Submodular Optimization Problem Research Articles

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.

Failure-Tolerant Monitoring Based on Spatial–Temporal Correlation via Mobile Sensors for Large-Scale Acyclic Flow Systems in Smart Cities

Large-scale acyclic flow systems (LSAFSs) are models of pipeline networks that are used to transport important resources, such as water, oil, and natural gas in smart cities. LSAFSs have features of complex topology and deep-underground deployment, which would cause accidents, such as leakages and pollution that are difficult to be detected in time. Mobile sensors (MSs)-based monitoring schemes appear as an effective solution in recent years to handle this situation. These schemes drop MSs into an LSAFS from specified locations, and the MSs will move along with fluid inside the LSAFS to collect data. When the MSs pass through a predeployed and activated receiver node (RN), they will upload their data to the RN. However, how to decide the optimal locations and timings for dropping of the MSs and the best activation periods and deployment locations of the RNs, such that total length of monitored pipelines in the LSAFS is maximized, as well as energy consumption of the RNs is minimized, is a challenging problem. The problem is more challenging if potential uploading failures and undetermined movement of the MSs are considered. In this article, we first formalize the problem as a multiobjective optimization problem. Then, we decompose the problem into a submodular optimization problem and an union set optimization problem, and prove they are NP-hard. Next, an approximate algorithm based on the Pigeonhole principle is proposed to solve the first problem, and a heuristic algorithm based on the inclusion-exclusion principle is proposed to solve the second problem. Extensive theoretical analyses and simulations show that the proposed algorithms outperform state-of-the-art algorithms.

Robust Adaptive Submodular Maximization

The goal of a sequential decision-making problem is to design an interactive policy that adaptively selects a group of items, each selection is based on the feedback from the past, to maximize the expected utility of selected items. It has been shown that the utility functions of many real-world applications are adaptive submodular. However, most of existing studies on adaptive submodular optimization focus on the average-case, that is, their objective is to find a policy that maximizes the expected utility over a known distribution of realizations. Unfortunately, a policy that has a good average-case performance may have very poor performance under the worst-case realization. In this study, we propose to study two variants of adaptive submodular optimization problems, namely, worst-case adaptive submodular maximization and robust submodular maximization. The first problem aims to find a policy that maximizes the worst-case utility and the latter one aims to find a policy, if any, that achieves both near optimal average-case utility and worst-case utility simultaneously. We introduce a new class of stochastic functions, called worst-case submodular function. For the worst-case adaptive submodular maximization problem subject to a p-system constraint, we develop an adaptive worst-case greedy policy that achieves a [Formula: see text] approximation ratio against the optimal worst-case utility if the utility function is worst-case submodular. For the robust adaptive submodular maximization problem subject to cardinality constraints (respectively, partition matroid constraints), if the utility function is both worst-case submodular and adaptive submodular, we develop a hybrid adaptive policy that achieves an approximation close to [Formula: see text] (respectively, 1/3) under both worst- and average-case settings simultaneously. We also describe several applications of our theoretical results, including pool-base active learning, stochastic submodular set cover, and adaptive viral marketing. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1239 .

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.

Anchor Selection for SLAM Based on Graph Topology and Submodular Optimization

This article considers simultaneous localization and mapping (SLAM) problem for robots in situations where accurate estimates for some of the robot poses, termed anchors, are available. These may be acquired through external means, for example, by either stopping the robot at some previously known locations or pausing for a sufficient period of time to measure the robot poses with an external measurement system. The main contribution is an efficient algorithm for selecting a fixed number of anchors from a set of potential poses that minimizes estimated error in the SLAM solution. Based on a graph-topological connection between the D-optimality design metric and the tree-connectivity of the pose-graph, the anchor selection problem can be formulated approximately as a submatrix selection problem for reduced weighted Laplacian matrix, leading to a cardinality-constrained submodular maximization problem. Two greedy methods are presented to solve this submodular optimization problem with a performance guarantee. These methods are complemented by Cholesky decomposition, approximate minimum degree permutation, order reuse, and rank-1 update that exploit the sparseness of the weighted Laplacian matrix. We demonstrate the efficiency and effectiveness of the proposed techniques on public-domain datasets, Gazebo simulations, and real-world experiments.

