A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of $2$. Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a $(0.385-\varepsilon)$-approximation, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest.
We consider the problem of maximizing a monotone submodular function under a knapsack constraint. We show that, for any fixed $ε> 0$, there exists a polynomial-time algorithm with an approximation ratio $1-c/e-ε$, where $c \in [0,1]$ is the (total) curvature of the input function. This approximation ratio is tight up to $ε$ for any $c \in [0,1]$. To the best of our knowledge, this is the first result for a knapsack constraint that incorporates the curvature to obtain an approximation ratio better than $1-1/e$, which is tight for general submodular functions. As an application of our result, we…
We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set $I$ of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function $f$ over subsets of the items. The objective is to find a packing of a subset of items $A \subseteq I$ in the bins such that $f(A)$ is maximized. Our main result is an almost optimal polynomial time $(1-e^{-1}-\varepsilon)$-approximation algorithm for the problem, for any $\varepsilon>0$. The algorithm relies on a…
We study the correlated stochastic knapsack problem of a submodular target function, with optional additional constraints. We utilize the multilinear extension of submodular function, and bundle it with an adaptation of the relaxed linear constraints from Ma [Mathematics of Operations Research, Volume 43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then solved by the stochastic continuous greedy algorithm, and rounded by a novel method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We obtain a pseudo-polynomial time $(1 - 1/\sqrt{e})/2 \simeq 0.…
We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Since a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a $1 - 1/e$ approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of a distributive lattice and define the multilinear e…
We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, $1 - 1/e - ε$ approximation, using $(1/ε)^{O(1/ε^4)} n \log^2{n}$ function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to $Ω(n^2)$ run…
In this work, we study the classic submodular maximization problem under knapsack constraints and beyond. We first present an $(7/16-\varepsilon)$-approximate algorithm for single knapsack constraint, which requires $O(n\cdot\max\{\varepsilon^{-1},\log\log n\})$ queries, and two passes in the streaming setting. This provides improvements in approximation ratio, query complexity and number of passes on the stream. We next show that there exists an $(1/2-\varepsilon)$-approximate deterministic algorithm for constant number of binary packing constraints, which achieves a query complexity of $O_{\…
We consider the maximization problem of monotone submodular functions under an uncertain knapsack constraint. Specifically, the problem is discussed in the situation that the knapsack capacity is not given explicitly and can be accessed only through an oracle that answers whether or not the current solution is feasible when an item is added to the solution. Assuming that cancellation of the last item is allowed when it overflows the knapsack capacity, we discuss the robustness ratios of adaptive policies for this problem, which are the worst case ratios of the objective values achieved by the …
In this paper, we study the non-monotone adaptive submodular maximization problem subject to a knapsack and a $k$-system constraints. The input of our problem is a set of items, where each item has a particular state drawn from a known prior distribution. However, the state of an item is initially unknown, one must select an item in order to reveal the state of that item. There is a utility function which is defined over items and states. Our objective is to sequentially select a group of items to maximize the expected utility. Although the cardinality-constrained non-monotone adaptive submodu…
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. For the problem of maximizing a non-monotone submodular function, Feige, Mirrokni, and Vondrák recently developed a $2\over 5$-approximation algorithm \cite{FMV07}, however, their algorithms do not handle side constraints.} In this pa…
We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between…
Submodular maximization problems belong to the family of combinatorial optimization problems and enjoy wide applications. In this paper, we focus on the problem of maximizing a monotone submodular function subject to a $d$-knapsack constraint, for which we propose a streaming algorithm that achieves a $\left(\frac{1}{1+2d}-ε\right)$-approximation of the optimal value, while it only needs one single pass through the dataset without storing all the data in the memory. In our experiments, we extensively evaluate the effectiveness of our proposed algorithm via two applications: news recommendation…
