Submodular Function Maximization Research Articles

A near-optimal content placement in D2D underlaid cellular networks

The rapid growth of mobile data traffic, especially video streaming traffic, places a serious burden on cellular networks. D2D caching has emerged as a promising paradigm to alleviate network congestions, in which contents are cached at user terminals proactively and then shared among neighbor requesting users via D2D communications. In this paper, we study the content placement problem to maximize cache hit probability (i.e., the probability that contents requested by users are successfully served by neighbor helpers through D2D communications) in D2D underlaid cellular networks. To decide where to cache and which contents to be pushed, we formulate our problem considering D2D communication probability of helpers and preference probability of requesting users. Then our problem is proved to be a submodular function maximization problem under a matroid constraint. To solve this problem, we present an improved greedy algorithm which can achieve an approximation guarantee of ${\min \limits } (1, \linebreak \frac {1}{v_{0}+\frac {1}{t_{min}}} )$ , based on the classic $\frac {1}{2}$ -approximation algorithm. Simulation results show that our proposed scheme achieves higher cache hit probability and lower energy consumption compared with existing caching schemes.

Social-Aware Collaborative Caching Based on User Preferences for D2D Content Sharing

With rapid growth of content demands, device-to-device (D2D) content sharing is exploited to effectively improve the service quality of users. Considering the limited storage space and various content demands of users, caching schemes are significant. However, most of them ignore the influence of the asynchronous content reuse and the selfishness of users. In this work, the user preferences are defined by exploiting the user-oriented content popularity and the current caching situation, and further, we propose the social-aware rate, which compre-hensively reflects the achievable contents download rate affected by the social ties, the caching indicators, and the user preferences. Guided by this, we model the collaborative caching problem by making a trade-off between the redundancy of caching contents and the cache hit ratio, with the goal of maximizing the sum of social-aware rate over the constraint of limited storage space. Due to its intractability, it is computationally reduced to the maximi-zation of a monotone submodular function, subject to a matroid constraint. Subsequently, two social-aware collaborative caching algorithms are designed by leveraging the standard and continuous greedy algorithms respectively, which are proved to achieve different approxima-tion ratios in unequal polynomial-time. We present the simulation results to illustratethe performance of our schemes.

Mobile Edge Cloud-Based Industrial Internet of Things: Improving Edge Intelligence With Hierarchical SDN Controllers

The industrial Internet of Things (IIoT), which integrates the key technologies of industrial communication, computing, and control, can implement flexible management and dynamic scheduling for manufacturing resources. To improve the edge intelligence of the IIoT, this article proposes a novel IIoT architecture with a hierarchical control structure in the mobile edge cloud (MEC). Massive remote radio heads (RRHs) are partitioned into several clusters, and each cluster is equipped with one or more servers for creating virtual machines (VMs) to execute the processing tasks of IIoT devices. The proposed IIoT architecture separates the control plane from the data plane based on software-defined networking (SDN). The hierarchical controllers improve the flexibility and intelligence of the control plane, while the RRHs and servers in the same cluster, forming an MEC-based radio access network (RAN) and supporting RAN function split, improve the scalability and cooperative gain of the data plane. A deep-learning technique is implemented in the MEC to further enhance the edge intelligence. In addition, we design two control schemes, one centralized and the other distributed, which provide a tradeoff between performance and overhead. Finally, aiming to minimize the system delay, we formulate a joint optimization problem of task scheduling, VM assignment, RRH allocation, and RAN function split as an example. To find solutions, a heuristic algorithm is proposed based on submodular function maximization.

CHASE: Charging and Scheduling Scheme for Stochastic Event Capture in Wireless Rechargeable Sensor Networks

In this paper, we consider the scenario in which a mobile charger (MC) periodically travels within a sensor network to recharge the sensors wirelessly. We design joint charging and scheduling schemes to maximize the Quality of Monitoring (QoM) for stochastic events, which arrive and depart according to known probability distributions of time. Information is considered captured if it is sensed by at least one sensor. We focus on two closely related research issues, i.e., how to choose the sensors for charging and decide the charging time for each of them, and how to schedule the sensors’ activation schedules according to their received energy. We formulate our problem as the maximum QoM CHA rging and S ch E duling problem (CHASE). We first ignore the MC's travel time and study the resulting relaxed version of the problem, which we call CHASE-R. We show that both CHASE and CHASE-R are NP-hard. For CHASE-R, we prove that it can be formulated as a submodular function maximization problem, which allows two algorithms to achieve $1/6$ 1 / 6 - and $1/(4 + \epsilon)$ 1 / ( 4 + e ) -approximation ratios. Then, for CHASE, we propose approximation algorithms to solve it by extending the CHASE-R results. We conduct simulations to validate our algorithm design.

