Submodular Maximization Problem Research Articles

An accelerated deterministic algorithm for maximizing monotone submodular minus modular function with cardinality constraint

Submodular optimization not only covers some classical combinatorial optimization problems, but also has a wide range of applications in fields such as machine learning and artificial intelligence. For submodular maximization problems with constraints, some work has been done including the design of approximation algorithms, the measurement of approximation algorithms in terms of quality and efficiency, etc. In this paper, we consider the problem of maximizing a non-negative monotone submodular function minus a non-negative modular function with the cardinality constraint. This model has been applied to many scenarios, such as team formation problem, influence maximization problem, recommender systems problem, etc. We propose a threshold algorithm that achieve a (1/2−O(ε),2)-bicriteria approximation ratio and query complexity O(nlog⁡n). Our algorithm makes a small sacrifice in the approximation ratio but improves the best query complexity result of existing deterministic algorithms from O(n2) to O(nlog⁡n) in the worst case.

Adaptive Influence Maximization: Adaptability via Nonadaptability

Adaptive influence maximization is an important research problem in computational social networks, which is also a typical problem in the study of adaptive processing of information and adaptive construction of objects. In this paper, we propose a new method that reduces the adaptive influence maximization problem into a nonadaptive one in a different social network, so that an adaptive optimization can be solved by those methods for nonadaptive optimization. In addition, we provide a new approximation algorithm for the submodular maximization problem with a knapsack constraint, which runs in [Formula: see text] time and has performance ratio [Formula: see text], where n is the number of nodes in the network. The ratio is better than the best known previous one with the same running time. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This research is supported in part by the National Natural Science Foundation of China [Grant U20A2068].

Addressing Heterogeneity in Federated Learning with Client Selection via Submodular Optimization

Federated learning (FL) has been proposed as a privacy-preserving distributed learning paradigm, which differs from traditional distributed learning in two main aspects: the systems heterogeneity, meaning that clients participating in training have significant differences in systems performance including CPU frequency, dataset size, and transmission power, and the statistical heterogeneity, indicating that the data distribution among clients exhibits Non-Independent Identical Distribution. Therefore, the random selection of clients will significantly reduce the training efficiency of FL. In this article, we propose a client selection mechanism considering both systems and statistical heterogeneity, which aims to improve the time-to-accuracy performance by trading off the impact of systems performance differences and data distribution differences among the clients on training efficiency. First, client selection is formulated as a combinatorial optimization problem that jointly optimizes systems and statistical performance. Then, we generalize it to a submodular maximization problem with knapsack constraint, and propose the Iterative Greedy with Partial Enumeration (IGPE) algorithm to greedily select the suitable clients. Then, the approximation ratio of IGPE is analyzed theoretically. Extensive experiments verify that the time-to-accuracy performance of the IGPE algorithm outperforms other compared algorithms in a variety of heterogeneous environments.

H-hop independently submodular maximization problem with curvature

The Connected Sensor Problem (CSP) presents a prevalent challenge in the realms of communication and Internet of Things (IoT) applications. Its primary aim is to maximize the coverage of users while maintaining connectivity among K sensors. Addressing the challenge of managing a large user base alongside a finite number of candidate locations, this paper proposes an extension to the CSP: the h-hop independently submodular maximization problem characterized by curvature α. We have developed an approximation algorithm that achieves a ratio of 1−e−α(2h+3)α. The efficacy of this algorithm is demonstrated on the CSP, where it shows superior performance over existing algorithms, marked by an average enhancement of 8.4%.

