Submodular Maximization Problem Research Articles

Submodularity of Distributed Join Computation.

We study distributed equi-join computation in the presence of join-attribute skew, which causes load imbalance. Skew can be addressed by more fine-grained partitioning, at the cost of input duplication. For random load assignment, e.g., using a hash function, fine-grained partitioning creates a tradeoff between load expectation and variance. We show that minimizing load variance subject to a constraint on expectation is a monotone submodular maximization problem with Knapsack constraints, hence admitting provably near-optimal greedy solutions. In contrast to previous work on formal optimality guarantees, we can prove this result also for self-joins and more general load functions defined as weighted sum of input and output. We further demonstrate through experiments that this theoretical result leads to an effective algorithm for the problem of minimizing running time, even when load is assigned deterministically.

Multimodal Hierarchical Dirichlet Process-Based Active Perception by a Robot.

In this paper, we propose an active perception method for recognizing object categories based on the multimodal hierarchical Dirichlet process (MHDP). The MHDP enables a robot to form object categories using multimodal information, e.g., visual, auditory, and haptic information, which can be observed by performing actions on an object. However, performing many actions on a target object requires a long time. In a real-time scenario, i.e., when the time is limited, the robot has to determine the set of actions that is most effective for recognizing a target object. We propose an active perception for MHDP method that uses the information gain (IG) maximization criterion and lazy greedy algorithm. We show that the IG maximization criterion is optimal in the sense that the criterion is equivalent to a minimization of the expected Kullback–Leibler divergence between a final recognition state and the recognition state after the next set of actions. However, a straightforward calculation of IG is practically impossible. Therefore, we derive a Monte Carlo approximation method for IG by making use of a property of the MHDP. We also show that the IG has submodular and non-decreasing properties as a set function because of the structure of the graphical model of the MHDP. Therefore, the IG maximization problem is reduced to a submodular maximization problem. This means that greedy and lazy greedy algorithms are effective and have a theoretical justification for their performance. We conducted an experiment using an upper-torso humanoid robot and a second one using synthetic data. The experimental results show that the method enables the robot to select a set of actions that allow it to recognize target objects quickly and accurately. The numerical experiment using the synthetic data shows that the proposed method can work appropriately even when the number of actions is large and a set of target objects involves objects categorized into multiple classes. The results support our theoretical outcomes.

QoE-Driven Mobile Edge Caching Placement for Adaptive Video Streaming

Caching at mobile edge servers can smooth temporal traffic variability and reduce the service load of base stations in mobile video delivery. However, the assignment of multiple video representations to distributed servers is still a challenging question in the context of adaptive streaming, since any two representations from different videos or even from the same video will compete for the limited caching storage. Therefore, it is important, yet challenging, to optimally select the cached representations for each edge server in order to effectively reduce the service load of base station while maintaining a high quality of experience (QoE) for users. To address this, we study a QoE-driven mobile edge caching placement optimization problem for dynamic adaptive video streaming that properly takes into account the different rate-distortion (R–D) characteristics of videos and the coordination among distributed edge servers. Then, by the optimal caching placement of representations for multiple videos, we maximize the aggregate average video distortion reduction of all users while minimizing the additional cost of representation downloading from the base station, subject not only to the storage capacity constraints of the edge servers, but also to the transmission and initial startup delay constraints of the users. We formulate the proposed optimization problem as an integer linear program to provide the performance upper bound, and as a submodular maximization problem with a set of knapsack constraints to develop a practically feasible cost benefit greedy algorithm. The proposed algorithm has polynomial computational complexity and a theoretical lower bound on its performance. Simulation results further show that the proposed algorithm is able to achieve a near-optimal performance with very low time complexity. Therefore, the proposed optimization framework reveals the caching performance upper bound for general adaptive video streaming systems, while the proposed algorithm provides some design guidelines for the edge servers to select the cached representations in practice based on both the video popularity and content information.

Multiobject Tracking by Submodular Optimization.

In this paper, we propose a new multiobject visual tracking algorithm by submodular optimization. The proposed algorithm is composed of two main stages. At the first stage, a new selecting strategy of tracklets is proposed to cope with occlusion problem. We generate low-level tracklets using overlap criteria and min-cost flow, respectively, and then integrate them into a candidate tracklets set. In the second stage, we formulate the multiobject tracking problem as the submodular maximization problem subject to related constraints. The submodular function selects the correct tracklets from the candidate set of tracklets to form the object trajectory. Then, we design a connecting process which connects the corresponding trajectories to overcome the occlusion problem. Experimental results demonstrate the effectiveness of our tracking algorithm. Our source code is available at https://github.com/shenjianbing/submodulartrack.

