Submodular Maximization Problem Research Articles

Streaming algorithms for robust submodular maximization

Submodular maximization is well studied in the fields of data mining and machine learning. We study the submodular maximization subject to a cardinality constraint k for large scale scenarios applications under two combined settings. One is that all elements arrive in a streaming fashion, and the other is that some elements may be invalid at last. For this problem, which is called streaming robust submodular maximization (SRSM) problem, we explore an approximation algorithm, returning a subset S from the ground set V with a limit size, such that it represents V and is robust to a broken set E well. Our algorithm only makes one pass over data, and achieves a constant-factor 0.1224 approximation guarantee, independent of the cardinality constraint parameter k. Based on the algorithm for SRSM problem, we continue to discuss this problem over sliding windows, in which we are asked to obtain a proper set that only considers the last W elements, and derive an algorithm with a constant (0.0612−ϵ)-approximation guarantee. At last we also propose numerical experiments on some applications to well demonstrate our algorithm for SRSM problem over sliding windows.

Kelly Cache Networks

We study networks of M/M/1 queues in which nodes act as caches that store objects. Exogenous requests for objects are routed towards nodes that store them; as a result, object traffic in the network is determined not only by demand but, crucially, by where objects are cached. We determine how to place objects in caches to attain a certain design objective, such as, e.g., minimizing network congestion or retrieval delays. We show that for a broad class of objectives, including minimizing both the expected network delay and the sum of network queue lengths, this optimization problem can be cast as an NP-hard submodular maximization problem. We show that so-called continuous greedy algorithm attains a ratio arbitrarily close to $1-1/e\approx 0.63$ using a deterministic estimation via a power series; this drastically reduces execution time over prior art, which resorts to sampling. Finally, we show that our results generalize, beyond M/M/1 queues, to networks of M/M/ $k$ and symmetric M/D/1 queues.

A constrained two-stage submodular maximization

In this paper, we investigate the two-stage submodular maximization problem, where there is a collection F={f1,...,fm} of m submodular functions which are defined on the same element ground set Ω. The goal is to select a subset S⊆Ω of size at most ℓ such that 1m∑f∈FmaxT⊆S,T∈I⁡f(T) is maximized, where I denotes a specifically-defined independence system. We consider the two-stage submodular maximization with a P-matroid constraint and present a (1/(P+1))(1−1/e(P+1))-approximation algorithm. Furthermore, we extend the algorithm to the two-stage submodular maximization with a more generalized P-exchange system constraint and show the approximation ratio can be maintained with slightly modifications of the algorithm.

Local Search Algorithms for the Maximum Carpool Matching Problem

The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph $$G = (V, A)$$ , a capacity function $$c: V \rightarrow {\mathbb {N}}$$ , and a weight function $$w: A \rightarrow {\mathbb {R}}^+$$ , a carpool matching is a subset of arcs, $$M \subseteq A$$ , such that every $$v \in V$$ satisfies: (1) $$d^{\text {in}}_{M}(v) \cdot d^{\text {out}}_{M}(v) = 0$$ , (2) $$d^{\text {in}}_{M}(v) \le c(v)$$ , and (3) $$d^{\text {out}}_{M}(v) \le 1$$ . A vertex v for which $$d^{\text {out}}_{M}(v) = 1$$ is a passenger, and a vertex for which $$d^{\text {out}}_{M}(v) = 0$$ is a driver who has $$d^{\text {in}}_{M}(v)$$ passengers. In the Maximum Carpool Matching problem the goal is to find a carpool matching M of maximum total weight. The problem arises when designing an online carpool service, such as Zimride (Zimride by enterprise. https://zimride.com/ ), which tries to connect between users based on a similarity function. The problem is known to be NP-hard, even in the unweighted and uncapacitated case. The Maximum Group Carpool Matching problem, is an extension of the Maximum Carpool Matching where each vertex represents an unsplittable group of passengers. Formally, each vertex $$u \in V$$ has a size $$s(u) \in {\mathbb {N}}$$ , and the constraint $$d^{\text {in}}_{M}(v) \le c(v)$$ is replaced with $$\sum _{u:(u,v) \in M} s(u) \le c(v)$$ . We show that Maximum Carpool Matching can be formulated as an unconstrained submodular maximization problem, thus it admits a $$\frac{1}{2}$$ -approximation algorithm. We show that the same formulation does not work for Maximum Group Carpool Matching, nevertheless, we present a local search $$(\frac{1}{2} - \varepsilon )$$ -approximation algorithm for Maximum Group Carpool Matching. For the unweighted variant of both problems when the maximum possible capacity, $$c_{\max }$$ , is bounded by a constant, we provide a local search $$(\frac{1}{2}+ \frac{1}{2c_{\max }} - \varepsilon )$$ -approximation algorithm. We also provide an APX-hardness result, even if the maximum degree and $$c_{\max }$$ are at most 3.