Data labeling through the centralities of co-reference networks improves the classification accuracy of scientific papers

Labeled data are fed to learning models of classification tasks to help them learn to classify unlabeled data. Massive papers are hinged by citations to a few influential papers, much smaller than the total size, which, if labeled, would cause the spread of label information to the most of the papers. We utilized the co-reference relationship between papers since the references cited by a paper dataset usually cannot be contained by the dataset. We stated the problem of optimal paper labeling as the problem of picking a given fraction of nodes from a co-reference network to maximize the number of their neighbors, which is a submodular maximization problem with a cardinality constraint, NP-hard for general networks. We approximately solved it by picking nodes according to the ranks of specific network centralities. We further proved that labeling papers according to the rank of degree, the lowest-complexity centrality, can give a near-optimal solution with specific constraints on the maximum degree of co-reference network and labeling proportion. Experimental results show that our method brings a significant improvement in the accuracy of classification.

A Data-Driven Approach for Optimizing Early-Stage Electric Vehicle Charging Station Placement

This paper presents a novel and practical data-driven approach to sub-optimally allocate charging stations for electric vehicles (EVs) in an early-stage setting. Specifically, we investigate the following problem: For a city with a limited budget for public EV charging infrastructure construction, where should the charging stations be deployed in order to promote the transition of EVs from traditional cars? We develop a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> -nearest model and a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> -nearest model that can capture people's satisfaction towards a certain design and formulate the early-stage EV charging station placement problem as a monotone submodular maximization problem utilizing fine-grained population, trip, transportation network and POI data. A greedy-based algorithm is proposed to solve the problem efficiently with a provable approximation ratio. A case study of Haikou is provided to demonstrate the effectiveness of our approach.

Learning automata-accelerated greedy algorithms for stochastic submodular maximization

Submodular function maximization is a typical combinatorial optimization problem that arises in many areas of computer science, such as data summarization, batch mode active learning, and recommendations. We focus on the stochastic submodular maximization (SSM) problems, where an unknown monotonic submodular function is to be maximized under cardinality constraints. Previous research commonly relied on the Monte-Carlo method to assess the stochasticity in submodular functions, which is challenged by the accuracy–efficiency dilemma. Achieving high accuracy requires a large number of Monte-Carlo simulations, while efficiency necessitates the minimization of simulation costs. In this paper, we propose a new family of greedy algorithms based on learning automata (LA) for SSM problems. By exploring the solution space more efficiently, the proposed LA-based greedy family accelerates conventional greedy algorithms. Instead of massively evaluating the expected benefits of each element indiscriminately, an LA-based learning strategy is designed to quickly identify the optimal element among all candidates. Based on the strategy, we present a primitive greedy algorithm and two improved variants, along with proofs of the same approximation guarantee as conventional greedy algorithms. We conducted experiments on three representative SSM tasks, nonparametric learning for structured data, exemplar-based clustering for unstructured data, and influence maximization for graph data. The results verified the advantage of the proposed LA-based greedy family against previous greedy algorithms.

Group Equality in Adaptive Submodular Maximization

In this paper, we study the classic submodular maximization problem subject to a group equality constraint under both nonadaptive and adaptive settings. It is shown that the utility function of many machine learning applications, including data summarization, influence maximization in social networks, and personalized recommendation, satisfies the property of submodularity. Hence, maximizing a submodular function subject to various constraints can be found at the heart of many of those applications. On a high level, submodular maximization aims to select a group of most representative items (e.g., data points). However, the design of most existing algorithms does not incorporate the fairness constraint, leading to underrepresentation or overrepresentation of some particular groups. This motivates us to study the submodular maximization problem with group equality, in which we aim to select a group of items to maximize a (possibly nonmonotone) submodular utility function subject to a group equality constraint. To this end, we develop the first constant-factor approximation algorithm for this problem. The design of our algorithm is robust enough to be extended to solving the submodular maximization problem under a more complicated adaptive setting. Moreover, we further extend our study to incorporating a global cardinality constraint and other fairness notations. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0384 .