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. Our algorithms combine existing local search and greedy based algorithms. Different constraints that we study are exact cardinality and multiple knapsack constraints. For the multiple-knapsack constraints we achieve a $(0.25-2ε)$-factor algorithm. We also show, as our main contribution, how to use the continuous greedy process for non-monotone functions and, as a result, obtain a $0.13$-…
Indiana University（印第安纳大学）；MIT（麻省理工学院）；The Alan Turing Institute（艾伦·图灵研究所）
We propose the first adversarially robust algorithm for monotone submodular maximization under single and multiple knapsack constraints with scalable implementations in distributed and streaming settings. For a single knapsack constraint, our algorithm outputs a robust summary of almost optimal (up to polylogarithmic factors) size, from which a constant-factor approximation to the optimal solution can be constructed. For multiple knapsack constraints, our approximation is within a constant-factor of the best known non-robust solution. We evaluate the performance of our algorithms by comparison…
We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. We propose the following streaming algorithms taking $O(\varepsilon^{-1})$ passes: ----a $(1-e^{-1}-\varepsilon)$-approximation algorithm for the cardinality-constrained problem ---- a $(0.5-\varepsilon)$-approximation algorithm for the knapsack-constrained problem. Both of our algorithms run in $O^\a…
Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to $d$ knapsack constraints, where $d$ is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through {\em extension by expectation} of the submodular function. Formally, we show that, for any non-negative submodular function, an $α$…
Hasso Plattner Institute（哈索普拉特纳研究所）；The University of Adelaide（阿德莱德大学）
We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints. We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions with bounded curvature. In contrast to other heuristics, this requires no problem relaxation to continuous domains and it maintains a constant-factor approximation guarantee in the problem size. In the case of a single knapsack, our analysis suggests that the standard greedy can be used in non-monotone settings. Additionally, we study this problem …
Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is g…
We study a variant of the \emph{generalized assignment problem} ({\sf GAP}) with group constraints. An instance of {\sf Group GAP} is a set $I$ of items, partitioned into $L$ groups, and a set of $m$ uniform (unit-sized) bins. Each item $i \in I$ has a size $s_i >0$, and a profit $p_{i,j} \geq 0$ if packed in bin $j$. A group of items is \emph{satisfied} if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total profit from satisfied groups is maximized. We point to central applications of {\sf Group GAP} in Video-on-Demand s…
Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: $(1)$ reduce the enumeration in the tight $(1-e^{-1})$-approximation of [Sviridenko 04] from subsets of size three to two; $(2)$ present an improved upper bound of $0.42945$ for the classic algorithm which returns the better between a single element and the output of the greedy heuristic.
Indiana University, Bloomington（印第安纳大学伯明顿分校）；The Alan Turing Institute（艾伦·图灵研究所）；Carnegie Mellon University（卡内基梅隆大学）
The problem of selecting a small-size representative summary of a large dataset is a cornerstone of machine learning, optimization and data science. Motivated by applications to recommendation systems and other scenarios with query-limited access to vast amounts of data, we propose a new rigorous algorithmic framework for a standard formulation of this problem as a submodular maximization subject to a linear (knapsack) constraint. Our framework is based on augmenting all partial Greedy solutions with the best additional item. It can be instantiated with negligible overhead in any model of comp…
We present a simple combinatorial $\frac{1 -e^{-2}}{2}$-approximation algorithm for maximizing a monotone submodular function subject to a knapsack and a matroid constraint. This classic problem is known to be hard to approximate within factor better than $1 - 1/e$. We show that the algorithm can be extended to yield a ratio of $\frac{1 - e^{-(k+1)}}{k+1}$ for the problem with a single knapsack and the intersection of $k$ matroid constraints, for any fixed $k > 1$. Our algorithms, which combine the greedy algorithm of [Khuller, Moss and Naor, 1999] and [Sviridenko, 2004] with local search, sho…
Breaking the 1/3 Barrier for constrained $\boldsymbol{k}$-Submodular Maximization: A Proportional Top-2 Randomized Framework
School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所处理器与系统安全国家重点实验室）；School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）；Zhongguancun Laboratory（中关村实验室）；Beijing Normal University-Zhuhai（北京师范大学珠海校区）；Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）