An exact method for constrained maximization of the conditional value-at-risk of a class of stochastic submodular functions

We consider a class of risk-averse submodular maximization problems (RASM) where the objective is the conditional value-at-risk (CVaR) of a random nondecreasing submodular function at a given risk level. We propose valid inequalities and an exact general method for solving RASM under the assumption that we have an efficient oracle that computes the CVaR of the random function. We demonstrate the proposed method on a stochastic set covering problem that admits an efficient CVaR oracle for the random coverage function.

Ranking an Assortment of Products Via Sequential Submodular Optimization

We study a submodular maximization problem motivated by applications in online retail. A platform displays a list of products to a user in response to a search query. The user inspects the first k items in the list for a k chosen at random from a given distribution, and decides whether to purchase an item from that set based on a choice model. The goal of the platform is to maximize the engagement of the shopper defined as the probability of purchase. This problem gives rise to a less-studied variation of submodular maximization in which we are asked to choose an ordering of a set of elements to maximize a linear combination of different submodular functions. First, using a reduction to maximizing submodular functions over matroids, we give an optimal (1-1/e)-approximation for this problem. We then consider a variant in which the platform cares not only about user engagement, but also about diversification across various groups of users, that is, guaranteeing a certain probability of purchase in each group. We characterize the polytope of feasible solutions and give a bi-criteria ((1-1/e)^2,(1-1/e)^2)-approximation for this problem by rounding an approximate solution of a linear programming relaxation. For rounding, we relay on our reduction and the particular rounding techniques for matroid polytopes. For the special case in which underlying submodular functions are coverage functions -- which is practically relevant in online retail -- we propose an alternative LP relaxation and a simpler randomized rounding for the problem. This approach yields to an optimal bi-criteria (1-1/e,1-1/e)-approximation algorithm for the special case of the problem with coverage functions.

Organizing an Influential Social Event Under a Budget Constraint

Recently, the proliferation of event-based social services has made it possible for organizing personalized offline events through the users’ information shared online. In this paper, we study the budget-constrained influential social event organization problem, where the goal is to select a group of influential users with required features to organize a social event under a budget $B$ B . We show that our problem is NP-hard and can be formulated as a submodular maximization problem with mixed packing and covering constraints. We then propose several polynomial time algorithms for our problem with provable approximation ratios, which adopt a novel “surrogate optimization” approach and the method of reverse-reachable set sampling. Moreover, we also consider the case where the influence spread function is unknown and can be arbitrarily selected from a set of candidate submodular functions, and extend our algorithms to address a “robust influential event organization” problem under this case. Finally, we conduct extensive experiments using real social networks to test the performance of our algorithms, and the experimental results demonstrate that our algorithms significantly outperform the prior studies both on the running time and on the influence spread.

