Monotone Submodular Function Research Articles

Fully Dynamic Submodular Maximization over Matroids

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this significant problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an \({\tilde{O}({k^{2}}{\varepsilon})}\) amortized update time (in the number of insertions and deletions) and yields a \({(4+O(\varepsilon))}\) -approximate solution with respect to the dynamic optimum, where \(k\) is the rank of the matroid.

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.

Runtime Analysis of Quality Diversity Algorithms

AbstractQuality diversity (QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the “number of ones” feature space, where the ith cell stores the best solution amongst those with a number of ones in $$[(i-1)k, ik-1]$$ [ ( i - 1 ) k , i k - 1 ] . Here k is a granularity parameter $$1 \le k \le n+1$$ 1 ≤ k ≤ n + 1 . We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all k and analyse the expected optimisation time of QD on OneMax and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a $${(1-1/e)}$$ ( 1 - 1 / e ) -approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of an edge-weighted graph, we show that QD finds a minimum spanning forest in expected polynomial time. We further consider QD’s performance on classes of transformed functions in which the feature space is not well aligned with the problem. The asymptotic performance is unaffected by transformations on easy functions like OneMax. Applying a worst-case transformation to a deceptive problem increases the expected optimisation time from $$O(n^2 \log n)$$ O ( n 2 log n ) to an exponential time. However, QD is still faster than a (1+1) EA by an exponential factor.

Deletion-Robust Submodular Maximization with Knapsack Constraints

Submodular maximization algorithms have found wide applications in various fields such as data summarization, recommendation systems, and active learning. In recent years, deletion-robust submodular maximization algorithms have garnered attention due to their significant implications in scenarios where some data points may be removed due to user preferences or privacy concerns, such as in recommendation systems and influence maximization. In this paper, we study the fundamental problem of submodular maximization with knapsack constraints and propose a robust streaming algorithm for it. To the best of our knowledge, our algorithm is the first to solve this problem for non-monotone submodular functions and can achieve an approximation ratio of 1/(6.82+2.63d)-ϵ under a near-optimal summary size of O(k+r), where k denotes the maximum cardinality of any feasible solution, d denotes the number of the knapsack constraints and r is the robustness parameter. For monotone submodular functions, our algorithm can achieve an approximation ratio of 1/(2+2d)-ϵ under a near-optimal summary size of O(k+r), significantly improving upon the best-known ratio of Ω((1/d-ϵ)^2). The empirical performance of our algorithm is extensively evaluated in several applications including influence maximization and recommendation systems, and the experimental results demonstrate the effectiveness of our algorithm.

Approximation algorithm of maximizing non-monotone non-submodular functions under knapsack constraint

The maximization of non-negative monotone submodular functions under a certain constraint is intensively studied. However, there are few works considering the maximization of non-monotone non-submodular functions, which even might be negative. These functions also have many applications, such as optimal marketing for revenue maximization over social networks, budget allocation problems, and Epidemic transmission, etc. In our paper, we discuss the maximization of the non-monotone non-submodular functions under a knapsack constraint, which even might be negative, and explore the performance under the greedy algorithm. We obtain some improved approximate ratios, for the functions with different properties, such as the non-negative weak-monotone weak-submodular functions, and the weak-submodular weak-supermodular functions that might be negative. Moreover, we generalize the results to the functions with weak subadditivity and curvature.

A Truthful Reverse Auction Mechanism for Federated Learning Utility Maximization Resource Allocation in Edge–Cloud Collaboration

Federated learning is a promising technique in cloud computing and edge computing environments, and designing a reasonable resource allocation scheme for federated learning is particularly important. In this paper, we propose an auction mechanism for federated learning resource allocation in the edge–cloud collaborative environment, which can motivate data owners to participate in federated learning and effectively utilize the resources and computing power of edge servers, thereby reducing the pressure on cloud services. Specifically, we formulate the federated learning platform data value maximization problem as an integer programming model with multiple constraints, develop a resource allocation algorithm based on the monotone submodular value function, devise a payment algorithm based on critical price theory and demonstrate that the mechanism satisfies truthfulness and individual rationality.

