial parity arguments on directed graphs (PPAD)). For a constant number of buyers or goods, we give a polynomial-time algorithm to compute an exact equilibrium. We show how (approximate) equilibria can be rounded and provide the first constant-factor approximation algorithm (with a factor of 2.404) for maximizing Nash social welfare when agents have capped linear (also known as budget-additive) valuations. Finally, we significantly improve the approximation hardness for additive valuations to 8/7>1.069. Funding: J. Garg was supported by the National Science Foundation [Grant CCF-1942321 (CAREER)]. M. Hoefer was supported by Deutsche Forschungsgemeinschaft [Grants Ho 3831/5-1, Ho 3831/6-1, and Ho 3831/7-1]. [ABSTRACT FROM AUTHOR]

We study a shipment consolidation problem commonly faced by companies that outsource logistics operations and operate in a commit-to-delivery mode. It involves delivering a given set of orders to their destinations by their committed due times using multiple shipping methods at the minimum total shipping and inventory cost. The shipping cost is generally nonlinear in shipping quantity and can be represented by a subadditive piecewise linear function. We investigate two shipping scenarios, one involving long-haul shipping only and the other involving joint long-haul and short-haul shipping. We develop analytical results and solution algorithms for the shipment consolidation problem under each shipping scenario. The problem under the first shipping scenario is shown to be strongly NP -hard. We find that a simple policy, called the First-Due-First-Delivered (FDFD) policy, which assigns orders with earlier delivery due times to shipping methods with earlier destination arrival times, is very effective. This policy enables us to develop a polynomial time algorithm, which not only solves the problem under the concave shipping cost structure optimally but also achieves a performance guarantee of 2 for the problem under the general subadditive shipping cost structure. For the problem under the second shipping scenario, we extend the FDFD policy for long-haul shipping and derive another policy, called the No-Wait policy, for short-haul shipping. We use these policies to develop a polynomial time algorithm and analyze its performance guarantee. Our computational experiments show that the algorithm significantly outperforms a commercial optimization solver, and its performance is robust across different parameter settings that reflect various practical situations. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The data and e-companion are available at https://doi.org/10.1287/mnsc.2023.4835. [ABSTRACT FROM AUTHOR]

Problem definition: Content promotion policies are crucial for online content platforms to improve content consumption and user engagement. However, traditional promotion policies generally neglect the diffusion effect within a crowd of users. In this paper, we study the candidate generation and promotion optimization (CGPO) problem for an online content platform, emphasizing the incorporation of the diffusion effect. Methodology/results: We propose a diffusion model that incorporates platform promotion decisions to characterize the adoption process of online content. Based on this diffusion model, we formulate the CGPO problem as a mixed-integer program with nonconvex and nonlinear constraints, which is proved to be NP-hard. Additionally, we investigate methods for estimating the diffusion model parameters using available online platform data and introduce novel double ordinary least squares (D-OLS) estimators. We prove the submodularity of the objective function for the CGPO problem, which enables us to find an efficient (1−1/e) -approximation greedy solution. Furthermore, we demonstrate that the D-OLS estimators are consistent and have smaller asymptotic variances than traditional ordinary least squares estimators. By utilizing real data from a large-scale video-sharing platform, we show that our diffusion model effectively characterizes the adoption process of online content. Compared with the policy implemented on the platform, our proposed promotion policy increases total adoptions by 49.90%. Managerial implications: Our research highlights the essential role of diffusion in online content and provides actionable insights for online content platforms to optimize their content promotion policies by leveraging our diffusion model. Funding: R. Zhang is grateful for the financial support from the Hong Kong Research Grants Council General Research Fund [Grants 14502722 and 14504123] and the National Natural Science Foundation of China [Grant 72293560/72293565]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0172. [ABSTRACT FROM AUTHOR]