Approximation Algorithms for Submodular Data Summarization with a Knapsack Constraint

Data summarization, i.e., selecting representative subsets of manageable size out of massive data, is often modeled as a submodular optimization problem. Although there exist extensive algorithms for submodular optimization, many of them incur large computational overheads and hence are not suitable for mining big data. In this work, we consider the fundamental problem of (non-monotone) submodular function maximization with a knapsack constraint, and propose simple yet effective and efficient algorithms for it. Specifically, we propose a deterministic algorithm with approximation ratio 6 and a randomized algorithm with approximation ratio 4, and show that both of them can be accelerated to achieve nearly linear running time at the cost of weakening the approximation ratio by an additive factor of ε. We then consider a more restrictive setting without full access to the whole dataset, and propose streaming algorithms with approximation ratios of 8+ε and 6+ε that make one pass and two passes over the data stream, respectively. As a by-product, we also propose a two-pass streaming algorithm with an approximation ratio of 2+ε when the considered submodular function is monotone. To the best of our knowledge, our algorithms achieve the best performance bounds compared to the state-of-the-art approximation algorithms with efficient implementation for the same problem. Finally, we evaluate our algorithms in two concrete submodular data summarization applications for revenue maximization in social networks and image summarization, and the empirical results show that our algorithms outperform the existing ones in terms of both effectiveness and efficiency.

Improving in-memory file system reading performance by fine-grained user-space cache mechanisms

Nowadays, as the memory capacity of servers become larger and larger, distributed in-memory file systems, which enable applications to interact with data at fast speed, have been widely used. However, the existing distributed in-memory file systems still face the problem of low data access performance in small data reading, which seriously reduce their usefulness in many important big data scenarios. In this paper, we analyze the factors that affect the performance of reading in-memory files and propose a two-layer user space cache management mechanism: in the first layer, we cache data packet references to reduce frequent page fault interruptions (packet-level cache); in the second layer, we cache and manage small file data units to avoid redundant inter-process communications (object-level cache). We further design a fine-grained caching model based on the submodular function optimization theory, for efficiently managing the variable-length cache units with partially overlapping fragments on the client side. Experimental results on synthetic and real-world workloads show that compared with the existing cutting-edge systems, the first level cache can double the reading performance on average, and the second level cache can improve random reading performance by more than 4 times. Our caching strategies also outperform the cutting-edge cache algorithms over 20% on hit ratio. Furthermore, the proposed client-side caching framework idea has been adopted by the Alluxio open source community, which shows the practical benefits of this work. • A packet-level cache policy for reducing page faults during sequential data reading. • An in-memory cache layer to cache the variable-length hot file fragments. • Proved that the variable-length cache problem is a submodular optimization problem theoretically. • Comprehensive performance evaluation of the client-side cache strategies.

Prioritizing transcriptomic and epigenomic experiments using an optimization strategy that leverages imputeddata.

MotivationSuccessful science often involves not only performing experiments well, but also choosing well among many possible experiments. In a hypothesis generation setting, choosing an experiment well means choosing an experiment whose results are interesting or novel. In this work, we formalize this selection procedure in the context of genomics and epigenomics data generation. Specifically, we consider the task faced by a scientific consortium such as the National Institutes of Health ENCODE Consortium, whose goal is to characterize all of the functional elements in the human genome. Given a list of possible cell types or tissue types (‘biosamples’) and a list of possible high-throughput sequencing assays, where at least one experiment has been performed in each biosample and for each assay, we ask ‘Which experiments should ENCODE perform next?’ResultsWe demonstrate how to represent this task as a submodular optimization problem, where the goal is to choose a panel of experiments that maximize the facility location function. A key aspect of our approach is that we use imputed data, rather than experimental data, to directly answer the posed question. We find that, across several evaluations, our method chooses a panel of experiments that span a diversity of biochemical activity. Finally, we propose two modifications of the facility location function, including a novel submodular–supermodular function, that allow incorporation of domain knowledge or constraints into the optimization procedure.Availability and implementationOur method is available as a Python package at https://github.com/jmschrei/kiwano and can be installed using the command pip install kiwano. The source code used here and the similarity matrix can be found at http://doi.org/10.5281/zenodo.3708538.Supplementary information Supplementary data are available at Bioinformatics online.