Submodular Maximization Through the Lens of Linear Programming

The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a 1/2-approximation, and we also present an appropriate extension to the nonmonotone setting. Moreover, we describe a very general local search procedure that applies to a wide range of constraint families and unifies as well as extends previous methods. In our framework, we match known approximation guarantees while disentangling and simplifying previous approaches. Moreover, despite its generality, we are able to show that our local search procedure is slightly faster than previous specialized methods. Furthermore, we negatively answer the open question whether a linear optimization oracle may be enough to obtain strong approximation algorithms for submodular maximization. We do this by providing an example of a constraint family on a ground set of size n for which, if only given a linear optimization oracle, any algorithm for submodular maximization with a polynomial number of calls to the linear optimization oracle has an approximation ratio of only [Formula: see text].

Price of dependence: stochastic submodular maximization with dependent items

In this paper, we study the stochastic submodular maximization problem with dependent items subject to packing constraints such as matroid and knapsack constraints. The input of our problem is a finite set of items, and each item is in a particular state from a set of possible states. After picking an item, we are able to observe its state. We assume a monotone and submodular utility function over items and states, and our objective is to select a group of items adaptively so as to maximize the expected utility. Previous studies on stochastic submodular maximization often assume that items' states are independent, however, this assumption may not hold in general. This motivates us to study the stochastic submodular maximization problem with dependent items. We first introduce the concept of \emph{degree of independence} to capture the degree to which one item's state is dependent on others'. Then we propose a non-adaptive policy that approximates the optimal adaptive policy within a factor of $\alpha(1-e^{-\frac{\kappa}{2}+\frac{\kappa}{18m^2}}-\frac{\kappa+2}{3m\kappa})$ where the value of $\alpha$ is depending on the type of constraints, e.g., $\alpha=1$ for matroid constraint, $\kappa$ is the degree of independence, e.g., $\kappa=1$ for independent items, and $m$ is the number of items. We also analyze the adaptivity gap, i.e., the ratio of the values of best adaptive policy and best non-adaptive policy, of our problem with prefix-closed constraints.

Submodular Optimization for Coupled Task Allocation and Intermittent Deployment Problems

In this paper, we demonstrate a formulation for optimizing coupled submodular maximization problems with provable sub-optimality bounds. In robotics applications, it is quite common that optimization problems are coupled with one another and therefore cannot be solved independently. Specifically, we consider two problems coupled if the outcome of the first problem affects the solution of a second problem that operates over a longer time scale. For example, in our motivating problem of environmental monitoring, we posit that multi-robot task allocation will potentially impact environmental dynamics and thus influence the quality of future monitoring, here modeled as a multi-robot intermittent deployment problem. The general theoretical approach for solving this type of coupled problem is demonstrated through this motivating example. Specifically, we propose a method for solving coupled problems modeled by submodular set functions with matroid constraints. A greedy algorithm for solving this class of problem is presented, along with sub-optimality guarantees. Finally, practical optimality ratios are shown through Monte Carlo simulations to demonstrate that the proposed algorithm can generate near-optimal solutions with high efficiency.

Constrained Submodular Maximization via a Nonsymmetric Technique

The study of combinatorial optimization problems with submodular objectives has attracted much attention in recent years. Such problems are important in both theory and practice because their objective functions are very general. Obtaining further improvements for many submodular maximization problems boils down to finding better algorithms for optimizing a relaxation of them known as the multilinear extension. In this work, we present an algorithm for optimizing the multilinear relaxation whose guarantee improves over the guarantee of the best previous algorithm (by Ene and Nguyen). Moreover, our algorithm is based on a new technique that is, arguably, simpler and more natural for the problem at hand. In a nutshell, previous algorithms for this problem rely on symmetry properties that are natural only in the absence of a constraint. Our technique avoids the need to resort to such properties, and thus seems to be a better fit for constrained problems.