An Approximation Algorithm for the h-Hop Independently Submodular Maximization Problem and Its Applications

An Approximation Algorithm for the h-Hop Independently Submodular Maximization Problem and Its Applications

Quality-Aware Joint Caching, Computing and Communication Optimization for Video Delivery in Vehicular Networks

As being clear that the joint optimization of caching, computing, and communication (3 C) can significantly improve vehicular network performance, this paper investigates the quality-aware 3 C optimization for video delivery in vehicular networks. By incorporating the quality-awareness into 3 C models, we formulate both the quality-aware joint 3 C optimization problem and the quality-aware joint computing and communication (2 C) optimization problem with predetermined caching. By observing that the proposed 2 C and 3 C problems can be interpreted as submodular maximization problems with constraints, we propose a solution methodology using the submodularity and knapsack interpretation. Based on the proposed methodology, we then propose efficient solution approaches assuming batch input and streaming input, respectively. Complexity of the proposed solution approaches is analyzed. In addition, extensive simulations are conducted to evaluate the proposed solution approaches. Results show that our proposed approaches can outperform the reference schemes and improve the video delivery of vehicular networks.

Improvement of Submodular Maximization Problems With Routing Constraints via Submodularity and Fourier Sparsity

Various robotic problems (e.g., map exploration, environmental monitoring and spatial search) can be formulated as submodular maximization problems with routing constraints. These problems involve two NP-hard problems, maximal coverage and traveling salesman problems. The generalized cost-benefit algorithm (GCB) is able to solve this problem with a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{1}{2}(1-\frac{1}{e})\widetilde{OPT}$</tex-math></inline-formula> guarantee, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\widetilde{OPT}$</tex-math></inline-formula> is the approximation of optimal performance. There is a gap between the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\widetilde{OPT}$</tex-math></inline-formula> and the optimal solution <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(OPT)$</tex-math></inline-formula> . In this research, the proposed algorithms, Tree-Structured Fourier Supports Set (TS-FSS), utilize the submodularity and sparsity of routing trees to boost GCB performance. The theorems show that the proposed algorithms have a higher optimum bound than GCB. The experiments demonstrate that the proposed approach outperforms benchmark approaches.

Group fairness in non-monotone submodular maximization

Maximizing a submodular function has a wide range of applications in machine learning and data mining. One such application is data summarization whose goal is to select a small set of representative and diverse data items from a large dataset. However, data items might have sensitive attributes such as race or gender, in this setting, it is important to design fairness-aware algorithms to mitigate potential algorithmic bias that may cause over- or under- representation of particular groups. Motivated by that, we propose and study the classic non-monotone submodular maximization problem subject to novel group fairness constraints. Our goal is to select a set of items that maximizes a non-monotone submodular function, while ensuring that the number of selected items from each group is proportionate to its size, to the extent specified by the decision maker. We develop the first constant-factor approximation algorithms for this problem. We also extend the basic model to incorporate an additional global size constraint on the total number of selected items.

The Impact of Measurement Passing in Sensor Network Measurement Selection

This article considers a set of sensors, which, as a group, are tasked with taking measurements of the environment and sending a small subset of the measurements to a centralized data fusion center, where the measurements will be used to estimate the overall state of the environment. The sensors’ goal is to send the most informative set of measurements so that the estimate is as accurate as possible. This problem is formulated as a submodular maximization problem, for which there exists a well-studied greedy algorithm, where each sensor sequentially chooses a set of measurements from its own local set and communicates its decision to the future sensors in the sequence. In this work, sensors can additionally share measurements with one another, in order to augment the decision set of each sensor. We explore how this increase in communication can be exploited to improve the results of the nominal greedy algorithm. Specifically, we show that this measurement passing can improve the quality of the resulting measurement set by up to a factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n+1$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of sensors.

Regularized nonmonotone submodular maximization

In this paper, we present a thorough study of the regularized submodular maximization problem, in which the objective f := g − ℓ can be expressed as the difference between a submodular function and a modular function. This problem has drawn much attention in recent years. While existing works focuses on the case of g being monotone, we investigate the problem with a nonmonotone g. The main technique we use is to introduce a distorted objective function, which varies weights of the submodular component g and the modular component ℓ during the iterations of the algorithm. By combining the weighting technique and measured continuous greedy algorithm, we present an algorithm for the matroid-constrained problem, which has a provable approximation guarantee. In the cardinality-constrained case, we utilize random greedy algorithm and sampling technique together with the weighting technique to design two efficient algorithms. Moreover, we consider the unconstrained problem and propose a much simpler and faster algorithm compared with the algorithms for solving the problem with a cardinality constraint.