Shortest Cycles with Monotone Submodular Costs

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function f defined on the edges (or the vertices) of an undirected graph G , we seek for a cycle C in G of minimum cost 𝖮𝖯𝖳 = f(C) . We give an algorithm that given an n -vertex graph G , parameter ɛ > 0, and the function f represented by an oracle, in time n 𝒪 (log 1/ɛ) finds a cycle C in G with f(C) ≤ (1+ɛ). 𝖮𝖯𝖳. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest ( s,t -Path problem, which requires exponentially many queries to the oracle for finding an n 2/3-ɛ -approximation Goel et al. [ 7 ], FOCS 2009. We complement our algorithm with a matching lower bound. We show that for every ɛ > 0, obtaining a (1+ɛ)-approximation requires at least n Ω (log 1/ ɛ) queries to the oracle. When the function f is integer-valued, our algorithm yields that a cycle of cost 𝖮𝖯𝖳 can be found in time n 𝒪(log 𝖮𝖯𝖳) . In particular, for 𝖮𝖯𝖳 = n 𝒪(1) this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that n 𝒪(log n ) queries are required even when 𝖮𝖯𝖳= 𝒪( n ). We also consider special cases of monotone submodular functions, corresponding to the number of different color classes needed to cover a cycle in an edge-colored multigraph G . For special cases of the corresponding minimization problem, we obtain fixed-parameter tractable algorithms and polynomial-time algorithms, when restricted to certain classes of inputs.

Practical Charger Placement Scheme for Wireless Rechargeable Sensor Networks with Obstacles

Benefitting from the maturation of Wireless Power Transfer technology, Wireless Rechargeable Sensor Networks have become a promising solution for prolonging network lifetime. In practical charging scenarios, obstacles are ubiquitous. However, most prior arts have failed to consider the combined impacts of the material, size, and location of obstacles on the charging performance, making these schemes unsuitable for real applications. In this article, we study a fundamental issue of W ireless ch A rger placement w I th obs T acles (WAIT), that is, how to place wireless chargers by comprehensively considering these parameters of obstacles, such that the overall charging utility is maximized. To tackle the WAIT problem, we first build a practical charging model with obstacles by introducing shadow fading, and conduct experiments to verify its correctness. Then, we design a piecewise constant function to approximate the nonlinear charging power. Afterwards, we develop a Dominating Coverage Set extraction algorithm to reduce the continuous solution space to a limited number. Finally, we prove the WAIT problem is a maximizing monotone submodular function problem, and propose a 1-1/e-ε approximation algorithm to address it. Extensive simulations and field experiments show that our scheme outperforms comparison algorithms by at least 20.6% in charging utility improvement.

User-Assisted Base Station Caching and Cooperative Prefetching for High-Speed Railway Systems

With the aim of reducing the transmission latency, this paper proposes a scheme of user-assisted base station (BS) caching and cooperative prefetching for high-speed railway (HSR) communications. In the model under consideration, neighboring BSs periodically exchange information, including the coverage area and communication rate, to facilitate content caching and prefetching. As an additional means, users can cache contents that are different from BSs. Specifically, we construct an optimization problem for content caching and prefetching that minimizes the overall transmission latency. Moreover, this problem is decomposed into two subproblems, namely the one aiming to maximize the available throughput of BSs and the other attempting to maximize the user cache utilization subject to the constraint of the BS cache. We demonstrate that the objective function of the first subproblem is equivalent to a monotone submodular function subject to matroid constraints. Therefore, it can be efficiently solved with a greedy algorithm. Meanwhile, the user cache issue can be viewed as a weighted sum maximization problem. Simulation results are presented to verify that the proposed scheme can not only reduce the average transmission latency but also improve the hit probability and system throughput.

The One-Way Communication Complexity of Submodular Maximization with Applications to Streaming and Robustness

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multiplayer model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two-player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the connections mentioned previously, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets (i.e., sets respecting the cardinality constraint) are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the link to the (non-streaming) robust setting mentioned previously, both of these algorithms improve on the current state of the art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2+ɛ hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best-known approximation algorithm.

Output–Input Ratio Maximization for Online Social Networks: Algorithms and Analyses

In the last few decades, profit maximization (PM), which is mainly considered for maximizing net profits, i.e., the difference between gain and cost, has been a prominent issue for online social networks (OSNs). However, the output-to-input ratio, which is an important metric in economics, is also worth studying for OSNs. In this article, we present a novel problem that considers the PM problem from the ratio of gain and cost, known as output-to-input ratio maximization (OIRM). Unfortunately, it is neither submodular nor supermodular. The hill-climbing greedy algorithm for solving this problem is a <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex>1-e^-(1-c_g)</tex> </formula> approximation algorithm, where c_g is the curvature of the monotone submodular set function g. To speed up the hill-climbing greedy algorithm, we propose the threshold decrease algorithm and prove that its approximation ratio is <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex>1-e^-(1-c_g)^2-ε</tex> </formula> . In addition, based on the relationship between classical net PM and OIRM, the algorithms for solving PM can also solve OIRM. Finally, we evaluate the performance of our algorithms using massive experiments on real datasets. To the best of our knowledge, this is the first time to study the OIRM in viral marketing.

