Non-monotone Submodular Functions Research Articles

On submodular prophet inequalities and correlation gap

Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla [31] developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions.In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

Non-monotone Sequential Submodular Maximization

In this paper, we study a fundamental problem in submodular optimization known as sequential submodular maximization. The primary objective of this problem is to select and rank a sequence of items to optimize a group of submodular functions. The existing research on this problem has predominantly concentrated on the monotone setting, assuming that the submodular functions are non-decreasing. However, in various real-world scenarios, like diversity-aware recommendation systems, adding items to an existing set might negatively impact the overall utility. In response, we propose to study this problem with non-monotone submodular functions and develop approximation algorithms for both flexible and fixed length constraints, as well as a special case with identical utility functions. The empirical evaluations further validate the effectiveness of our proposed algorithms in the domain of video recommendations.

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.

On Maximizing Sums of Non-monotone Submodular and Linear Functions

We study the problem of Regularized Unconstrained SubmodularMaximization (RegularizedUSM) as defined by Bodek and Feldman (Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412, 2022): given query access to a non-negative submodular function f:2N→R≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:2^{{\\mathcal {N}}}\\rightarrow {\\mathbb {R}}_{\\ge 0}$$\\end{document} and a linear function ℓ:2N→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell :2^{{\\mathcal {N}}}\\rightarrow {\\mathbb {R}}$$\\end{document} over the same ground set N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}$$\\end{document}, output a set T⊆N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\subseteq {\\mathcal {N}}$$\\end{document} approximately maximizing the sum f(T)+ℓ(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(T)+\\ell (T)$$\\end{document}. An algorithm is said to provide an (α,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\alpha ,\\beta )$$\\end{document}-approximation for RegularizedUSM if it outputs a set T such that E[f(T)+ℓ(T)]≥maxS⊆N[α·f(S)+β·ℓ(S)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}[f(T)+\\ell (T)]\\ge \\max _{S\\subseteq {\\mathcal {N}}}[\\alpha \\cdot f(S)+\\beta \\cdot \\ell (S)]$$\\end{document}. We also consider the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized ConstrainedSubmodular Maximization (RegularizedCSM). The special case of RegularizedCSM with monotone f has been extensively studied (Sviridenko et al. in Math Oper Res 42(4):1197–1218, 2017; Feldman in Algorithmica 83(3):853–878, 2021; Harshaw et al., in: International conference on machine learning, PMLR, 2634–2643, 2019), whereas we are aware of only one prior work that studies RegularizedCSM with non-monotone f (Lu et al. in Optimization 1–27, 2023), and that work constrains ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} to be non-positive. In this work, we provide improved (α,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\alpha ,\\beta )$$\\end{document}-approximation algorithms for both RegularizedUSM and RegularizedCSM with non-monotone f. Specifically, we are the first to provide nontrivial (α,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\alpha ,\\beta )$$\\end{document}-approximations for RegularizedCSM where the sign of ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is unconstrained, and the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} we obtain for RegularizedUSM improves over (Bodek and Feldman in Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412, 2022) for all β∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in (0,1)$$\\end{document}. We also prove new inapproximability results for RegularizedUSM and RegularizedCSM, as well as 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to Oveis Gharan and Vondrak (in: Proceedings of the twenty-second annual ACM-SIAM symposium on discrete algorithms, SIAM, pp 1098–1116, 2011).

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.

Packet-in request redirection: A load-balancing mechanism for minimizing control plane response time in SDNs

A distributed control plane is more scalable and robust in software defined networking. This paper focuses on controller load balancing using packet-in request redirection, that is, given the instantaneous state of the system, determining whether to redirect packet-in requests for each switch, such that the overall control plane response time (CPRT) is minimized. To address the above problem, we propose a framework based on Lyapunov optimization. First, we use the drift-plus-penalty algorithm to combine CPRT minimization problem with controller capacity constraints, and further derive a non-linear program, whose optimal solution is obtained with brute force using standard linearization techniques. Second, we present a greedy strategy to efficiently obtain a solution with a bounded approximation ratio. Third, we reformulate the program as a problem of maximizing a non-monotone submodular function subject to matroid constraints. We implement a controller prototype for packet-in request redirection, and conduct trace-driven simulations to validate our theoretical results. The results show that our algorithms can reduce the average CPRT by 81.6% compared to static assignment, and achieve a 3× improvement in maximum controller capacity violation ratio.

An Optimal Streaming Algorithm for Submodular Maximization with a Cardinality Constraint

We study the problem of maximizing a nonmonotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi) streaming algorithm that uses roughly [Formula: see text] memory, where k is the size constraint. At the end of the stream, our algorithm postprocesses its data structure using any off-line algorithm for submodular maximization and obtains a solution whose approximation guarantee is [Formula: see text], where α is the approximation of the off-line algorithm. If we use an exact (exponential time) postprocessing algorithm, this leads to [Formula: see text] approximation (which is nearly optimal). If we postprocess with the state-of-the-art offline approximation algorithm, whose guarantee is [Formula: see text], we obtain a 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715. It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for nonmonotone submodular maximization, and enjoys a fast update time of [Formula: see text] per element.