The Impact of Information in Distributed Submodular Maximization

The maximization of submodular functions is an NP-Hard problem for certain subclasses of functions, for which a simple greedy algorithm has been shown to guarantee a solution whose quality is within 1/2 of the optimal. When this algorithm is implemented in a distributed way, agents sequentially make decisions based on the decisions of all previous agents. This work explores how limited access to the decisions of previous agents affects the quality of the solution of the greedy algorithm. Specifically, we provide tight upper and lower bounds on how well the algorithm performs, as a function of the information available to each agent. Intuitively, the results show that performance roughly degrades proportionally to the size of the largest group of agents which make decisions independently. Additionally, we consider the case where a system designer is given a set of agents and a global limit on the amount of information that can be accessed. Our results show that the best designs partition the agents into equally-sized sets and allow agents to access the decisions of all previous agents within the same set.

Trajectory Planning for Data Collection of Energy-Constrained Heterogeneous UAVs.

Nowadays, Unmanned Aerial Vehicles (UAVs) have received growing popularity in the Internet-of-Things (IoT) which often deploys many sensors in a relatively wide region. Since the battery capacity is limited, sensors cannot transmit over a long distance. It is necessary for designing efficient sensor data collection mechanisms to prolong the lifetime of the IoT and enhance data collection efficiency. In this paper, we consider a UAV-enabled data collection scenario, where multiple heterogeneous UAVs with different energy constraints are employed to collect data from sensors. The value of data depends on the importance of the monitoring area of the sensor and the freshness of collected data. Our objective is to maximize the data collection utility by jointly optimizing the communication scheduling and trajectory of each UAV. The data collection utility is determined by the amount and value of the collected data. This problem is a variant of multiple knapsack problem, which is a classical NP-hard problem. First, we transform the initial problem into a submodular function maximization problem under energy constraints, and then we design a novel trajectory planning algorithm to maximize the data collection utility, while accounting for different values of data and different energy constraints of heterogeneous UAVs. Finally, under different network settings, the performance of the proposed trajectory planning algorithm is evaluated via extensive simulations. The results show that the proposed algorithm can obtain maximum data collection utility.

Approximating Robust Parameterized Submodular Function Maximization in Large-Scales

We study a robust parameterized submodular function maximization inspired by [Mitrović, S, I Bogunovic, A Norouzi-Fard and Jakub Tarnawski (2017). Streaming robust submodular maximization: A partitioned thresholding approach. In Proc. NIPS, pp. 4560–4569] and [Bogunovic, I, J Zhao and V Cevher (2018). Robust maximization of nonsubmodular objectives. In Proc. AISTATS, pp. 890–899]. In our setting, given a parameterized set function, there are two additional twists. One is that elements arrive in a streaming style, and the other is that there are at most [Formula: see text] items deleted from the algorithm’s memory when the stream is finished. The goal is to choose a robust set from the stream such that the robust ratio is maximized. We propose a two-phase algorithm for maximizing a normalized monotone robust parameterized submodular function with a cardinality constraint and show the robust ratio is close to a constant as [Formula: see text]. In the end, we empirically demonstrate the performance of our algorithm on deletion robust support selection problem.

Greedy Maximization of Functions with Bounded Curvature under Partition Matroid Constraints

We investigate the performance of a deterministic GREEDY algorithm for the problem of maximizing functions under a partition matroid constraint. We consider non-monotone submodular functions and monotone subadditive functions. Even though constrained maximization problems of monotone submodular functions have been extensively studied, little is known about greedy maximization of non-monotone submodular functions or monotone subadditive functions.
 We give approximation guarantees for GREEDY on these problems, in terms of the curvature. We find that this simple heuristic yields a strong approximation guarantee on a broad class of functions.
 We discuss the applicability of our results to three real-world problems: Maximizing the determinant function of a positive semidefinite matrix, and related problems such as the maximum entropy sampling problem, the constrained maximum cut problem on directed graphs, and combinatorial auction games.
 We conclude that GREEDY is well-suited to approach these problems. Overall, we present evidence to support the idea that, when dealing with constrained maximization problems with bounded curvature, one needs not search for (approximate) monotonicity to get good approximate solutions.