Optimization of Chance-Constrained Submodular Functions

Submodular optimization plays a key role in many real-world problems. In many real-world scenarios, it is also necessary to handle uncertainty, and potentially disruptive events that violate constraints in stochastic settings need to be avoided. In this paper, we investigate submodular optimization problems with chance constraints. We provide a first analysis on the approximation behavior of popular greedy algorithms for submodular problems with chance constraints. Our results show that these algorithms are highly effective when using surrogate functions that estimate constraint violations based on Chernoff bounds. Furthermore, we investigate the behavior of the algorithms on popular social network problems and show that high quality solutions can still be obtained even if there are strong restrictions imposed by the chance constraint.

Submodular optimization problems and greedy strategies: A survey

The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. How good is the greedy strategy compared to the optimal solution? In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. The first is set submodular optimization, which is to choose a set of actions to optimize a set submodular objective function, and the second is string submodular optimization, which is to choose an ordered set of actions to optimize a string submodular function. Our emphasis here is on performance bounds for the greedy strategy in submodular optimization problems. Specifically, we review performance bounds for the greedy strategy, more general and improved bounds in terms of curvature, performance bounds for the batched greedy strategy, and performance bounds for Nash equilibria.

Broadcast Your Weaknesses: Cooperative Active Pose-Graph SLAM for Multiple Robots

In this letter, we propose a low-cost, high-efficiency framework for cooperative active pose-graph simultaneous localization and mapping (SLAM) for multiple robots in three-dimensional (3D) environments based on graph topology. Based on the selection of weak connections in pose graphs, this method aims to find the best trajectories for optimal information exchange to repair these weaknesses opportunistically when robots move near them. Based on tree-connectivity, which is greatly related to the D-optimality metric of the Fisher information matrix (FIM), we explore the relationship between measurement (edge) selection and pose-measurement (node-edge) selection, which often occurs in active SLAM, in terms of information increment. The measurement selection problem is formulated as a submodular optimization problem and solved by an exhaustive method using rank-1 updates. We decide which robot takes the selected measurements through a bidding framework where each robot computes its predicted cost. Finally, based on a novel continuous trajectory optimization method, these additional measurements collected by the winning robot are sent to the requesting robot to strengthen its pose graph. In simulations and experiments, we validate our approach by comparing against existing methods. Further, we demonstrate online communication based on offline planning results using two unmanned aerial vehicles (UAVs).

Mobility-Aware Traffic Offloading via Cooperative Coded Edge Caching

With caching popular contents at the small-cell base stations (SBSs), cooperative edge caching has emerged as an effective approach to offload explosively increasing network traffic from a massive number of users in mobile edge networks (MENs). However, most previous works ignored impact of user mobility on caching strategy, thus having limited practical applications. How to improve the caching strategy by exploiting user mobility is still a challenging problem. Accordingly, in this paper, we propose a mobility-aware and cooperative coded caching (MoCoCoCa) for achieving traffic offloading in MENs, by considering user mobility and hard service deadline constraints. Specifically, we first develop the MoCoCoCa framework to satisfy users' requests locally and conduct the construction of access trajectory set via random walks on a Markov chain, which takes the randomness of contact into account. Then, based on user mobility predictions, we formulate the caching problem into a mixed integer nonlinear programming (MINLP) problem to minimize the load of core network, subject to finiteness of network resources (e.g., storage capacity, transmission rate) and non-uniformity of content popularity constraints. The objective function's submodular property is proved and a cooperative placement algorithm is given, which can achieve (1-1/e) -optimality. Furthermore, to adapt the given content placement configuration to the spatio-temporal dynamics of realtime request flow, the original placement problem is decomposed into some independent subproblems of content placement at individual SBSs. Each subproblem is converted into a submodular optimization problem, and a request-aware distributed replacement algorithm with linear computational complexity is proposed. Trace-based simulations and numerical results demonstrate that our proposed caching strategy can offload up to 69% network traffic than existing caching strategies.