Non-monotone submodular function maximization under k-system constraint

The problems of maximizing constrained monotone submodular functions have many practical applications, most recently in the context of combinatorial optimization, operations research, economics and especially machine learning, with constant approximation algorithms known under a variety of constraints. Unfortunately, non-monotone submodular functions maximization is less well studied; the first approximation algorithm for the non-monotone case was studied by Feige et al. (Proceedings of the 48th IEEE symposium on foundations of computer science (FOCS’07), 2007) about unconstrained non-monotone submodular maximization in 2007. In this paper, we extend the work of Lee et al. (Proceedings of the 41st ACM-SIAM symposium on theory of computing (STOC’09), pp 323–332, 2009) for maximizing a non-monotone submodular function under k-matroid constraint to k-system constraint. We first propose a Modified-Greedy algorithm that works no worse than that of Gupta et al. (Proceedings of the 6th international workshop on internet and network economics (WINE’10), vol 6484, pp 246–257, 2010). Based on this, then we provide the NMSFMk algorithm for maximizing a non-monotone submodular function subject to k-system constraint (which generalizes the k-matroid constraint), using Modified-Greedy algorithm combined with USFM algorithm (USFM algorithm is the random linear time 1/2-approximation algorithm proposed by Buchbinder et al. (Proceedings of the 53rd IEEE symposium on foundations of computer science (FOCS’12), pp 649–658, 2012) for unconstrained non-monotone submodular function maximization problem.) iteratively. Finally, we show that NMSFMk algorithm achieves a $$\frac{1}{2k+3+1/k}$$ -approximation ratio with running time of O(nmk) (where m is the size of largest set returned by the NMSFMk algorithm), which beats the existing algorithms in many aspects.

Budgeted Coupon Advertisement Problem: Algorithm and Robust Analysis

Coupon advertisement is everywhere in people's daily lives, and it is a common marketing strategy adopted by merchants. A problem, Budget Profit Maximization with Coupon Advertisement, is formulated in this article, which is a branch of classical profit maximization problem in social networks. Profit maximization has been researched intensively before, but its theoretical bound is not satisfactory usually because of NP-hardness, budget constraint and non-monotonicity. Learned from the lastest results, we proposed the B-Framework, which combines the ideas of Random Greedy and Continuous Double Greedy to obtain a more acceptable approximation ratio for this problem. For Continuous Double Greedy, it can be implemented by multilinear extension and discretized techniques. In addition, most of existing researches consider only maximizing total profit, however, in real scenarios, the diffusion probability is hard to determine due to the uncertainty. Then, we study the robustness for budgeted profit maximization, which can be used as a general strategy to analyze the robustness of non-monotone submodular function. It aims to obtain the best solution maximizing the ratio between the profit of any feasible seed set and the optimal seed set. We design LU-B-Framework first, and then we apply the method of uniform sampling to improve the robustness by reducing the uncertainty. The effectiveness and correctness of our algorithms are evaluated by heavy simulation on real-world social networks eventually.

Approximation Algorithm for the Single Machine Scheduling Problem with Release Dates and Submodular Rejection Penalty

In this paper, we consider the single machine scheduling problem with release dates and nonmonotone submodular rejection penalty. We are given a single machine and multiple jobs with probably different release dates and processing times. For each job, it is either accepted and processed on the machine or rejected. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a nonmonotone submodular function. We design a combinatorial algorithm based on the primal-dual framework to deal with the problem, and study its property under two cases. For the general case where the release dates can be different, the proposed algorithm have an approximation ratio of 2. When all the jobs release at the same time, the proposed algorithm becomes a polynomial-time exact algorithm.

Greedy Maximization of Functions with Bounded Curvature under Partition Matroid Constraints

We investigate the performance of a deterministic GREEDY algorithm for the problem of maximizing functions under a partition matroid constraint. We consider non-monotone submodular functions and monotone subadditive functions. Even though constrained maximization problems of monotone submodular functions have been extensively studied, little is known about greedy maximization of non-monotone submodular functions or monotone subadditive functions.
 We give approximation guarantees for GREEDY on these problems, in terms of the curvature. We find that this simple heuristic yields a strong approximation guarantee on a broad class of functions.
 We discuss the applicability of our results to three real-world problems: Maximizing the determinant function of a positive semidefinite matrix, and related problems such as the maximum entropy sampling problem, the constrained maximum cut problem on directed graphs, and combinatorial auction games.
 We conclude that GREEDY is well-suited to approach these problems. Overall, we present evidence to support the idea that, when dealing with constrained maximization problems with bounded curvature, one needs not search for (approximate) monotonicity to get good approximate solutions.