Cooperative caching for content dissemination in vehicular networks

SummaryThe explosive growth of mobile data traffic has made cellular operators to seek low‐cost alternatives for cellular traffic off‐loading. In this paper, we consider a content delivery network where a vehicular communication network composed of roadside units (RSUs) is integrated into a cellular network to serve as an off‐loading platform. Each RSU subjecting to its storage capacity caches a subset of the contents of the central content server. Allocating the suitable subset of contents in each RSU cache such that maximizes the hit ratio of vehicles requests is a problem of paramount value that is targeted in this study. First, we propose a centralized solution in which, we model the cache content placement problem as a submodular maximization problem and show that it is NP‐hard. Second, we propose a distributed cooperative caching scheme, in which RSUs in an area periodically share information about their contents locally and thus update their cache. To this end, we model the distributed caching problem as a strategic resource allocation game that achieves at least 50% of the optimal solution. Finally, we evaluate our scheme using simulation for urban mobility simulator under realistic conditions. On average, the results show an improvement of 8% in the hit ratio of the proposed method compared with other well‐known cache content placement approaches.

Fast Budgeted Influence Maximization Over Multi-Action Event Logs

In a social network, influence maximization is the problem of identifying a set of users that own the maximum influence ability across the network. In this paper, a novel credit distribution (CD)-based model, termed as the multiaction CD (mCD) model, is introduced to quantify the influence ability of each user, which works with practical datasets where one type of action could be recorded for multiple times. Based on this model, influence maximization is formulated as a submodular maximization problem under a general knapsack constraint, which is NP hard. An efficient streaming algorithm with one-round scan over the user set is developed to find a suboptimal solution. Specifically, we first solve a special case of knapsack constraints, i.e., a cardinality constraint, and show that the developed streaming algorithm can achieve ( $({1}/{2})-\epsilon $ ) approximation of the optimality. Furthermore, for the general knapsack case, we show that the modified streaming algorithm can achieve ( $({1}/{3})-\epsilon $ ) approximation of the optimality. Finally, experiments are conducted over real Twitter dataset and demonstrate that the mCD model enjoys high accuracy compared to the conventional CD model in estimating the total number of people who get influenced in a social network. Moreover, through the comparison to the conventional CD, non-CD models, and the mCD model with the greedy algorithm on the performance of the influence maximization problem, we show the effectiveness and efficiency of the proposed mCD model with the streaming algorithm.

Streaming Algorithms for News and Scientific Literature Recommendation: Monotone Submodular Maximization With a <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math> </inline-formula>-Knapsack Constraint

Submodular optimization plays a significant role in combinatorial problems, since it captures the structure of the edge cuts in graphs, the coverage of sets, and so on. Many data mining and machine learning problems can be cast as submodular maximization problems with applications in recommendation systems and data diversification. In this paper, we focus on the problem of maximizing a monotone submodular function subject to a $d$ -knapsack constraint, for which we propose a streaming algorithm that achieves a $(({1}/{1+2d})-\epsilon)$ -approximation of the optimal value, while it only needs one single pass through the data set without storing all the data in the memory. In our experiments, we extensively evaluate the effectiveness of our proposed algorithm via two applications: news recommendation and scientific literature recommendation. It is observed that the proposed streaming algorithm achieves both execution speedup and memory saving by several orders of magnitude, compared with existing approaches.

Cache Placement in Fog-RANs: From Centralized to Distributed Algorithms

To deal with the rapid growth of high-speed and/or ultra-low latency data traffic for massive mobile users, fog radio access networks (Fog-RANs) have emerged as a promising architecture for next-generation wireless networks. In Fog-RANs, the edge nodes and user terminals possess storage, computation and communication functionalities to various degrees, which provides high flexibility for network operation, i.e., from fully centralized to fully distributed operation. In this paper, we study the cache placement problem in Fog-RANs, by taking into account flexible physical-layer transmission schemes and diverse content preferences of different users. We develop both centralized and distributed transmission aware cache placement strategies to minimize users' average download delay subject to the storage capacity constraints. In the centralized mode, the cache placement problem is transformed into a matroid constrained submodular maximization problem, and an approximation algorithm is proposed to find a solution within a constant factor to the optimum. In the distributed mode, a belief propagation based distributed algorithm is proposed to provide a suboptimal solution, with iterative updates at each BS based on locally collected information. Simulation results show that by exploiting caching and cooperation gains, the proposed transmission aware caching algorithms can greatly reduce the users' average download delay.