Partial-monotone adaptive submodular maximization

Many AI/Machine learning problems require adaptively selecting a sequence of items, each selected item might provide some feedback that is valuable for making better selections in the future, with the goal of maximizing an adaptive submodular function. Most of existing studies in this field focus on either monotone case or non-monotone case. Specifically, if the utility function is monotone and adaptive submodular, Golovin and Krause (J Artif Intell Res 42:427–486, 2011) developed \((1-1/e)\) approximation solution subject to a cardinality constraint. For the cardinality-constrained non-monotone case, Tang (Theor Comput Sci 850:249–261, 2021) showed that a random greedy policy attains an approximation ratio of 1/e. In this work, we generalize the above mentioned results by studying the partial-monotone adaptive submodular maximization problem. To this end, we introduce the notation of adaptive monotonicity ratio \(m\in [0,1]\) to measure the degree of monotonicity of a function. Our main result is to show that for the case of cardinality constraints, if the utility function has an adaptive monotonicity ratio of m and it is adaptive submodular, then a random greedy policy attains an approximation ratio of \(m(1-1/e)+(1-m)(1/e)\). Notably this result recovers the aforementioned \((1-1/e)\) and 1/e approximation ratios when \(m = 1\) and \(m = 0\), respectively. We further extend our results to consider a knapsack constraint and develop a \((m+1)/10\) approximation solution for this general case. One important implication of our results is that even for a non-monotone utility function, we still can attain an approximation ratio close to \((1-1/e)\) if this function is “close” to a monotone function. This leads to improved performance bounds for many machine learning applications whose utility functions are almost adaptive monotone.

Practical Budgeted Submodular Maximization

We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (Operat Res Lett 32:41–43, 2004) showed that by guessing 3 appropriate elements of an optimal solution, and then executing a greedy algorithm, one can obtain the optimal approximation ratio of $$\alpha =1-{1}/{e} \approx 0.632$$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko (Operat Res Lett 32:41–43, 2004) impractical as it leads to a time complexity of roughly $$O(n^5)$$ (this time complexity can be slightly improved using the thresholding technique of Badanidiyuru & Vondrák (in: SODA, 1497–1514, 2014) but only to roughly $$O(n^4)$$ ). Our main results in this paper show that fewer guesses suffice. Specifically, by making only 2 guesses (and using the thresholding technique of Badanidiyuru & Vondrák (in: SODA, 1497–1514, 2014), we get the same optimal approximation ratio of $$\alpha $$ with an improved time complexity of roughly $$O(n^3)$$ . Furthermore, by making only a single guess, we get an almost as good approximation ratio of $$0.6174 > 0.9767\alpha $$ in roughly $$O(n^2)$$ time. Prior to our work, the only approximation algorithms that were known to obtain an approximation ratio close to $$\alpha $$ for BSM were the algorithm of Sviridenko (Operat Res Lett 32:41–43, 2004) and an algorithm of Ene & Nguyen (in: ICALP, 53:1–53:12, 2019) that achieves $$(\alpha -\varepsilon )$$ -approximation. However, the algorithm of Ene & Nguyen (in: ICALP, 53:1–53:12, 2019) requires $${(1/\varepsilon )}^{O(1/\varepsilon ^4)}n\log ^2 n$$ time, and hence, is of theoretical interest only since $${(1/\varepsilon )}^{O(1/\varepsilon ^4)}$$ is huge even for moderate values of $$\varepsilon $$ . In contrast, all the algorithms we analyze are simple and parallelizable, which makes them good candidates for practical use. Recently, Tang et al. (in: Proc ACM Meas Anal Comput Syst, 5(1): 08:1–08:22, 2021) studied a simple greedy algorithm that already has a long research history, and proved that it admits an approximation ratio of at least 0.405 (without any guesses). The last part of this paper improves over the result of Tang et al. (in: Proc ACM Meas Anal Comput Syst, 5(1): 08:1–08:22, 2021) and shows that the approximation ratio of this algorithm is within the range [0.427, 0.462].