Profit Maximization problem with Coupons in social networks

Viral marketing has become one of the most effective marketing strategies. In the process of real commercialization, in order to let some seed individuals know the products, companies can provide free samples to them. However, for some companies, especially famous ones, they are more willing to offer coupons than give samples. In this paper, we consider the Profit Maximization problem with Coupons (PM-C) in our new diffusion model named the Independent Cascade Model with Coupons and Valuations (IC-CV). To solve this problem, we propose the PMCA algorithm which can return a (13−ε)-approximate solution with at least 1−2n−l probability, and runs in O(log⁡(np)⋅mn3log⁡n(llog⁡n+nlog⁡2)/ε3) expected time. Furthermore, during the analysis we provide a method to estimate the non-monotone submodular function.

Streaming Non-Monotone Submodular Maximization: Personalized Video Summarization on the Fly

The need for real time analysis of rapidly producing data streams (e.g., video and image streams) motivated the design of streaming algorithms that can efficiently extract and summarize useful information from massive data "on the fly." Such problems can often be reduced to maximizing a submodular set function subject to various constraints. While efficient streaming methods have been recently developed for monotone submodular maximization, in a wide range of applications, such as video summarization, the underlying utility function is non-monotone, and there are often various constraints imposed on the optimization problem to consider privacy or personalization. We develop the first efficient single pass streaming algorithm, Streaming Local Search, that for any streaming monotone submodular maximization algorithm with approximation guarantee α under a collection of independence systems I, provides a constant 1/(1+2/√α+1/α+2d(1+√α)) approximation guarantee for maximizing a non-monotone submodular function under the intersection of I and d knapsack constraints. Our experiments show that for video summarization, our method runs more than 1700 times faster than previous work, while maintaining practically the same performance.

Interpretable Decision Sets: A Joint Framework for Description and Prediction.

One of the most important obstacles to deploying predictive models is the fact that humans do not understand and trust them. Knowing which variables are important in a model's prediction and how they are combined can be very powerful in helping people understand and trust automatic decision making systems. Here we propose interpretable decision sets, a framework for building predictive models that are highly accurate, yet also highly interpretable. Decision sets are sets of independent if-then rules. Because each rule can be applied independently, decision sets are simple, concise, and easily interpretable. We formalize decision set learning through an objective function that simultaneously optimizes accuracy and interpretability of the rules. In particular, our approach learns short, accurate, and non-overlapping rules that cover the whole feature space and pay attention to small but important classes. Moreover, we prove that our objective is a non-monotone submodular function, which we efficiently optimize to find a near-optimal set of rules. Experiments show that interpretable decision sets are as accurate at classification as state-of-the-art machine learning techniques. They are also three times smaller on average than rule-based models learned by other methods. Finally, results of a user study show that people are able to answer multiple-choice questions about the decision boundaries of interpretable decision sets and write descriptions of classes based on them faster and more accurately than with other rule-based models that were designed for interpretability. Overall, our framework provides a new approach to interpretable machine learning that balances accuracy, interpretability, and computational efficiency.

Maximizing non-monotone submodular set functions subject to different constraints: Combined algorithms

We study the problem of maximizing constrained non-monotone submodular functions and provide approximation algorithms that improve existing algorithms in terms of either the approximation factor or simplicity. Different constraints that we study are exact cardinality and multiple knapsack constraints for which we achieve ( 0.25 − ϵ ) -factor algorithms. We also show, as our main contribution, how to use the continuous greedy process for non-monotone functions and, as a result, obtain a 0.13-factor approximation algorithm for maximization over any solvable down-monotone polytope.

Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including max cut in digraphs, graphs, and hypergraphs; certain constraint satisfaction problems; maximum entropy sampling; and maximum facility location problems. Our main result is that for any k ≥ 2 and any ε > 0, there is a natural local search algorithm that has approximation guarantee of 1/(k + ε) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves upon the 1/(k + 1)-approximation of Fisher, Nemhauser, and Wolsey obtained in 1978 [Fisher, M., G. Nemhauser, L. Wolsey. 1978. An analysis of approximations for maximizing submodular set functions—II. Math. Programming Stud. 8 73–87]. Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general nonmonotone submodular function subject to k matroid constraints. We show that, in these cases, the approximation guarantees of our algorithms are 1/(k − 1 + ε) and 1/(k + 1 + 1/(k − 1) + ε), respectively. Our analyses are based on two new exchange properties for matroids. One is a generalization of the classical Rota exchange property for matroid bases, and another is an exchange property for two matroids based on the structure of matroid intersection.

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 $\frac{1}{k+1}$.