Matroid-constrained vertex cover

In this paper, we introduce the Matroid-Constrained Vertex Cover problem: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied maxk-vertex cover problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint.In the first part of this work, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if k is the rank of the matroid, we obtain (1−ε) approximation using (1/ε)O(k)nO(1) time for partition and laminar matroids and using (1/ε+k)O(k)nO(1) time for transversal matroids. This extends a result of Manurangsi for uniform matroids [33]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics.In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees 2/3≈0.66 approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a 2/3⋅(1−1/(p+1)) approximation in nO(p) time, for any integer p. We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of 2/3 is likely the best possible, using the currently known techniques of local search.

DCEC: D2D-Enabled Cost-Aware Cooperative Caching in MEC Networks

Various kinds of powerful intelligent mobile devices (MDs) need to access multimedia content anytime and anywhere, which places enormous pressure on mobile wireless networks. Fetching content from remote sources may introduce overly long accessing delays, which will result in a poor quality of experience (QoE). In this article, we considered the advantages of combining mobile/multi-access edge computing (MEC) with device-to-device (D2D) technologies. We propose a D2D-enabled cooperative edge caching (DCEC) architecture to reduce the delay of accessing content. We designed the DCEC caching management scheme through the maximization of a monotone submodular function under matroid constraints. The DCEC scheme includes a proactive cache placement algorithm and a reactive cache replacement algorithm. Thus, we obtained an optimal content caching and content update, which minimized the average delay cost of fetching content files. Finally, simulations compared the DCEC network architecture with the MEC and D2D networks and the DCEC caching management scheme with the least-frequently used and least-recently used scheme. The numerical results verified that the proposed DCEC scheme was effective at improving the cache hit ratio and the average delay cost. Therefore, the users’ QoE was improved.

A Differentially Private Approximation Algorithm for Submodular Maximization Under a Polymatroid Constraint Over the Integer Lattice

We consider the problem of local privacy where actions are subsets of a ground multiset and expectation rewards are modeled by a [Formula: see text]decomposable monotone submodular function. For the DR-submodular maximization problem under a polymatroid constraint, Soma and Yoshida [26] provide a continuous greedy algorithm for no-privacy setting. In this paper, we obtain the first differentially private algorithm for DR-submodular maximization subject to a polymatroid constraint. Our algorithm achieves a [Formula: see text]approximation with a little loss and runs in [Formula: see text][Formula: see text] times where [Formula: see text] is the rank of the base polymatroid and [Formula: see text] is the size of ground set. Along the way, we analyze the utility and privacy of our algorithm. A concrete experiment to simulate the privacy Uber pickups location problem is provided, and our algorithm performs well within the agreed range.

A Submodular Optimization Framework for Imbalanced Text Classification With Data Augmentation

In the domain of text classification, imbalanced datasets are a common occurrence. The skewed distribution of the labels of these datasets poses a great challenge to the performance of text classifiers. One popular way to mitigate this challenge is to augment underwhelmingly represented labels with synthesized items. The synthesized items are generated by data augmentation methods that can typically generate an unbounded number of items. To select the synthesized items that maximize the performance of text classifiers, we introduce a novel method that selects items that jointly maximize the likelihood of the items belonging to their respective labels and the diversity of the selected items. Our proposed method formulates the joint maximization as a monotone submodular objective function, whose solution can be approximated by a tractable and efficient greedy algorithm. We evaluated our method on multiple real-world datasets with different data augmentation techniques and text classifiers and compared results with several baselines. The experimental results demonstrate the effectiveness and efficiency of the proposed method.

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].