Problem definition: We investigate the effects of waiting time, mainly due to production in a make-to-batch-order (MTBO) system, on consumer choice behavior, pricing, assortment, and model estimation. In an MTBO system, the seller/manufacturer first collects orders placed within a certain period of time into a batch and then starts the production process. After the production of all orders in a batch are complete, the products are then shipped and delivered to individual consumers. Because of batch production and shipping, the disutility of the waiting time exhibits negative externality. Methodology/results: We adopt the widely used multinomial logit (MNL) model as a starting point and incorporate the anticipated wait into consumers' decision making. The derived model, referred to as the MNL with wait model, is a solution of the rational expectation equilibrium and is capable of capturing the effects of negative externality induced by anticipated wait that may change the substitution patterns dramatically. We characterize the multiproduct price optimization problem under the MNL with wait model by establishing a one-to-one mapping between the price vector and the choice probability vector. We find that firms tend to charge higher prices for time-consuming items and charge lower prices for time-saving items compared with the optimal prices under the standard MNL model. In addition to price competition, we also study the Cournot-type competition, in which the decision is the choice probability for each firm and establish the existence of a Nash equilibrium. For assortment optimization, we identify mild conditions under which the optimality of revenue-ordered assortments still holds. However, the assortment problem under the MNL with wait model is generally NP-hard, so we develop approximation algorithms with performance guarantees and provide an easy-to-compute tight upper bound. Moreover, we develop an efficient maximum likelihood-based algorithm for model calibration and further conduct numerical studies to showcase the importance of incorporating disutility due to wait in estimation, pricing, and assortment planning problems. Managerial implications: The MNL with wait model can increase prediction accuracy for consumers' choice behavior especially when they are aware of the potential wait. Failure to take into account the effects of anticipated wait in firms' decision making may lead to substantial losses. Funding: The research of C. Ke is supported in part by the National Natural Science Foundation of China [Grant 72101113]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2020.0346. [ABSTRACT FROM AUTHOR]

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

Under an area-level random regression coefficient Poisson model, this article derives small area predictors of counts and proportions and introduces bootstrap estimators of the mean squared errors (MSEs). The maximum likelihood estimators of the model parameters and the mode predictors of the random effects are calculated by a Laplace approximation algorithm. Simulation experiments are implemented to investigate the behavior of the fitting algorithm, the predictors, and the MSE estimators with and without bias correction. The new statistical methodology is applied to data from the Spanish Living Conditions Survey. The target is to estimate the proportions of women and men under the poverty line by province. [ABSTRACT FROM AUTHOR]

Modeling longitudinal and survival data jointly offers many advantages such as addressing measurement error and missing data in the longitudinal processes, understanding and quantifying the association between the longitudinal markers and the survival events, and predicting the risk of events based on the longitudinal markers. A joint model involves multiple submodels (one for each longitudinal/survival outcome) usually linked together through correlated or shared random effects. Their estimation is computationally expensive (particularly due to a multidimensional integration of the likelihood over the random effects distribution) so that inference methods become rapidly intractable, and restricts applications of joint models to a small number of longitudinal markers and/or random effects. We introduce a Bayesian approximation based on the integrated nested Laplace approximation algorithm implemented in the R package R-INLA to alleviate the computational burden and allow the estimation of multivariate joint models with fewer restrictions. Our simulation studies show that R-INLA substantially reduces the computation time and the variability of the parameter estimates compared with alternative estimation strategies. We further apply the methodology to analyze five longitudinal markers (3 continuous, 1 count, 1 binary, and 16 random effects) and competing risks of death and transplantation in a clinical trial on primary biliary cholangitis. R-INLA provides a fast and reliable inference technique for applying joint models to the complex multivariate data encountered in health research. [ABSTRACT FROM AUTHOR]

The projected gradient method for matrix completion is generalized towards the higher-dimensional case of low-rank Tucker tensors. It is shown that an operation order rearrangement in the common projected gradient approach provides a complexity improvement. An even better algorithm complexity can be obtained by replacing the completion operator by a general operator that satisfies restricted isometry property; however, such a replacement transforms the completion algorithm into an approximation algorithm. [ABSTRACT FROM AUTHOR]

In 2015, Haviv introduced the Remote set problem (RSP) and studied the complexity of the covering radius problem (CRP), which is a classical problem in lattices. The RSP aims to identify a set containing a point that is sufficiently distant from a given lattice L . It introduced a new method for analyzing the complexity of CRP. An open question in RSP is whether we can obtain the approximation factor γ = 1 / 2 . This paper investigates this question and proposes a probabilistic polynomial-time algorithm for RSP with an approximation factor of 1 / 2 - 1 / (c λ n (p)) , where c ∈ Z + and λ n (p) is the n-th successive minima in lattice under l p -norm. For a given lattice L with rank n and positive integer d, our algorithm outputs a set S of size d in polynomial time. This set S includes a point at least (1 2 - 1 c λ n (p) ) ρ (p) (L) from lattice L with a probability greater than 1 - 1 / 2 d . Here, c is a positive integer and ρ (p) (L) denotes the covering radius of L in l p -norm( 1 ≤ p ≤ ∞ ). Based on this, we obtain that GAPCRP 2 + 1 / 2 O (n) belongs to the complexity class coRP, and we provide new reductions from GAPCRP to GAPCVP. [ABSTRACT FROM AUTHOR]

