Submodular Function Maximization Research Articles

Predictive Service Provisioning With Online Learning in Wireless Edge Networks

Mobile Edge Computing (MEC) technology can be implemented at cellular base stations, enabling flexible and configurable provisions of services for mobile users to access. Nevertheless, the conventional solutions mainly focus on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">statical</i> service provisioning, which ignores the dynamic nature of the arriving service requests. In this work, we first conduct comprehensive data-driven observations on over 4 million service requests throughout 9,800 base stations. Our key findings suggest that users' demands intrinsically exhibit spatial and temporal patterns, which inevitably lead to performance degradation in statical service provisioning. Motivated by that, we design and implement MobiEdge, a predictive service provisioning system with online learning in wireless edge networks. We propose a graph embedding learning-based model for representation learning, thus to achieve accurate request prediction at different base stations. Then, based on the prediction of incoming service requests, we study the service provisioning reconfiguration problem, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , how to jointly optimize service placement and corresponding request scheduling across dual timescales, under constraints of network resources and the total budget. By leveraging the submodular technique, we transform the research issue into a submodular function maximization problem under the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -independence system constraint, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> is a positive constant related to the ratio of coefficients in constraint conditions. On this basis, we propose a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1/(1+q)$</tex-math></inline-formula> approximation algorithm with rigorous theoretical analysis on the bounded maximum utility. Extensive trace-driven evaluations are conducted over networks of different scales, and MobiEdge shows remarkable performance enhancements by achieving the accuracy of up to 98% in service prediction and an average utility of 92.9% to the optimal solution in service provisioning.

Approximation algorithm of maximizing non-monotone non-submodular functions under knapsack constraint

The maximization of non-negative monotone submodular functions under a certain constraint is intensively studied. However, there are few works considering the maximization of non-monotone non-submodular functions, which even might be negative. These functions also have many applications, such as optimal marketing for revenue maximization over social networks, budget allocation problems, and Epidemic transmission, etc. In our paper, we discuss the maximization of the non-monotone non-submodular functions under a knapsack constraint, which even might be negative, and explore the performance under the greedy algorithm. We obtain some improved approximate ratios, for the functions with different properties, such as the non-negative weak-monotone weak-submodular functions, and the weak-submodular weak-supermodular functions that might be negative. Moreover, we generalize the results to the functions with weak subadditivity and curvature.

: Channel Pruning via Class-Aware Trace Ratio Optimization.

Deep convolutional neural networks are shown to be overkill with high parametric and computational redundancy in many application scenarios, and an increasing number of works have explored model pruning to obtain lightweight and efficient networks. However, most existing pruning approaches are driven by empirical heuristics and rarely consider the joint impact of channels, leading to unguaranteed and suboptimal performance. In this article, we propose a novel channel pruning method via class-aware trace ratio optimization () to reduce the computational burden and accelerate the model inference. Utilizing class information from a few samples, measures the joint impact of multiple channels by feature space discriminations and consolidates the layerwise impact of preserved channels. By formulating channel pruning as a submodular set function maximization problem, solves it efficiently via a two-stage greedy iterative optimization procedure. More importantly, we present theoretical justifications on convergence of and performance of pruned networks. Experimental results demonstrate that achieves higher accuracy with similar computation cost or lower computation cost with similar accuracy than other state-of-the-art channel pruning algorithms. In addition, because of its class-aware property, is suitable to prune efficient networks adaptively for various classification subtasks, enhancing handy deployment and usage of deep networks in real-world applications.

Maximizing Charging Efficiency With Fresnel Zones

Benefitting from the discovery of wireless power transfer (WPT) technology, the wireless rechargeable sensor network (WRSN) has become a promising way for lifetime extension for wireless sensor networks. In practical WRSN scenarios, obstacles can be found almost everywhere. Most state-of-the-art researches believe that obstacles will always degrade signal strength, and omit the influence of obstacles for simplifying the computation process. However, overlooking the positive impacts of obstacles on signal propagation is inconsistent with the intrinsic features of electromagnetic waves. To address this issue, in this paper, we explore the wireless signal propagation process and provide a theoretical charging model to enhance the charging efficiency by leveraging obstacles. Through utilizing the concept of the Fresnel Zone model, we re-formalize the wireless charging model and discretize the charging area and charging time to determine the best charging locations as well as charging duration. We model the charging <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> fficiency <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> aximization with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> bstacles (EMO) problem as a submodular function maximization problem and propose a cost-efficient algorithm to solve it. Finally, test-bed experiments and extensive simulations are both conducted to verify that our schemes outperform baseline algorithms by 33.46% on average in charging efficiency improvement.