An exact solution approach for the mobile multi‐agent sensing problem

AbstractMulti‐agent systems are generally applicable in various fields and aim to coordinate the decisions from agents, each of which makes a local decision. In particular, research on operations management with mobile multi‐agents such as drones has gained prominence in recent years. In this article, we present a mobile multi‐agent sensing problem, which is formulated as a submodular maximization problem under a partition matroid constraint. When events detected by agents lead to severe and catastrophic consequences, obtaining (near)‐optimal solutions by using exact algorithms is crucial to reducing the probability of harmful situations. Therefore, in this article we propose an exact solution approach, using two valid inequalities for our problem to find a (near)‐optimal solution. Moreover, we deal with the risk‐averse decision and its cutting‐plane algorithm for our problem, which is to maximize conditional value‐at‐risk (CVaR). Finally, we show the performance of our algorithms through numerical experiments, including a case study on forest fires.

Beyond pointwise submodularity: Non-monotone adaptive submodular maximization subject to knapsack and k-system constraints

Although the knapsack-constrained and k-system-constrained non-monotone adaptive submodular maximization have been well studied in the literature, it has only been settled given the additional assumption of pointwise submodularity. In this paper, we remove the common assumption on pointwise submodularity and propose the first approximation solutions for both knapsack and k-system constrained adaptive submodular maximization problems. Inspired by two recent studies on non-monotone adaptive submodular maximization, we develop a sampling-based randomized algorithm that achieves a 110 approximation ratio for the case of a knapsack constraint and that achieves a 12k+4 approximation ratio for the case of a k-system constraint.

Stochastic submodular probing with state-dependent costs

In this paper, we study a new stochastic submodular maximization problem with state-dependent costs and rejections. The input of our problem is a budget constraint B, and a set of items whose states (i.e., the marginal contribution and the cost of an item) are drawn from a known probability distribution. The only way to know the realized state of an item is to probe that item. We allow rejections, i.e., after probing an item and knowing its actual state, we must decide immediately and irrevocably whether to add that item to our solution or not. Our objective is to sequentially probe/select a best group of items subject to a budget constraint on the total cost of the selected items. We present a constant approximate solution to this problem. We show that our solution can be extended to an online setting.

Happiness maximizing sets under group fairness constraints

Finding a happiness maximizing set (HMS) from a database, i.e., selecting a small subset of tuples that preserves the best score with respect to any nonnegative linear utility function, is an important problem in multi-criteria decision-making. When an HMS is extracted from a set of individuals to assist data-driven algorithmic decisions such as hiring and admission, it is crucial to ensure that the HMS can fairly represent different groups of candidates without bias and discrimination. However, although the HMS problem was extensively studied in the database community, existing algorithms do not take group fairness into account and may provide solutions that under-represent some groups. In this paper, we propose and investigate a fair variant of HMS (FairHMS) that not only maximizes the minimum happiness ratio but also guarantees that the number of tuples chosen from each group falls within predefined lower and upper bounds. Similar to the vanilla HMS problem, we show that FairHMS is NP-hard in three and higher dimensions. Therefore, we first propose an exact interval cover-based algorithm called IntCov for FairHMS on two-dimensional databases. Then, we propose a bicriteria approximation algorithm called BiGreedy for FairHMS on multi-dimensional databases by transforming it into a submodular maximization problem under a matroid constraint. We also design an adaptive sampling strategy to improve the practical efficiency of BiGreedy. Extensive experiments on real-world and synthetic datasets confirm the efficacy and efficiency of our proposal.