We investigate the seemingly untouched yet the most general parallel identical k-stage flow-shops scheduling, in which we are given an arbitrary number of indistinguishable k-stage flow-shops and a set of jobs each to be processed on one of the flow-shops, and the goal is to schedule these jobs to the flow-shops so as to minimize the makespan. Here the number k of stages is a fixed constant, but the number of flow-shops is part of the input. This scheduling problem is strongly NP-hard. To the best of our knowledge all previously presented approximation algorithms are for the special case where k = 2 , including a 17/6-approximation in 2017, a (2 + ϵ) -approximation in 2018, and the most recent polynomial-time approximation scheme in 2020; they all take advantage of the number of stages being two. We deal with an arbitrary constant k ≥ 3 , where the k-stage flow-shop scheduling is already strongly NP-hard. To define a configuration that summarizes the key information about the job assignments in a feasible schedule, we present novel concepts of big job type, big job assignment pair type, and flow-shop type. We show that the total number of distinct configurations is a polynomial in the input number of flow-shops. We then present how to compute a schedule for each configuration that assigns all the big jobs, followed by how to allocate all the small jobs into the schedule at a cost only a fraction of the makespan. These together lead to a polynomial-time approximation scheme for the problem. [ABSTRACT FROM AUTHOR]

In this work, we introduce a multi-vehicle (or multi-postman) extension of the classical Mixed Rural Postman Problem, which we call the Min–Max Mixed Rural Postmen Cover Problem (MRPCP). The MRPCP is defined on a mixed graph G = (V , E , A) , where V is the vertex set, E denotes the (undirected) edge set and A represents the (directed) arc set. Let F ⊆ E ( H ⊆ A ) be the set of required edges (required arcs). There is a nonnegative weight associated with each edge and arc. The objective is to determine no more than k closed walks to cover all the required edges in F and all the required arcs in H such that the weight of the maximum weight closed walk is minimized. By replacing closed walks with (open) walks in the MRPCP, we obtain the Min–Max Mixed Rural Postmen Walk Cover Problem (MRPWCP). The Min–Max Mixed Chinese Postmen Cover Problem (MCPCP) is a special case of the MRPCP where F = E and H = A . The Min–Max Stacker Crane Cover Problem (SCCP) is another special case of the MRPCP where F = ∅ and H = A For the MRPCP with the input graph satisfying the weakly symmetric condition, i.e., for each arc there exists a parallel edge whose weight is not greater than this arc, we devise a 27 4 -approximation algorithm. This algorithm achieves an approximation ratio of 33 5 for the SCCP with the weakly symmetric condition. Moreover, we obtain the first 5-approximation algorithm (4-approximation algorithm) for the MRPWCP (MCPCP) with the weakly symmetric condition. [ABSTRACT FROM AUTHOR]

We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time O(1)-approximation algorithm for the general case and a pseudopolynomial time O(1)-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Additionally, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation—that is, we allow to increase the size of the knapsack by a factor of 1+δ for some δ > 0 but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles. [ABSTRACT FROM AUTHOR]

Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fréchet distance. A shortcoming of existing map matching algorithms under the Fréchet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this article, we investigate map matching queries under the Fréchet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in O((pq)1-δ) query time for any δ > 0, where p and q are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this article. We show that for c-packed graphs, one can construct a data structure of \(\tilde{O}(cp)\) size that can answer (1+ε)-approximate map matching queries in \(\tilde{O}(c^4 q \log ^4 p)\) time, where \(\tilde{O}(\cdot)\) hides lower-order factors and dependence on ε. [ABSTRACT FROM AUTHOR]