Learning automata-accelerated greedy algorithms for stochastic submodular maximization

Submodular function maximization is a typical combinatorial optimization problem that arises in many areas of computer science, such as data summarization, batch mode active learning, and recommendations. We focus on the stochastic submodular maximization (SSM) problems, where an unknown monotonic submodular function is to be maximized under cardinality constraints. Previous research commonly relied on the Monte-Carlo method to assess the stochasticity in submodular functions, which is challenged by the accuracy–efficiency dilemma. Achieving high accuracy requires a large number of Monte-Carlo simulations, while efficiency necessitates the minimization of simulation costs. In this paper, we propose a new family of greedy algorithms based on learning automata (LA) for SSM problems. By exploring the solution space more efficiently, the proposed LA-based greedy family accelerates conventional greedy algorithms. Instead of massively evaluating the expected benefits of each element indiscriminately, an LA-based learning strategy is designed to quickly identify the optimal element among all candidates. Based on the strategy, we present a primitive greedy algorithm and two improved variants, along with proofs of the same approximation guarantee as conventional greedy algorithms. We conducted experiments on three representative SSM tasks, nonparametric learning for structured data, exemplar-based clustering for unstructured data, and influence maximization for graph data. The results verified the advantage of the proposed LA-based greedy family against previous greedy algorithms.

Research on the effect of animation IP dissemination based on social network analysis

Abstract This paper constructs a GCN-based animation IP propagation algorithm based on social network analysis. Through the analysis of social network influence, a topology-based social network influence model is constructed. Combined with the ranking algorithm, the influence size of animation IP nodes is judged. The LDA model was used to construct a huge document set with the character IPs in the animation as documents. The clustering algorithm is used to classify the propagation effect of animation IP nodes, and the interval of animation IP propagation maximization is found by combining the submodular function maximization. The feature extraction of animation IP propagation effect maximization is performed by graph convolutional neural network. Evaluation indexes are constructed to assess the spreading ability of animation IP under social networks. The results show that the spreading power of animation IP with negative celebrity endorsement is −0.1, and the spreading effect of animation IP with positive communication content is 0.6.

Constraint generation approaches for submodular function maximization leveraging graph properties

Constraint generation approaches for submodular function maximization leveraging graph properties

Robust Wireless Rechargeable Sensor Networks

Wireless rechargeable sensor networks have become a hot research issue as it can overcome the limited energy bottleneck of wireless sensor networks owing to the recent breakthrough of wireless power transfer technology. Though network lifetime is prolonged and sensor nodes can sustain immortally, the issue of network robustness is overlooked, yielding most theoretical work unsuitable for practical applications when confronting with unpredictable packet loss. In this paper, we address the network robustness issue by maximizing the charging utility in a risk-averse view. First, we build a risk-averse model based on the concept of CVaR (Conditional Value at Risk), which trades-off charging utility and risk aversion for quantifying robustness. Then, we propose a spatial discretization scheme to construct a charging route for mobile charger, which can reduce computational overhead. Afterwards, a path optimization scheme is designed to further improve the charging utility. We convert the original problem into the submodular function maximization problem and propose a method with a performance guarantee while maximizing the system robustness. Finally, testbed experiments and simulations are conducted, and the results demonstrate that our schemes outperform comparison algorithms by at least 22.4% in effective energy in the presence of risks to guarantee system robustness.

Optimizing Comprehensive Cost of Charger Deployment in Multi-hop Wireless Charging

The multi-hop wireless charging technology has attracted a lot of attention, as it largely extends the charging range of chargers. Different from the existing work with single cost optimization, the objective of this article is to optimize the comprehensive cost, which is the combination of energy cost and deployment cost. We decompose the target problem into two sub-problems. The first sub-problem aims to minimize the deployment cost with energy capacity constraints. The proposed algorithm follows the greedy strategy, where the subset of sensor nodes for any charger is determined by finding the capacitated minimum spanning tree. The second sub-problem, which aims to maximize the reduction of comprehensive cost by adding chargers to the solution of the first sub-problem, is proved to be an unconstrained submodular set function maximization problem and can be solved by a 1/2-approximation randomized linear time algorithm for its equivalent problem. Through extensive simulations, we demonstrate that the proposed solution can reduce the comprehensive cost by 57.55% comparing with the benchmark algorithms.