Multi-objective evolutionary algorithms are generally good: Maximizing monotone submodular functions over sequences

Evolutionary algorithms (EAs) are general-purpose optimization algorithms, inspired by natural evolution. Recent theoretical studies have shown that EAs can achieve good approximation guarantees for solving the problem classes of submodular optimization, which have a wide range of applications, such as maximum coverage, sparse regression, influence maximization, document summarization and sensor placement, just to name a few. Though they have provided some theoretical explanation for the general-purpose nature of EAs, the considered submodular objective functions are defined only over sets or multisets. To complement this line of research, this paper studies the problem class of maximizing monotone submodular functions over sequences, where the objective function depends on the order of items. We prove that for each kind of previously studied monotone submodular objective functions over sequences, i.e., prefix monotone submodular functions, weakly monotone and strongly submodular functions, and DAG monotone submodular functions, a simple multi-objective EA, i.e., GSEMO, can always reach or improve the best known approximation guarantee after running polynomial time in expectation. Note that these best-known approximation guarantees can be obtained only by different greedy-style algorithms before. Empirical studies on various applications, e.g., accomplishing tasks, maximizing information gain, search-and-tracking and recommender systems, show the excellent performance of the GSEMO.

Online Learning via Offline Greedy Algorithms: Applications in Market Design and Optimization

Motivated by online decision making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have [Formula: see text] (approximate) regret under the full information setting. We further introduce a bandit extension of Blackwell approachability that we call Bandit Blackwell approachability. We leverage this notion to transform greedy robust offline algorithms into a [Formula: see text] (approximate) regret in the bandit setting. Demonstrating the flexibility of our framework, we apply our offline-to-online transformation to several problems at the intersection of revenue management, market design, and online optimization, including product ranking optimization in online platforms, reserve price optimization in auctions, and submodular maximization. We also extend our reduction to greedy-like first-order methods used in continuous optimization, such as those used for maximizing continuous strong DR monotone submodular functions subject to convex constraints. We show that our transformation, when applied to these applications, leads to new regret bounds or improves the current known bounds. We complement our theoretical studies by conducting numerical simulations for two of our applications, in both of which we observe that the numerical performance of our transformations outperforms the theoretical guarantees in practical instances. This paper was accepted by George Shanthikumar, data science. Funding: R. Niazadeh was supported by research funding from University of Chicago Booth School of Business. N. Golrezaei was supported in part by the Young Investigator Program Award from the Office of Naval Research [N00014-21-1-2776] and the MIT Research Support Award. Supplemental Material: The Online Appendix is available at https://doi.org/10.1287/mnsc.2022.4558

Analyzing Residual Random Greedy for monotone submodular maximization

Residual Random Greedy (RRGreedy) is a natural randomized version of the greedy algorithm for submodular maximization. It was introduced to address non-monotone submodular maximization [1] and plays an important role in the deterministic algorithm for monotone submodular maximization that beats the (1/2)-factor barrier [2]. In this work, we analyze RRGreedy for monotone submodular functions along two fronts: (1) For matroid constrained maximization of monotone submodular functions with bounded curvature α, we show that RRGreedy achieves a (1/(1+α))-approximation in the worst-case (i.e., irrespective of the randomness in the algorithm). In particular, this implies that it achieves a (1/2)-approximation in the worst-case (not just in expectation). (2) We generalize RRGreedy to k matroid intersection constraints and show that the generalization achieves a (1/(k+1))-approximation in expectation relative to the optimum value of the Lovász relaxation over the intersection of k matroid polytopes. Our results suggest that RRGreedy is at least as good as Greedy for matroid and matroid intersection constraints.

Efficient Streaming Algorithms for Maximizing Monotone DR-Submodular Function on the Integer Lattice

In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most submodular functions are specified in a set function. Meanwhile, recent advancements have been studied for maximizing a diminishing return submodular (DR-submodular) function on the integer lattice. Because plenty of publications show that the DR-submodular function has wide applications in optimization problems such as sensor placement impose problems, optimal budget allocation, social network, and especially machine learning. In this research, we propose two main streaming algorithms for the problem of maximizing a monotone DR-submodular function under cardinality constraints. Our two algorithms, which are called StrDRS1 and StrDRS2, have (1/2−ϵ), (1−1/e−ϵ) of approximation ratios and O(nϵlog(logBϵ)logk), O(nϵlogB), respectively. We conducted several experiments to investigate the performance of our algorithms based on the budget allocation problem over the bipartite influence model, an instance of the monotone submodular function maximization problem over the integer lattice. The experimental results indicate that our proposed algorithms not only provide solutions with a high value of the objective function, but also outperform the state-of-the-art algorithms in terms of both the number of queries and the running time.