This paper investigates the potential of active reconfigurable intelligent surfaces (RIS) to enhance wireless-powered communication networks (WPCNs), addressing the evolving connectivity needs of the internet of things (IoT). Active RIS, capable of amplifying and reflecting signals, offers a solution to surpass the limitations of passive RIS, such as double-fading attenuation, aiming to significantly improve network throughput and coverage. Our research focuses on exploiting the amplification capabilities of active RIS to boost the overall network sum throughput, engaging in a comprehensive optimization of critical network parameters, including time allocation, reflection coefficients, and phase shift matrices specific to active RIS. The formulated problem is non-convex and highly complex due to the coupling of optimization variables. We employed a successive convex approximation algorithm to solve the throughput maximization problem by converting the non-convex constraints into approximated convex constraints and solving them iteratively. Through extensive simulations, we demonstrate that our active RIS-assisted network substantially outperforms networks facilitated by passive RIS, marking a significant advancement in WPCN performance. These findings underscore the potential of active RIS technology in realizing the full capabilities of IoT connectivity. [ABSTRACT FROM AUTHOR]

We consider single-machine scheduling with a non-renewable resource. In this setting, we are given a set of jobs, each characterized by a processing time, a weight, and a resource requirement. At fixed points in time, certain amounts of the resource are made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that each job has enough resource available at the start of its processing. The objective that we consider is the minimization of the sum of weighted completion times. The main contribution of the paper is a PTAS for the case of 0 processing times ( 1 | r m = 1 , p j = 0 | ∑ w j C j ). In addition, we show strong NP-hardness of the case of unit resource requirements and weights ( 1 | r m = 1 , a j = 1 | ∑ C j ), thus answering an open question of Györgyi and Kis. We also prove that the schedule corresponding to the Shortest Processing Time First ordering provides a 3/2-approximation for the latter problem. Finally, we investigate a variant of the problem where processing times are 0 and the resource arrival times are unknown. We present a (4 + ϵ) -approximation algorithm, together with a (4 - ε) -inapproximability result, for any ε > 0 . [ABSTRACT FROM AUTHOR]

The surge in technology is driving demands for real-time interactive applications and high-speed transmissions, necessitating improved network throughput and energy efficiency (EE) for immersive experiences. The rise in industrial automation has led to higher connectivity needs, straining fifth-generation networks. Sixth-generation networks aim to address these demands, potentially maximizing throughput and EE through enhanced coverage. This paper introduces innovative techniques like phone user clustering-based downlink hybrid multiple access in unmanned aerial vehicle-assisted heterogeneous networks (HetNets) to jointly optimize phone user (PU) admission, cell association, throughput, and EE while ensuring PU fair association with cell (PUFAC) and quality of service (QoS), i.e., minimum rate requirement of PUs. An outer approximation algorithm solves the mixed integer non-linear programming (MINLP) optimization problem arising from the transformation of the concave fractional programming optimization problem using the Charnes-Cooper transformation. The method's effectiveness is assessed, showcasing its superiority over existing macro-cell-only networks and HetNets concerning throughput, EE, PU admission, PU-cell association, PUFAC, and QoS. [ABSTRACT FROM AUTHOR]

"Sim Sala Bim!" —Silvan, https://en.wikipedia.org/wiki/Silvan%5f(illusionist) Betweenness centrality is a popular centrality measure with applications in several domains and whose exact computation is impractical for modern-sized networks. We present SILVAN, a novel, efficient algorithm to compute, with high probability, accurate estimates of the betweenness centrality of all nodes of a graph and a high-quality approximation of the top-k betweenness centralities. SILVAN follows a progressive sampling approach and builds on novel bounds based on Monte Carlo Empirical Rademacher Averages, a powerful and flexible tool from statistical learning theory. SILVAN relies on a novel estimation scheme providing non-uniform bounds on the deviation of the estimates of the betweenness centrality of all the nodes from their true values and a refined characterisation of the number of samples required to obtain a high-quality approximation. Our extensive experimental evaluation shows that SILVAN extracts high-quality approximations while outperforming, in terms of number of samples and accuracy, the state-of-the-art approximation algorithm with comparable quality guarantees. [ABSTRACT FROM AUTHOR]

The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient compression algorithm. We demonstrate experimentally, that the new algorithm can achieve better compression ratios than standard JPEG compression with no visible loss of quality on many images. The mathematical core of this work shows that Binev's optimal tree approximation algorithm is applicable to image compression with high probability, when we assume small additive Gaussian noise on the pixels of the image. [ABSTRACT FROM AUTHOR]