Enabling Cost-effective Wireless Data Collection by Piggybacking on Delivery Drones in Agriculture

Drones have drawn considerable attention as the agents in wireless data collection for agricultural applications, by virtue of their three-dimensional mobility and dominant line-of-sight communication channels. Existing works mainly exploit dedicated drones via deployment and maintenance, which is insufficient regarding resource and cost-efficiency. In contrast, leveraging existing delivery drones for the data collection on their way of delivery, called delivery drones’ piggybacking , is a promising solution. For achieving such cost-efficiency, drone scheduling inevitably stands in front, but the delivery missions involved have escalated it to a wholly different and unexplored problem. As an attempt, we first survey 514 delivery workers and conduct field experiments; noticeably, the collection cost, which mostly comes from the energy consumption of drones’ piggybacking, is determined by the decisions on package-route scheduling and data collection time distribution . Based on such findings, we build a new model that jointly optimizes these two decisions to maximize data collection amount, subject to the collection budget and delivery constraints. Further model analysis finds it a Mixed Integer Non-Linear Programming problem, which is NP-hard. The major challenge stems from interdependence entangling the two decisions. For this point, we propose Delta , a \(\frac{1}{9+\delta } \) -approximation delivery drone scheduling algorithm. The key idea is to devise an approximate collection time distribution scheme leveraging energy slicing, which transforms the complex problem with two interdependent variables into a submodular function maximization problem only with one variable . The theoretical proofs and extensive evaluations verify the effectiveness and the near-optimal performance of Delta .

DCEC: D2D-Enabled Cost-Aware Cooperative Caching in MEC Networks

Various kinds of powerful intelligent mobile devices (MDs) need to access multimedia content anytime and anywhere, which places enormous pressure on mobile wireless networks. Fetching content from remote sources may introduce overly long accessing delays, which will result in a poor quality of experience (QoE). In this article, we considered the advantages of combining mobile/multi-access edge computing (MEC) with device-to-device (D2D) technologies. We propose a D2D-enabled cooperative edge caching (DCEC) architecture to reduce the delay of accessing content. We designed the DCEC caching management scheme through the maximization of a monotone submodular function under matroid constraints. The DCEC scheme includes a proactive cache placement algorithm and a reactive cache replacement algorithm. Thus, we obtained an optimal content caching and content update, which minimized the average delay cost of fetching content files. Finally, simulations compared the DCEC network architecture with the MEC and D2D networks and the DCEC caching management scheme with the least-frequently used and least-recently used scheme. The numerical results verified that the proposed DCEC scheme was effective at improving the cache hit ratio and the average delay cost. Therefore, the users’ QoE was improved.

Distributed strategy selection: A submodular set function maximization approach

Joint utility-maximization problems for multi-agent systems often should be addressed by distributed strategy-selection formulation. Constrained by discrete feasible strategy sets, these problems are broadly formulated as NP-hard combinatorial optimization problems. In many cases, these problems can be cast as constrained submodular set function maximization problems, which also belong to the NP-hard domain of problems. A prominent example is the problem of multi-agent mobile sensor dispatching over a discrete domain. This paper considers a class of submodular optimization problems that consist of maximization of a monotone and submodular set function subject to a partition matroid constraint over a group of networked agents that communicate over a connected undirected graph. We work with the value oracle model. Consequently, the only access of the agents to the utility function is through a black box that returns the utility function value given a specific strategy set. We propose a distributed suboptimal polynomial-time algorithm that enables each agent to obtain its respective strategy via local interactions with its neighboring agents. Our solution is a fully distributed gradient-based algorithm using the submodular set functions’ multilinear extension followed by a distributed stochastic Pipage rounding procedure. This algorithm results in a strategy set that when the team utility function is evaluated at the worst case, the utility function value is in 1c(1−e−c−O(1/T)) of the optimal solution with c being the curvature of the submodular function. An example demonstrates our results.