Stochastic Submodular Maximization with Performance-Dependent Item Costs

We formulate a new stochastic submodular maximization problem by introducing the performance-dependent costs of items. In this problem, we consider selecting items for the case where the performance of each item (i.e., how much an item contributes to the objective function) is decided randomly, and the cost of an item depends on its performance. The goal of the problem is to maximize the objective function subject to a budget constraint on the costs of the selected items. We present an adaptive algorithm for this problem with a theoretical guaran-√ tee that its expected objective value is at least (1−1/ 4 e)/2 times the maximum value attained by any adaptive algorithms. We verify the performance of the algorithm through numerical experiments.

A budgeted maximum multiple coverage model for cybersecurity planning and management

This article studies how to identify strategies for mitigating cyber-infrastructure vulnerabilities. We propose an optimization framework that prioritizes the investment in security mitigations to maximize the coverage of vulnerabilities. We use multiple coverage to reflect the implementation of a layered defense, and we consider the possibility of coverage failure to address the uncertainty in the effectiveness of some mitigations. Budgeted Maximum Multiple Coverage (BMMC) problems are formulated, and we demonstrate that the problems are submodular maximization problems subject to a knapsack constraint. Other variants of the problem are formulated given different possible requirements for selecting mitigations, including unit cost cardinality constraints and group cardinality constraints. We design greedy approximation algorithms for identifying near-optimal solutions to the models. We demonstrate an optimal (1–1/e)-approximation ratio for BMMC and a variation of BMMC that considers the possibility of coverage failure, and a 1/2-approximation ratio for a variation of BMMC that uses a cardinality constraint and group cardinality constraints. The computational study suggests that our models yield robust solutions that use a layered defense and provide an effective mechanism to hedge against the risk of possible coverage failure. We also find that the approximation algorithms efficiently identify near-optimal solutions, and that a Benders branch-and-cut algorithm we propose can find provably optimal solutions to the vast majority of our test instances within an hour for the variations of the proposed models that consider coverage failures.

Programmable Hierarchical C-RAN: From Task Scheduling to Resource Allocation

Traffic delay is a key metric to measure the quality-of-service of next-generation wireless communication networks. In this paper, we consider a cloud radio access network architecture with a hierarchical structure of virtual controllers and multiple clusters of remote radio heads (RRHs). A high-level controller coordinates control plane decisions among local controllers and each local controller is in charge of a cluster of RRHs. Moreover, each local controller is equipped with one server for creating virtual machines (VMs) to execute the users’ baseband processing tasks. Then, under the considered architecture, we aim to minimize the average delay consisting of task execution delay and signal transmission delay under total power constraint, by joint optimization of task scheduling and resource allocation, including VM allocation and RRH assignment. Due to the non-deterministic polynomial-time hardness (NP-hardness) of the joint optimization problem, we translate it into a matroid constrained submodular maximization problem and propose heuristic algorithms to find solutions with 0.5-approximation. Besides, both centralized and distributed control schemes are considered. In the centralized control scheme, all decisions about task scheduling, VM allocation, and RRH assignment are made in the high-level controller. But in the distributed control scheme, the high-level controller is only in charge of task scheduling based on graph theory and the local controllers are responsible for their respective VM allocation and RRH assignment. The simulation results show that the proposed algorithms can achieve better performance than the separate optimization of VM allocation and RRH assignment.

Submodular Function Optimization for Motion Clustering and Image Segmentation.

In this paper, we propose a framework of maximizing quadratic submodular energy with a knapsack constraint approximately, to solve certain computer vision problems. The proposed submodular maximization problem can be viewed as a generalization of the classic 0/1 knapsack problem. Importantly, maximization of our knapsack constrained submodular energy function can be solved via dynamic programing. We further introduce a range-reduction step prior to dynamic programing as a two-stage procedure for more efficient maximization. In order to demonstrate the effectiveness of the proposed energy function and its maximization algorithm, we apply it to two representative computer vision tasks: image segmentation and motion trajectory clustering. Experimental results of image segmentation demonstrate that our method outperforms the classic segmentation algorithms of graph cuts and random walks. Moreover, our framework achieves better performance than state-of-the-art methods on the motion trajectory clustering task.

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.