Distributed Submodular Minimization via Block-Wise Updates and Communications

In this paper we deal with a network of computing agents with local processing and neighboring communication capabilities that aim at solving (without any central unit) a submodular optimization problem. The cost function is the sum of many local submodular functions and each agent in the network has access to one function in the sum only. In this distributed set-up, in order to preserve their own privacy, agents communicate with neighbors but do not share their local cost functions. We propose a distributed algorithm in which agents resort to the Lovàsz extension of their local submodular functions and perform local updates and communications in terms of single blocks of the entire optimization variable. Updates are performed by means of a greedy algorithm which is run only until the selected block is computed, thus resulting in a reduced computational burden. The proposed algorithm is shown to converge in expected value to the optimal cost of the problem, and an approximate solution to the submodular problem is retrieved by a thresholding operation. As an application, we consider a distributed image segmentation problem in which each agent has access only to a portion of the entire image. While agents cannot segment the entire image on their own, they correctly complete the task by cooperating through the proposed distributed algorithm.

Automatically Design Convolutional Neural Networks by Optimization With Submodularity and Supermodularity.

The architecture of convolutional neural networks (CNNs) is a key factor of influencing their performance. Although deep CNNs perform well in many difficult problems, how to intelligently design the architecture is still a challenging problem. Focusing on two practical architectural design problems: to maximize the accuracy with a given forward running time and to minimize the forward running time with a given accuracy requirement, we innovatively utilize prior knowledge to convert architecture optimization problems into submodular optimization problems. We propose efficient Greedy algorithms to solve them and give theoretical bounds of our algorithms. Specifically, we employ the techniques on some public data sets and compare our algorithms with some other hyperparameter optimization methods. Experiments show our algorithms' efficiency.

Stochastic Block-Coordinate Gradient Projection Algorithms for Submodular Maximization

We consider a stochastic continuous submodular huge-scale optimization problem, which arises naturally in many applications such as machine learning. Due to high-dimensional data, the computation of the whole gradient vector can become prohibitively expensive. To reduce the complexity and memory requirements, we propose a stochastic block-coordinate gradient projection algorithm for maximizing continuous submodular functions, which chooses a random subset of gradient vector and updates the estimates along the positive gradient direction. We prove that the estimates of all nodes generated by the algorithm converge to some stationary points with probability 1. Moreover, we show that the proposed algorithm achieves the tight (pmin/2F⁎-ϵ) approximation guarantee after O(1/ϵ2) iterations for DR-submodular functions by choosing appropriate step sizes. Furthermore, we also show that the algorithm achieves the tight (γ2/1+γ2pminF⁎-ϵ) approximation guarantee after O(1/ϵ2) iterations for weakly DR-submodular functions with parameter γ by choosing diminishing step sizes.

QoE-driven social aware caching placement for terrestrial-satellite networks

In remote terrestrial-satellite networks, caching is a very promising technique to alleviate the burden of space cloudlet (e.g., cache-enabled satellite user terminal) and to improve subscribers' quality of experience (QoE) in terms of buffering delay and achievable video streaming rate. In this paper, we studied a QoE-driven caching placement optimization problem for video streaming that takes into account the required video streaming rate and the social relationship among users. Social ties between users are used to designate a set of helpers with caching capability, which can cache popular files pro-actively when the cloudlet is idle. We model the utility function of QoE as a logarithmic function. Then, the caching placement problem is formulated as an optimization problem to maximize the user's average QoE subject to the storage capacity constraints of the helpers and the cloudlets. Furthermore, we reformulate the problem into a monotone submodular optimization problem with a partition matroid constraint, and an efficient greedy algorithm with 1 − 1/6 approximation ratio is proposed to solve it. Simulation results show that the proposed caching placement approach significantly outperforms the traditional approaches in terms of QoE, while yields about the same delay and hit ratio performance compare to the delay-minimized scheme.