Data-Driven Omniscient Bounds and Greedy Policies for Ambulance Allocation and Dynamic Redeployment

We present an efficient data-driven computational bounding and solution approach for static allocation of ambulance fleet and its dynamic redeployment, where the goal is to position (or re-position) ambulances to bases to maximize the systems service level. We leverage the framework presented, to find data-driven omniscient bounds on the performance of a static location or dynamic redeployment policy. Central to our approach, is a discrete-event simulator to evaluate the impact of ambulance deployments to given call logs of emergency requests. We model ambulance allocation as an approximate submodular maximization problem, and devise a simple and efficient greedy algorithm that produces good solutions for static allocation, and can also be repeatedly employed in real-time for dynamic repositioning. Although the objective is not submodular, we generate data-driven guarantees on solution quality by formulating an omniscient upper bound (in both static and dynamic cases) using integer programming. We show that omniscient dispatch, as well as omniscient location combined with dispatch, are submodular upper bounds. Our experiments based on real data from an Asian city’s EMS, demonstrate how optimality gaps can be computed, and that they are small in this setting. The bound formulation that we present is general and can be applied for any alternative data-driven simulation framework or solution. We also test settings with customer abandonment and demonstrate the working of our algorithmic and bounding approach in that setting.

Maximizing Symmetric Submodular Functions

Symmetric submodular functions are an important family of submodular functions capturing many interesting cases, including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems that have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state-of-the-art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function f :2 N → R + subject to a down-monotone solvable polytope P ⊆ [0, 1] N . For this problem, we describe an algorithm producing a fractional solution of value at least 0.432 ċ f ( OPT ), where OPT is the optimal integral solution. Our second result considers the problem max{ f ( S ): | S | = k } for a non-negative symmetric submodular function f :2 N → R + . For this problem, we give an approximation ratio that depends on the value k /| N | and is always at least 0.432. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces 1/e − o (1) approximation, improving over the best-known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function, we describe a deterministic linear-time 1/2-approximation algorithm. Finally, we give a [1 − (1 − 1/ k ) k − 1 ]-approximation algorithm for Submodular Welfare with k players having identical non-negative submodular utility functions and show that this is the best possible approximation ratio for the problem.

Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

Fast algorithms for submodular maximization problems have a vast potential use in applicative settings, such as machine learning, social networks, and economics. Though fast algorithms were known for some special cases, only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint. The known fast algorithm matches the best possible approximation guarantee, while trying to reduce the number of value oracle queries the algorithm performs. Our main result is a new algorithm for this general case that establishes a surprising trade-off between two seemingly unrelated quantities: the number of value oracle queries and the number of matroid independence queries performed by the algorithm. Specifically, one can decrease the former by increasing the latter, and vice versa, while maintaining the best possible approximation guarantee. Such a trade-off is very useful since various applications might incur significantly different costs in querying the value and matroid independence oracles. Furthermore, in case the rank of the matroid is O(nc), where n is the size of the ground set and c is an absolute constant smaller than 1, the total number of oracle queries our algorithm uses can be made to have a smaller magnitude compared to that needed by the current best known algorithm. We also provide even faster algorithms for the well-studied special cases of a cardinality constraint and a partition matroid independence constraint, both of which capture many real-world applications and have been widely studied both theoretically and in practice.

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.

Parallel algorithms for anomalous subgraph detection

SummaryFor the many application domains concerning entities and their connections, often their data can be formally represented as graphs and an important problem is detecting an anomalous subgraph within it. Numerous methods have been proposed to speed‐up anomalous subgraph detection; however, each incurs non‐trivial costs on detection accuracy. In this paper, we formulate the anomalous subgraph detection problem as the maximization of a non‐parametric scan statistic and then approximate it to a submodular maximization problem. We propose two parallel algorithms: non‐coordination anomalous subgraph detection (NCASD) and under‐coordination anomalous subgraph detection (UCASD)for the anomalous subgraph detection. To the best of our knowledge, this paper is the first to solve this problem in parallel. NCASD emphasizes speed at the expense of approximation guarantees, while UCASD achieves a higher approximation factor through additional coordination controls and reduced parallelism. The experiments demonstrate the effectiveness and efficiency of our proposed approaches in a real‐world application domain (water pollution detection), comparing them with five other state‐of‐the‐art methods. Copyright © 2016 John Wiley & Sons, Ltd.

A Submodular Optimization Framework for Outage-Aware Cell Association in Heterogeneous Cellular Networks

In cellular heterogeneous networks (HetNets), offloading users to small cell base stations (SBSs) leads to a degradation in signal to interference plus noise ratio (SINR) and results in high outage probabilities for offloaded users. In this paper, we propose a novel framework to solve the cell association problem with the intention of improving user outage performance while achieving load balancing across different tiers of BSs. We formulate a combinatorial utility maximization problem with weighted BS loads that achieves proportional fairness among users and also takes into account user outage performance. A formulation of the weighting parameters is proposed to discourage assigning users to BSs with high outage probabilities. In addition, we show that the combinatorial optimization problem can be reformulated as a monotone submodular maximization problem and it can be readily solved via a greedy algorithm with lazy evaluations. The obtained solution offers a constant performance guarantee to the cell association problem. Simulation results show that our proposed approach leads to over 30% reduction in outage probabilities for offloaded users and achieves load balancing across macrocell and small cell BSs.