Distributed Submodular Maximization With Limited Information

We consider a class of distributed submodular maximization problems in which each agent must choose a single strategy from its strategy set. The global objective is to maximize a submodular function of the strategies chosen by each agent. When choosing a strategy, each agent has access to only a limited number of other agents’ choices. For each of its strategies, an agent can evaluate its marginal contribution to the global objective given its information. The main objective is to investigate how this limitation of information about the strategies chosen by other agents affects the performance when agents make choices according to a local greedy algorithm. In particular, we provide lower bounds on the performance of greedy algorithms for submodular maximization, which depend on the clique number of a graph that captures the information structure. We also characterize graph-theoretic upper bounds in terms of the chromatic number of the graph. Finally, we demonstrate how certain graph properties limit the performance of the greedy algorithm. Simulations on several common models for random networks demonstrate our results.

Online Submodular Maximization with Free Disposal

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k -uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α ∞ ≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k -uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α ∞ ≈ 0.3178.

Efficient Representative Subset Selection over Sliding Windows

Representative subset selection (RSS) is an important tool for users to draw insights from massive datasets. Existing literature models RSS as the submodular maximization problem to capture the "diminishing returns" property of the representativeness of selected subsets, but often only has a single constraint (e.g., cardinality), which limits its applications in many real-world problems. To capture the data recency issue and support different types of constraints, we formulate dynamic RSS in data streams as maximizing submodular functions subject to general $d$-knapsack constraints (SMDK) over sliding windows. We propose a \textsc{KnapWindow} framework (KW) for SMDK. KW utilizes the \textsc{KnapStream} algorithm (KS) for SMDK in append-only streams as a subroutine. It maintains a sequence of checkpoints and KS instances over the sliding window. Theoretically, KW is $\frac{1-\varepsilon}{1+d}$-approximate for SMDK. Furthermore, we propose a \textsc{KnapWindowPlus} framework (KW$^{+}$) to improve upon KW. KW$^{+}$ builds an index \textsc{SubKnapChk} to manage the checkpoints and KS instances. \textsc{SubKnapChk} deletes a checkpoint whenever it can be approximated by its successors. By keeping much fewer checkpoints, KW$^{+}$ achieves higher efficiency than KW while still guaranteeing a $\frac{1-\varepsilon'}{2+2d}$-approximate solution for SMDK. Finally, we evaluate the efficiency and solution quality of KW and KW$^{+}$ in real-world datasets. The experimental results demonstrate that KW achieves more than two orders of magnitude speedups over the batch baseline and preserves high-quality solutions for SMDK over sliding windows. KW$^{+}$ further runs 5-10 times faster than KW while providing solutions with equivalent or even better utilities.

Delay-Power-Rate-Distortion Optimization of Video Representations for Dynamic Adaptive Streaming

Dynamic adaptive streaming addresses user heterogeneity by providing multiple encoded representations at different rates and/or resolutions for the same video content. For delay-sensitive applications, such as live streaming, there is however a stringent requirement on the encoding delay, and usually the encoding power (or rate) budget is also limited by the computational (or storage) capacity of the server. It is therefore important, yet challenging, to optimally select the source coding parameters for each encoded representation in order to minimize the resource consumption while maintaining a high quality of experience for the users. To address this, we propose an optimization framework with an optimal representation selection problem for delay, power, and rate constrained adaptive video streaming. Then, by the optimal selection of source coding parameters for each selected representation, we maximize the overall expected user satisfaction, subject not only to the encoding rate constraint, but also to the delay and power constraints at the server. We formulate the proposed optimization problem as an integer linear program formulation to provide the performance upper bound, and as a submodular maximization problem with two knapsack constraints to develop a practically feasible algorithm. Simulation results show that the proposed weighted rate and power cost benefit greedy algorithm is able to achieve a near-optimal performance with very low time complexity. In addition, it can strike the best tradeoff both between the rate and power cost, and between the algorithm’s performance and the delay requirements proposed by delay sensitive applications.