Set Function Optimization Research Articles

On Zone-Differentiated Time-Constrained Flow Capacity Intelligent Monitoring for Large-Scale Urban Pipeline Systems by Mobile Sensors

Large-scale urban pipeline systems (LSUPSs) are complex pipeline networks of flows (e.g., water, oil, or gas). Flow capacity intelligent monitoring, e.g., automatically monitor the sum of flows in an LSUPS, is an important task in smart cities. Recently, mobile sensors and static receiver nodes are used to perform the task. Mobile sensors are released into the network at selected entrances to collect data, and upload the data to receiver nodes deployed at selected locations of the network for further analysis. Due to cost constraints, the numbers of mobile sensors and receiver nodes are limited, which cause the problem that some pipelines may not be monitored. However, applications normally require that some Zones of Interest (ZoIs) in the network have to be monitored. Therefore, how to select optimal entrances and locations for given numbers of mobile sensors and receiver nodes, so that the capacity of monitored flow is maximized within a given time under the constraint that all ZoIs are also monitored with expected probabilities, is a challenging problem. First, we prove the problem is NP-complete. Then, we design two algorithms based on submodular set function optimization to solve it. The first algorithm can obtain an approximate optimal solution with high time complexity, while the second algorithm can obtain a suboptimal solution with much lower time complexity. Finally, we analyze time complexity and approximate ratio of the two algorithms. Theoretical analyses and simulation results show that the proposed algorithms outperform the state-of-the-art algorithms.

Service Placement and Request Scheduling for Data-Intensive Applications in Edge Clouds

Mobile edge computing provides the opportunity for wireless users to exploit the power of cloud computing without a large communication delay. To serve data-intensive applications (e.g., video analytics, machine learning tasks) from the edge, we need, in addition to computation resources, storage resources for storing server code and data as well as network bandwidth for receiving user-provided data. Moreover, due to time-varying demands, the code and data placement needs to be adjusted over time, which raises concerns of system stability and operation cost. In this paper, we address these issues by proposing a two-time-scale framework that jointly optimizes service (code and data) placement and request scheduling, while considering storage, communication, computation, and budget constraints. First, by analyzing the hardness of various cases, we completely characterize the complexity of our problem. Next, we develop a polynomial-time service placement algorithm by formulating our problem as a set function optimization, which attains a constant-factor approximation under certain conditions. Furthermore, we develop a polynomial-time request scheduling algorithm by computing the maximum flow in a carefully constructed auxiliary graph, which satisfies hard resource constraints and is provably optimal in the special case where requests have homogeneous resource demands. Extensive synthetic and trace-driven simulations show that the proposed algorithms achieve 90% of the optimal performance.

Novel algorithms for maximum DS decomposition

DS decomposition plays an important role in set function optimization problem, because there is DS decomposition for any set function. How to design an efficient and effective algorithm to solve maximizing DS decomposition is a heated problem. In this work, we propose a framework called Parameter Conditioned Greedy Algorithm which has a deterministic version and two random versions. In more detail, this framework uses the difference with parameter decomposition function and combines non-negative condition. Besides, if we set the different parameters, the framework can return solution with different approximation ratio. Also, we choose two special case to show our deterministic algorithm gets f(Sk)−(e−1−cg)g(Sk)≥(1−e−1)[f(OPT)−g(OPT)] and f(Sk)−(1−cg)g(Sk)≥(1−e−1)f(OPT)−g(OPT) respectively for cardinality constrained problem, where cg is the curvature of monotone submodular set function. To speed the deterministic algorithm, we introduce a random sample set whose intersection with the optimal solution is as nonempty as possible. Importantly, it also can get the same approximation ratio as deterministic algorithm under expectation. Further, for maximization DS decomposition without constraint, our another random algorithm gets E[f(Sk)−(e−1−cg)g(Sk)]≥(1−e−1)[f(OPT)−g(OPT)] and E[f(Sk)−(1−cg)g(Sk)]≥(1−e−1)f(OPT)−g(OPT) respectively. Because the Parameter Conditioned Algorithm is the general framework, different users can choose the parameters that fit their problem to get a better approximation.

Actuator Placement Under Structural Controllability Using Forward and Reverse Greedy Algorithms

Actuator placement is an active field of research, which has received significant attention for its applications in complex dynamical networks. In this article, we study the problem of finding a set of actuator placements minimizing the metric that measures the average energy consumed for state transfer by the controller, while satisfying a structural controllability requirement and a cardinality constraint on the number of actuators allowed. As no computationally efficient methods are known to solve such combinatorial set function optimization problems, two greedy algorithms, forward and reverse, are proposed to obtain approximate solutions. We first show that the constraint sets these algorithms explore can be characterized by matroids. We then obtain performance guarantees for the forward and reverse greedy algorithms applied to the general class of matroid optimization problems by exploiting properties of the objective function such as the submodularity ratio and the curvature. Finally, we propose feasibility check methods for both algorithms based on maximum flow problems on certain auxiliary graphs originating from the network graph. Our results are verified with case studies over large networks.