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization

We consider the \sf Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function $f:2^{\mathcal{N}}\rightarrow \mathbb{R}^+$, and the objective is to find a subset $S\subseteq \mathcal{N}$ maximizing $f(S)$. This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by \sf Unconstrained Submodular Maximization include \sf Max-Cut, \sf Max-DiCut, and variants of \sf Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige, Mirrokni, and Vondrák [SIAM J. Comput., 40 (2011), pp. 1133--1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.

Simultaneous Approximation of Multi-criteria Submodular Function Maximization

Recently intensive interest has been raised on approximation of the NP-hard submodular maximization problem due to their theoretical and practical significance. In this work, we extend this line of research by focusing on the simultaneous approximation of multiple submodular function maximization. We address the existence and nonexistence results for both deterministic and randomized approximation when the submodular functions are symmetric and asymmetric, respectively, along with algorithmic corollaries. We offer complete characterization of the symmetric case and partial results on the asymmetric case.

Symmetry and Approximability of Submodular Maximization Problems

A number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem. These results hold typically in the value oracle model, where the objective function is accessible via a black box returning $f(S)$ for a given $S$. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of symmetry gap. Our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g., the optimality of $(1-1/e)$-approximation for monotone submodular maximization under a cardinality constraint and the impossibility of $(\frac12+\epsilon)$-approximation for unconstrained (nonmonotone) submodular maximization. As a new applica...

Precoder Design for Physical Layer Multicasting

This paper studies the instantaneous rate maximization and the weighted sum delay minimization problems over a K-user multicast channel, where multiple antennas are available at the transmitter as well as at all the receivers. Motivated by the degree of freedom optimality and the simplicity offered by linear precoding schemes, we consider the design of linear precoders using the aforementioned two criteria. We first consider the scenario wherein the linear precoder can be any complex-valued matrix subject to rank and power constraints. We propose cyclic alternating ascent based precoder design algorithms and establish their convergence to respective stationary points. Simulation results reveal that our proposed algorithms considerably outperform known competing solutions. We then consider a scenario in which the linear precoder can be formed by selecting and concatenating precoders from a given finite codebook of precoding matrices, subject to rank and power constraints. We show that under this scenario, the instantaneous rate maximization problem is equivalent to a robust submodular maximization problem which is strongly NP hard. We propose a deterministic approximation algorithm and show that it yields a bicriteria approximation. For the weighted sum delay minimization problem we propose a simple deterministic greedy algorithm, which at each step entails approximately maximizing a submodular set function subject to multiple knapsack constraints, and establish its performance guarantee.

Maximum-Parsimony Haplotype Inference Based on Sparse Representations of Genotypes

The haplotypes of an individual can be used to predict diseases and help designing drugs. However, experimentally determining haplotypes is expensive and time-consuming, so genotypes are usually measured instead. Given the set of genotypes for a group of unrelated individuals, it is possible to infer the haplotype pair for each subject based on the maximum parsimony principle. Finding the exact solution to this problem is NP-hard. We propose two related formulations of the haplotype inference problem that translate the maximum parsimony principle into the sparse representation of genotypes. In the first formulation we look for the set of haplotypes that explain the genotypes such that the resulting frequency vector of haplotypes is as sparse as possible. The sparseness condition is achieved by minimizing the Tsallis entropy of the frequency vector, which is still an NP-hard problem. We propose a method that enumerates all local minima with high probability by solving a set of integer linear programs of low dimensionality. The minimizer is then found by identifying the local minimum point that achieves the lowest Tsallis entropy. In the second formulation, we state the haplotypes inference as a sparse dictionary selection problem. Each genotype is reconstructed by a haplotype pair selected from a set of available haplotypes that needs to be sparse. This leads to an approximately submodular maximization problem and therefore, can be solved with a fast greedy method. We test the proposed solutions with different data sets and compare the performance with the state-of-the-art methods, achieving similar or better results.

Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant k, we present a $(\frac{1}{k+2+\frac{1}{k}+\epsilon})$-approximation for the submodular maximization problem under k matroid constraints, and a $(\frac{1}{5}-\epsilon)$-approximation algorithm for this problem subject to k knapsack constraints ($\epsilon>0$ is any constant). We improve the approximation guarantee of our algorithm to $\frac{1}{k+1+\frac{1}{k-1}+\epsilon}$ for $k\geq2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon})$-approximation for maximizing a monotone submodular function subject to $k\geq2$ partition matroids, which is an improvement over the previously best known guarantee of $\frac{1}{k+1}$.