Profit maximization in social networks and non-monotone DR-submodular maximization

In this paper, we study the non-monotone DR-submodular function maximization over integer lattice. Functions over integer lattice have been defined submodular property that is similar to submodularity of set functions. DR-submodular is a further extended submodular concept for functions over the integer lattice, which captures the diminishing return property. Such functions find many applications in machine learning, social networks, wireless networks, etc. The techniques for submodular set function maximization can be applied to DR-submodular function maximization, e.g., the double greedy algorithm has a 1/2-approximation ratio, whose running time is O(nB), where n is the size of the ground set, B is the integer bound of a coordinate. In our study, we design a 1/2-approximate binary search double greedy algorithm, and we prove that its time complexity is O(nlog⁡B), which significantly improves the running time. Specifically, we consider its application to the profit maximization problem in social networks with a bipartite model, the goal of this problem is to maximize the net profit gained from a product promoting activity, which is the difference of the influence gain and the promoting cost. We prove that the objective function is DR-submodular over integer lattice. We apply binary search double greedy algorithm to this problem and verify the effectiveness.

A Novel Interactive Image Segmentation Algorithm Based on Maximization of Submodular Function

In this paper, an efficient interactive image segmentation method based on maximization of submodular function under user’s scribble constraint is presented. The problem of interactive image segmentation is formulated as the maximum entropy rate under user’s constraints. The objective function is submodular, and we solve the constrained submodular function maximization by incorporating a new data structure and some aggregating rules into the greedy algorithm. The main steps of our algorithm are as follows. First, the pixels scribbled by the user are clustered separately as target foreground and background clusters. Second, in the process of greedy algorithm, unscribbled pixels are aggregated to the corresponding target cluster according to the proposed aggregating rules. Finally, the segmentation result is presented by the two target clusters. The experiments and comparisons on three standard benchmarks show that our method has good performance. Our method is straightforward and efficient, and the time complexity of our method is between linear and polynomial. Furthermore, we analyze the influence of different scribbles, and propose some optimal scribble strategies.

QoE-Driven Adaptive Streaming for Point Clouds

With increasing popularity of virtual reality and augmented reality, application of point clouds is in critical demand as it enables users to freely navigate in an immersive scene with six degrees of freedom. However, point clouds usually comprise large amounts of data, and are thus difficult to stream in bandwidth-constrained networks. It is therefore important, yet challenging, to efficiently stream the resource-intensive point clouds, such that the user's quality of experience (QoE) is guaranteed on a high-level but with a low bandwidth consumption. To this end, we propose a QoE-driven adaptive streaming approach for the tile-based point cloud transmission, to maximize the user's QoE while reducing the transmission redundancy. By exploiting the perspective projection, we specifically model the QoE of a 3D tile as a function of the bitrate of its representation, user's view frustum and spatial position, occlusion between tiles, and the resolution of rendering device. Based on this QoE model, we then formulate the QoE-optimized rate adaptation problem as a multiple-choice knapsack problem, which allocates bitrates for different tiles under a given transmission capacity. It is equivalently converted to a submodular function maximization problem subject to knapsack constraints, and solved by a practical greedy-based algorithm with a theoretical worst-case performance guarantee. The proposed algorithm is able to achieve a near-optimal performance, but with a very low computational complexity. Experimental results further demonstrate superiority of the proposed rate adaptation algorithm over existing schemes, in terms of both user's visual quality and transmission efficiency.

A semi-automated material exploration scheme to predict the solubilities of tetraphenylporphyrin derivatives

Acceleration of material discovery has been tackled by informatics and laboratory automation. Here we show a semi-automated material exploration scheme to modelize the solubility of tetraphenylporphyrin derivatives. The scheme involved the following steps: definition of a practical chemical search space, prioritization of molecules in the space using an extended algorithm for submodular function maximization without requiring biased variable selection or pre-existing data, synthesis & automated measurement, and machine-learning model estimation. The optimal evaluation order selected using the algorithm covered several similar molecules (32% of all targeted molecules, whereas that obtained by random sampling and uncertainty sampling was ~7% and ~4%, respectively) with a small number of evaluations (10 molecules: 0.13% of all targeted molecules). The derived binary classification models predicted ‘good solvents’ with an accuracy >0.8. Overall, we confirmed the effectivity of the proposed semi-automated scheme in early-stage material search projects for accelerating a wider range of material research.