Submodular Optimization over Streams with Inhomogeneous Decays

Cardinality constrained submodular function maximization, which aims to select a subset of size at most k to maximize a monotone submodular utility function, is the key in many data mining and machine learning applications such as data summarization and maximum coverage problems. When data is given as a stream, streaming submodular optimization (SSO) techniques are desired. Existing SSO techniques can only apply to insertion-only streams where each element has an infinite lifespan, and sliding-window streams where each element has a same lifespan (i.e., window size). However, elements in some data streams may have arbitrary different lifespans, and this requires addressing SSO over streams with inhomogeneous-decays (SSO-ID). This work formulates the SSO-ID problem and presents three algorithms: BASIC-STREAMING is a basic streaming algorithm that achieves an (1/2 − ɛ) approximation factor; HISTAPPROX improves the efficiency significantly and achieves an (1/3 − ɛ) approximation factor; HISTSTREAMING is a streaming version of HISTAPPROX and uses heuristics to further improve the efficiency. Experiments conducted on real data demonstrate that HISTSTREAMING can find high quality solutions and is up to two orders of magnitude faster than the naive GREEDY algorithm.

Mobility-Aware Content Placement for Device-to-Device Caching Systems

User mobility has a large effect on optimal content placement in device-to-device (D2D) caching networks. Since a typical user can communicate neighboring users who stay in the D2D communication area of the typical user, the optimal content placement should be changed according to the user mobility. Under consideration of randomness of incoming and outgoing users, we formulate an optimization problem to minimize the average data load of a BS. It is proved that minimization of the average data load of a BS can be transformed to maximization of a monotonic submodular function with a matroid constraint, for which a greedy algorithm can find near-optimal solutions. Moreover, when motions of neighboring users are rapid, the optimal content placement is derived in closed-form, aided by reasonable approximation and relaxation. In the high mobility regime, the optimal content placement is shown to cache partial amounts of the most popular contents.

Online Submodular Maximization with Preemption

Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one by one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is to maximize a submodular function over the set of elements in the solution. We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a 1/ e -competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most k elements in its solution.

Edge Caching and Resource Allocation Scheme of Downlink Cloud Radio Access Networks With Fronthaul Compression

With the rapid development of real-time applications in the Internet of Things, people have paid more and more attention on the traffic delay. Cloud radio access networks (C-RANs) are seen as a novel network architecture with significant advantages in reducing latency on control and data planes. In this paper, we aim to minimize the average user delay in a downlink C-RAN with a hierarchical structure of virtual controllers and high-speed but limited-capacity fronthaul links, by simultaneously considering edge caching, user association, and computing resource allocation. We first propose a caching scheme based on user preference and user mobility. Then the user association scheme is carried out according to the distances between users and remote radio heads. Finally, we reformulate the computing resource allocation problem as a matroid-constrained submodular function maximization problem and propose a heuristic scheme to find a sub-optimal solution. Besides, fronthaul compression technique is adopted to alleviate the capacity constraint of fronthaul links. Simulation results reveal that the proposed scheme achieves a better performance in terms of average system delay than three baseline schemes.

Robust monotone submodular function maximization

We consider a robust formulation, introduced by Krause et al. (2008), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to $\tau$ elements from the chosen set. For the fundamental case of $\tau=1$, we give a deterministic $(1-1/e)-1/\Theta(m)$ approximation algorithm, where $m$ is an input parameter and number of queries scale as $O(n^{m+1})$. In the process, we develop a deterministic $(1-1/e)-1/\Theta(m)$ approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (2010), we show a randomized $(1-1/e)-\epsilon$ approximation for constant $\tau$ and $\epsilon\leq \frac{1}{\tilde{\Omega}(\tau)}$, making $O(n^{1/\epsilon^3})$ queries. Further, for $\tau\ll \sqrt{k}$, we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.

Multi-label active learning based on submodular functions

Abstract In the data collection task, it is more expensive to annotate the instance in multi-label learning problem, since each instance is associated with multiple labels. Therefore it is more important to adopt active learning method in multi-label learning to reduce the labeling cost. Recent researches indicate submodular function optimization works well on subset selection problem and provides theoretical performance guarantees while simultaneously retaining extremely fast optimization. In this paper, we propose a query strategy by constructing a submodular function for the selected instance-label pairs, which can measure and combine the informativeness and representativeness. Thus the active learning problem can be formulated as a submodular function maximization problem, which can be solved efficiently and effectively by a simple greedy lazy algorithm. Experimental results show that the proposed approach outperforms several state-of-the-art multi-label active learning methods.

Deterministic Algorithms for Submodular Maximization Problems

Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area, most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios. In this work, we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing a submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic 1/2-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.