In this paper, we consider the $ k $-prize-collecting minimum power cover problem ($ k $-PCPC), which is defined as follows. Given a set $ X = \{x_1, x_2, $ $ \cdots, x_n\} $ of $ n $ points and a set $ S = \{s_1, s_2, \cdots, s_m\} $ of $ m $ sensors on the plane $ \mathbb{R}^2 $, an integer $ k $ satisfying $ 0 \leq k \leq n $ and a penalty function $ \pi: X \rightarrow \mathbb{R}_0^{+} $, each sensor $ s\in S $ can adjust its power $ p(s) $ and the covering range, which is a disk of radius $ r(s) $ satisfying $ p(s) = c\cdot r(s)^\alpha $, where $ \alpha\geq 1 $ and $ c>0 $ are constants. We are asked to determine a set of disks $ \mathcal{F} $ such that at least $ k $ points are covered, where the center of any disk in $ \mathcal{F} $ is a sensor. The objective is to minimize the total power of the set of disks $ \mathcal{F} $ plus the penalty of $ R $, where $ R $ is the set of points that are not covered by $ \mathcal{F} $. It has been proven that this problem is NP-hard and admits a $ 3^\alpha $-approximation algorithm. By using some properties of relaxed independent sets and combining the guessing technique and the primal-dual method, we design an $ O(\alpha) $-approximation algorithm. Numerical experiments are tested to evaluate the performance of the $ 3^\alpha $-approximation algorithm and our new algorithm. [ABSTRACT FROM AUTHOR]

In solving the whole process of interaction between soft rock and yielding support in high-stress environments in tunnels using mechanical analysis methods, it is challenging to simultaneously satisfy both displacement coordination and static equilibrium at the contact surface between the rock and the support structure. This paper, based on the mechanical analysis of rock and rigid support, considers the impact of the circumferential installation of yielding elements on radial displacement, and proposes displacement approximation and support force approximation methods using displacement coordination and static equilibrium as approximation conditions. The study fits curves of numerical simulation results and laboratory test results of yielding elements, and attempts to directly use the laboratory test data set of yielding elements as computational data. By calculating two circular tunnel examples and comparing the effects of the trisection method, bisection method, and substitution method on the convergence of the displacement approximation method, the effectiveness of the methods proposed in this paper is verified. The research results show that the two approximation algorithms proposed in this paper have good accuracy and reliability in calculating the relative displacement of rock and yielding support structure contact surfaces, and the support force of yielding support. The bisection method outperforms the trisection and substitution methods in terms of stability and convergence. However, there are certain limitations in this study, such as the effectiveness of the algorithm may be influenced by geological conditions; the complexity of actual geological conditions may exceed the assumptions of the current rock-support mechanical analysis model. [ABSTRACT FROM AUTHOR]

We abstract and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph G=(V,E) and a set of demand pairs P⊆VxV, a reachability preserver is a sparse subgraph H that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an n-node graph and demand pairs of the form P⊆SxV for a small node subset S, there is always a reachability preserver on O(n+√n/P//S/) edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which O(n) size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the O(n0.6+ε) of Chlamatac, et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534-443] (n4/7+ε). [ABSTRACT FROM AUTHOR]

In this paper, we study the classic submodular maximization problem subject to a group equality constraint under both nonadaptive and adaptive settings. It is shown that the utility function of many machine learning applications, including data summarization, influence maximization in social networks, and personalized recommendation, satisfies the property of submodularity. Hence, maximizing a submodular function subject to various constraints can be found at the heart of many of those applications. On a high level, submodular maximization aims to select a group of most representative items (e.g., data points). However, the design of most existing algorithms does not incorporate the fairness constraint, leading to underrepresentation or overrepresentation of some particular groups. This motivates us to study the submodular maximization problem with group equality, in which we aim to select a group of items to maximize a (possibly nonmonotone) submodular utility function subject to a group equality constraint. To this end, we develop the first constant-factor approximation algorithm for this problem. The design of our algorithm is robust enough to be extended to solving the submodular maximization problem under a more complicated adaptive setting. Moreover, we further extend our study to incorporating a global cardinality constraint and other fairness notations. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0384. [ABSTRACT FROM AUTHOR]