Boundary-Guided Optimization Framework for Saliency Refinement

Salient object detection has made a rapid progress in recent years. To improve the quality of initial saliency maps, existing algorithms typically refine them via a neighbor-constrained smoothing model, which assigns similar saliency values to neighboring regions. Since the adjacent regions could also cross the boundary between the salient object and background, these spatial distance-based methods easily cause false detection by involving the background regions that are close to the salient objects. To address this problem, we propose a boundary-guided optimization framework to jointly improve the region smoothness and correct the false detect regions. Specifically, we introduce a latent segmentation variable to regularize the consistency between the refined saliency map and the latent segmentation mask, which penalizes high (low) saliency values of the regions lying outside (inside) the estimated object boundary. To optimize the proposed objective function, we decompose the primary problem into two subproblems: submodular optimization problem and convex optimization problem. The submodular optimization problem can be quickly optimized using the off-the-shelf technique while the convex optimization can be solved with a closed-form solution. The experimental results show that the proposed method consistently improve the performances of eight state-of-the-art salient object detection algorithms on three datasets, including latest deep convolutional neural network-based algorithms. Meanwhile, our method outperforms state-of-the-art saliency refinement algorithms.

Joint Rate Control and Scheduling for Real-Time Wireless Networks

This paper studies wireless networks with multiple real-time flows that have stringent requirements on both per-packet delay and long-term average delivery ratio. Each flow dynamically adjusts its traffic load based on its observation of network status. When the requirements of per-packet delay and delivery ratio are satisfied, each flow obtains some utility based on its traffic load. We aim to design joint rate control and scheduling policies that maximize the total utility in the system. We first show that the problem of maximizing total utility can be formulated as a submodular optimization problem with exponentially many constraints. We then propose two simple distributed policies that require almost no coordination between different entities in the network. The total utilities under these two policies can be made arbitrarily close to the theoretical upper-bound. Extensive simulations also show that they achieve much better performance than state-of-the-art policies.

Input Selection for Performance and Controllability of Structured Linear Descriptor Systems

A common approach to controlling complex networks is to directly control a subset of input nodes, which then controls the remaining nodes via network interactions. Current approaches for selecting input nodes assume that either all system matrix entries are known and fixed, or are independent free parameters, and focus either on performance or controllability. In this paper, we make two contributions towards input selection in networked systems. First, we propose polynomial-time algorithms for input selection in structured linear descriptor systems, which are systems with dependencies between free parameters due to physical laws or design constraints. Second, we develop a framework for input selection based on joint consideration of controllability and performance. We make both contributions by mapping input selection to a submodular optimization problem under two matroid constraints, which enables development of polynomial-time algorithms with provable optimality guarantees. We provide improved optimalit...

Submodular Attribute Selection for Visual Recognition

In real-world visual recognition problems, low-level features cannot adequately characterize the semantic content in images, or the spatio-temporal structure in videos. In this work, we encode objects or actions based on attributes that describe them as high-level concepts. We consider two types of attributes. One type of attributes is generated by humans, while the second type is data-driven attributes extracted from data using dictionary learning methods. Attribute-based representation may exhibit variations due to noisy and redundant attributes. We propose a discriminative and compact attribute-based representation by selecting a subset of discriminative attributes from a large attribute set. Three attribute selection criteria are proposed and formulated as a submodular optimization problem. A greedy optimization algorithm is presented and its solution is guaranteed to be at least (1-1/e)-approximation to the optimum. Experimental results on four public datasets demonstrate that the proposed attribute-based representation significantly boosts the performance of visual recognition and outperforms most recently proposed recognition approaches.