National University of Singapore（新加坡国立大学）；Nanyang Technological University（南洋理工大学）；University of Science and Technology of China（中国科学技术大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）
Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of $0.405$, which significantly improves the known factors of $0.357$ given by Wolsey and $(1-1/\mathrm{e})/2\approx 0.316$ given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of $(1-1/\sqrt{\mathrm{e}})\a…
The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function $f:\mathbb{Z}_+^n \to \mathbb{R}_+$ is given via an evaluation oracle. Assume further that $f$ satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time $(1-1/e-ε)$-approximation algorithms for …
We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint, a matroid independence constraint, or any combination of such constraints. Our algorithm make $O(n^3)$ oracle calls to the submodular function where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ (Svitkina …
Balkanski and Singer [5] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Very recent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal $(1-1/e-ε)$-approximation in $O(\log n/ε^2)$ rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subj…
Ohio State University（俄亥俄州立大学）；University of Haifa（海法大学）；Google（谷歌）；Yale University（耶鲁大学）
In this paper, we provide the first deterministic algorithm that achieves the tight $1-1/e$ approximation guarantee for submodular maximization under a cardinality (size) constraint while making a number of queries that scales only linearly with the size of the ground set $n$. To complement our result, we also show strong information-theoretic lower bounds. More specifically, we show that when the maximum cardinality allowed for a solution is constant, no algorithm making a sub-linear number of function evaluations can guarantee any constant approximation ratio. Furthermore, when the constrain…
Contention resolution schemes have proven to be an incredibly powerful concept which allows to tackle a broad class of problems. The framework has been initially designed to handle submodular optimization under various types of constraints, that is, intersections of exchange systems (including matroids), knapsacks, and unsplittable flows on trees. Later on, it turned out that this framework perfectly extends to optimization under uncertainty, like stochastic probing and online selection problems, which further can be applied to mechanism design. We add to this line of work by showing how to cr…
We consider the problem of maximizing a non-negative submodular set function $f:2^N \rightarrow \mathbb{R}_+$ over a ground set $N$ subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when $f$ may be a non-monotone function. Our algorithms are based on (approximately) maximizing the multilinear extension $F$ of $f$ over a polytope $P$ that represents the constraints, and then effectively rounding the f…
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 impo…
KTH Royal Institute of Technology（KTH皇家理工学院）；HIIT（赫尔辛基信息技术研究所）；University of Helsinki（赫尔辛基大学）
Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a …
In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements. We first present an algorithm that guarantees an approximation factor of $\frac{1}{2}\left(1-\frac{1}{e^α}\right)$, where $…
Robust optimization is becoming increasingly important in machine learning applications. In this paper, we study a unified framework of robust submodular optimization. We study this problem both from a minimization and maximization perspective (previous work has only focused on variants of robust submodular maximization). We do this under a broad range of combinatorial constraints including cardinality, knapsack, matroid as well as graph-based constraints such as cuts, paths, matchings and trees. Furthermore, we also study robust submodular minimization and maximization under multiple submodul…
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 (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 $α=1-1/e\approx 0.632$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko impractical as it leads to a time complexity of $O(n^5)$ (which can be slightly improved using the thresholdin…
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' st…
University of Texas at Dallas（德克萨斯大学达拉斯分校）；University of North Texas（北德克萨斯大学）
Many sequential decision making problems, including pool-based active learning and adaptive viral marketing, can be formulated as an adaptive submodular maximization problem. Most of existing studies on adaptive submodular optimization focus on either monotone case or non-monotone case. Specifically, if the utility function is monotone and adaptive submodular, \cite{golovin2011adaptive} developed a greedy policy that achieves a $(1-1/e)$ approximation ratio subject to a cardinality constraint. If the utility function is non-monotone and adaptive submodular, \cite{tang2021beyond} showed that a …