Submodular Function Maximization Over Graphs via Zero-Suppressed Binary Decision Diagrams

Submodular function maximization (SFM) has attracted much attention thanks to its applicability to various practical problems. Although most studies have considered SFM with size or budget constraints, more complex constraints often appear in practice. In this paper, we consider a very general class of SFM with such complex constraints (e.g., an s-t path constraint on a given graph). We propose a novel algorithm that takes advantage of zero-suppressed binary decision diagrams, which store all feasible solutions efficiently thus enabling us to circumvent the difficulty of determining feasibility. Theoretically, our algorithm is guaranteed to achieve (1-c)-approximations, where c is the curvature of a submodular function. Experiments show that our algorithm runs much faster than exact algorithms and finds better solutions than those obtained by an existing approximation algorithm in many instances. Notably, our algorithm achieves better than a 90%-approximation in all instances for which optimal values are available.

FPT approximation schemes for maximizing submodular functions

We investigate the existence of approximation algorithms for maximization of submodular functions, that run in a fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function v:2X→R the goal is to select a subset S of K elements from X such that v(S) is maximized. We identify three properties of set functions, referred to as p-separability properties, and we argue that many real-life problems can be expressed as maximization of submodular, p-separable functions, with low values of the parameter p. We present FPT approximation schemes for the minimization and maximization variants of the problem, for several parameters that depend on characteristics of the optimized set function, such as p and K.

Optimizing Energy Efficiency Over Energy-Harvesting LTE Cellular Networks

We consider the problem of downlink scheduling in an LTE network powered by energy harvesting devices. We formulate optimization problems that seek to optimize popular energy efficiency metrics subject to mandatory LTE network constraints along with energy harvesting causality constraints. We identify a key sub-problem pertaining to maximizing the weighted sum rate that is common for all optimization problems that we consider, and is also of independent interest. We show that the latter sub-problem can be reformulated as a constrained submodular set function maximization problem which enables us to design a constant-factor approximation algorithm for maximizing the weighted sum rate. We leverage this algorithm to design approximation algorithms (with provable performance guarantees) for maximizing all considered energy efficiency metrics over an energy harvesting LTE downlink. Our proposed algorithms are simple to implement and offer superior performance.

Revenue maximization in incentivized social advertising

Incentivized social advertising, an emerging marketing model, provides monetization opportunities not only to the owners of the social networking platforms but also to their influential users by offering a "cut" on the advertising revenue. We consider a social network (the host) that sells ad-engagements to advertisers by inserting their ads, in the form of promoted posts, into the feeds of carefully selected "initial endorsers" or seed users: these users receive monetary incentives in exchange for their endorsements. The endorsements help propagate the ads to the feeds of their followers. Whenever any user engages with an ad, the host is paid some fixed amount by the advertiser, and the ad further propagates to the feed of her followers, potentially recursively. In this context, the problem for the host is is to allocate ads to influential users, taking into account the propensity of ads for viral propagation, and carefully apportioning the monetary budget of each of the advertisers between incentives to influential users and ad-engagement costs, with the rational goal of maximizing its own revenue. We show that, taking all important factors into account, the problem of revenue maximization in incentivized social advertising corresponds to the problem of monotone submodular function maximization, subject to a partition matroid constraint on the ads-to-seeds allocation, and submodular knapsack constraints on the advertisers' budgets. We show that this problem is NP-hard and devise two greedy algorithms with provable approximation guarantees, which differ in their sensitivity to seed user incentive costs. Our approximation algorithms require repeatedly estimating the expected marginal gain in revenue as well as in advertiser payment. By exploiting a connection to the recent advances made in scalable estimation of expected influence spread, we devise efficient and scalable versions of our two greedy algorithms. An extensive experimental assessment confirms the high quality of our proposal.

Submodular Function Maximization for Group Elevator Scheduling

We propose a novel approach for group elevator scheduling by formulating it as the maximization of submodular function under a matroid constraint. In particular, we propose to model the total waiting time of passengers using a quadratic Boolean function. The unary and pairwise terms in the function denote the waiting time for single and pairwise allocation of passengers to elevators, respectively. We show that this objective function is submodular. The matroid constraints ensure that every passenger is allocated to exactly one elevator. We use a greedy algorithm to maximize the submodular objective function, and derive provable guarantees on the optimality of the solution. We tested our algorithm using Elevate 8, a commercial-grade elevator simulator that allows simulation with a wide range of elevator settings. We achieve significant improvement over the existing algorithms.