A short proof for stronger version of DS decomposition in set function optimization

Using a short proof, we show that every set function f can be decomposed into the difference of two monotone increasing and strictly submodular functions g and h, i.e., $$f=g-h$$ , and every set function f can also be decomposed into the difference of two monotone increasing and strictly supermodular functions g and h, i.e., $$f=g-h$$ .

Corrections to “Rigid Network Design Via Submodular Set Function Optimization”

We provide a correction to our paper Rigid Network Design Via Submodular Set Function Optimization, which appeared in Volume 2, Issue 3 of the IEEE Transactions on Network Science and Engineering.

A variation of DS decomposition in set function optimization

Any set function can be decomposed into the difference of two monotone nondecreasing submodular functions. This theorem plays an important role in the set function optimization theory. In this paper, we show a variation that any set function can be decomposed into the difference of two monotone nondecreasing supermodular functions. Meanwhile, we give an example in social network optimization and construct algorithmic solutions for the maximization problem of set functions with this variation of DS decomposition.

Optimal Formation of Autonomous Vehicles in Mixed Traffic Flow

Platooning of multiple autonomous vehicles has attracted significant attention in both academia and industry. Despite its great potential, platooning is not the only choice for the formation of autonomous vehicles in mixed traffic flow, where autonomous vehicles and human-driven vehicles (HDVs) coexist. In this paper, we investigate the optimal formation of autonomous vehicles that can achieve an optimal system-wide performance in mixed traffic flow. Specifically, we consider the optimal $\mathcal{H}_2$ performance of the entire traffic flow, reflecting the potential of autonomous vehicles in mitigating traffic perturbations. Then, we formulate the optimal formation problem as a set function optimization problem. Numerical results reveal two predominant optimal formations: uniform distribution and platoon formation, depending on traffic parameters. In addition, we show that 1) the prevailing platoon formation is not always the optimal choice; 2) platoon formation might be the worst choice when HDVs have a poor string stability behavior. These results suggest more opportunities for the formation of autonomous vehicles, beyond platooning, in mixed traffic flow.

Set Function Optimization

This article is an introduction to recent development of optimization theory on set functions, the nonsubmodular optimization, which contains two interesting results, DS (difference of submodular) functions decomposition and sandwich theorem, together with iterated sandwich method and data-dependent approximation. Some potential research problems will be mentioned.

Multiuser Joint Task Offloading and Resource Optimization in Proximate Clouds

Proximate cloud computing enables computationally intensive applications on mobile devices, providing a rich user experience. However, remote resource bottlenecks limit the scalability of offloading, requiring optimization of the offloading decision and resource utilization. To this end, in this paper, we leverage the variability in capabilities of mobile devices and user preferences. Our system utility metric is a measure of quality of experience (QoE) based on task completion time and energy consumption of a mobile device. We propose a heuristic offloading decision algorithm (HODA), which is semidistributed and jointly optimizes the offloading decision, and communication and computation resources to maximize system utility. Our main contribution is to reduce the problem to a submodular maximization problem and prove its NP-hardness by decomposing it into two subproblems: 1) optimization of communication and computation resources solved by quasiconvex and convex optimization and 2) offloading decision solved by submodular set function optimization. HODA reduces the complexity of finding the local optimum to $O(K^{3})$ , where $K$ is the number of mobile users. Simulation results show that HODA performs within 5% of the optimal on average. Compared with other solutions, HODA's performance is significantly superior as the number of users increases.

Rigid Network Design Via Submodular Set Function Optimization

We consider the problem of constructing networks that exhibit desirable algebraic rigidity properties, which can provide significant performance improvements for associated formation shape control and localization tasks. We show that the network design problem can be formulated as a submodular set function optimization problem and propose greedy algorithms that achieve global optimality or an established near-optimality guarantee. We also consider the separate but related problem of selecting anchors for sensor network localization to optimize a metric of the error in the localization solutions. We show that an interesting metric is a modular set function, which allows a globally optimal selection to be obtained using a simple greedy algorithm. The results are illustrated via numerical examples, and we show that the methods scale to problems well beyond the capabilities of current state-of-the-art convex relaxation techniques.