Analyzing Residual Random Greedy for monotone submodular maximization

Residual Random Greedy (RRGreedy) is a natural randomized version of the greedy algorithm for submodular maximization. It was introduced to address non-monotone submodular maximization [1] and plays an important role in the deterministic algorithm for monotone submodular maximization that beats the (1/2)-factor barrier [2]. In this work, we analyze RRGreedy for monotone submodular functions along two fronts: (1) For matroid constrained maximization of monotone submodular functions with bounded curvature α, we show that RRGreedy achieves a (1/(1+α))-approximation in the worst-case (i.e., irrespective of the randomness in the algorithm). In particular, this implies that it achieves a (1/2)-approximation in the worst-case (not just in expectation). (2) We generalize RRGreedy to k matroid intersection constraints and show that the generalization achieves a (1/(k+1))-approximation in expectation relative to the optimum value of the Lovász relaxation over the intersection of k matroid polytopes. Our results suggest that RRGreedy is at least as good as Greedy for matroid and matroid intersection constraints.

Efficient Streaming Algorithms for Maximizing Monotone DR-Submodular Function on the Integer Lattice

In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most submodular functions are specified in a set function. Meanwhile, recent advancements have been studied for maximizing a diminishing return submodular (DR-submodular) function on the integer lattice. Because plenty of publications show that the DR-submodular function has wide applications in optimization problems such as sensor placement impose problems, optimal budget allocation, social network, and especially machine learning. In this research, we propose two main streaming algorithms for the problem of maximizing a monotone DR-submodular function under cardinality constraints. Our two algorithms, which are called StrDRS1 and StrDRS2, have (1/2−ϵ), (1−1/e−ϵ) of approximation ratios and O(nϵlog(logBϵ)logk), O(nϵlogB), respectively. We conducted several experiments to investigate the performance of our algorithms based on the budget allocation problem over the bipartite influence model, an instance of the monotone submodular function maximization problem over the integer lattice. The experimental results indicate that our proposed algorithms not only provide solutions with a high value of the objective function, but also outperform the state-of-the-art algorithms in terms of both the number of queries and the running time.

Improved algorithms for non-submodular function maximization problem

The concept of submodularity finds wide applications in data science, artificial intelligence, and machine learning, providing a boost to the investigation of new ideas, innovative techniques, and creative algorithms to solve different submodular optimization problems arising from a diversity of applications. However pure submodular or supermodular problems only represent a small portion of the problems we are facing in real life applications. The main focus of this work is to consider a non-submodular function maximization problem subject to a cardinality constraint, where the objective function is the sum of a monotone γ-weakly submodular function and a supermodular function. This problem includes some previously studied problems as special cases, such as the submodular+supermodular maximization problem when γ=1, and the γ-weakly submodular function maximization problem when the supermodular function is void. We present greedy algorithms for this generalized problem under both offline and streaming models, improving existing results.

A new performance bound for submodular maximization problems and its application to multi-agent optimal coverage problems

Several important problems in multi-agent systems, machine learning, data mining, scheduling and others, may be formulated as set function maximization problems subject to cardinality constraints. In such problems, the set (objective) functions of interest often have monotonicity and submodularity properties. Hence, the class of monotone submodular set function maximization problems has been widely studied in the literature. Owing to its challenging nature, almost all existing solutions for this class of problems are based on greedy algorithms. A seminal work on this topic has exploited the submodularity property to prove a (1-1/e) performance bound for such greedy solutions. More recent literature on this topic has been focused on exploiting different curvature properties to establish improved (tighter) performance bounds. However, such improvements come at the cost of enforcing additional assumptions and increasing computational complexity while facing significant inherent limitations. In this paper, first, a brief review of existing performance bounds is provided. Then, a new performance bound that does not require any additional assumptions and is both practical and computationally inexpensive is proposed. In particular, this new performance bound is established based on a series of upper bounds derived for the objective function that can be computed in parallel with the execution of the greedy algorithm. Finally, to highlight the effectiveness of the proposed performance bound, extensive numerical results obtained from a well-known class of multi-agent coverage problems are provided.