The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions. However, when such applications involve sensitive data about individuals, their privacy concerns should be addressed. In this paper, we study the problem of maximizing monotone submodular functions subject to matroid constraints in the framework of differential privacy. We provide $(1-\frac{1}{\mathrm{e}})$-approximation algorithm which improves upon the previous results in terms of approximation guarantee. This is done with an almost cubic number of function evaluations in our algorithm.
Moreover, we study $k$-submodularity, a natural generalization of submodularity. We give the first $\frac{1}{2}$-approximation algorithm that preserves differential privacy for maximizing monotone $k$-submodular functions subject to matroid constraints. The approximation ratio is asymptotically tight and is obtained with an almost linear number of function evaluations.
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
We study the problem of differentially private constrained maximization of decomposable submodular functions. A submodular function is decomposable if it takes the form of a sum of submodular functions. The special case of maximizing a monotone, decomposable submodular function under cardinality constraints is known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al., 2008]. Previous work by Gupta et al. [2010] gave a differentially private algorithm for the CPP problem. We extend this work by designing differentially private algorithms for both monotone and non-monotone d…
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
In this work, we study the problem of privately maximizing a submodular function in the streaming setting. Extensive work has been done on privately maximizing submodular functions in the general case when the function depends upon the private data of individuals. However, when the size of the data stream drawn from the domain of the objective function is large or arrives very fast, one must privately optimize the objective within the constraints of the streaming setting. We establish fundamental differentially private baselines for this problem and then derive better trade-offs between privac…
In this work, we consider robust submodular maximization with matroid constraints. We give an efficient bi-criteria approximation algorithm that outputs a small family of feasible sets whose union has (nearly) optimal objective value. This algorithm theoretically performs less function calls than previous works at cost of adding more elements to the final solution. We also provide significant implementation improvements showing that our algorithm outperforms the algorithms in the existing literature. We finally assess the performance of our contributions in three real-world applications.
We study the problem of maximizing a monotone submodular function subject to a matroid constraint and present a deterministic algorithm that achieves (1/2 + ε)-approximation for the problem. This algorithm is the first deterministic algorithm known to improve over the 1/2-approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsely and Fisher in 1978.
A $k$-submodular function is an extension of a submodular function in that its input is given by $k$ disjoint subsets instead of a single subset. For unconstrained nonnegative $k$-submodular maximization, Ward and Živný proposed a constant-factor approximation algorithm, which was improved by the recent work of Iwata, Tanigawa and Yoshida presenting a $1/2$-approximation algorithm. Iwata et al. also provided a $k/(2k-1)$-approximation algorithm for monotone $k$-submodular maximization and proved that its approximation ratio is asymptotically tight. More recently, Ohsaka and Yoshida proposed co…
We study the matroid secretary problems with submodular valuation functions. In these problems, the elements arrive in random order. When one element arrives, we have to make an immediate and irrevocable decision on whether to accept it or not. The set of accepted elements must form an {\em independent set} in a predefined matroid. Our objective is to maximize the value of the accepted elements. In this paper, we focus on the case that the valuation function is a non-negative and monotonically non-decreasing submodular function. We introduce a general algorithm for such {\em submodular matroid…
Sharif University（谢里夫大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4 γ^{2}}{\sqrt{γr} + 1}$, where $γ$ is the submodu…
Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid c…
In this work, we study the problem of monotone non-submodular maximization with partition matroid constraint. Although a generalization of this problem has been studied in literature, our work focuses on leveraging properties of partition matroid constraint to (1) propose algorithms with theoretical bound and efficient query complexity; and (2) provide better analysis on theoretical performance guarantee of some existing techniques. We further investigate those algorithms' performance in two applications: Boosting Influence Spread and Video Summarization. Experiments show our algorithms return…
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…
Many important problems in discrete optimization require maximization of a monotonic submodular function subject to matroid constraints. For these problems, a simple greedy algorithm is guaranteed to obtain near-optimal solutions. In this article, we extend this classic result to a general class of adaptive optimization problems under partial observability, where each choice can depend on observations resulting from past choices. Specifically, we prove that a natural adaptive greedy algorithm provides a $1/(p+1)$ approximation for the problem of maximizing an adaptive monotone submodular funct…
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a \emph{general matroid constraint} with a $1 - 1/e - ε$ approximation that achieves a fast running time provided we have a fast data structure for maintaining a maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroid…
We consider the problem of maximizing a non-negative submodular function under the $b$-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of $2+\varepsilon$, $3 + 2 \sqrt{2} \approx 5.828$, and $4 + 2 \sqrt{3} \approx 7.464$, respectively. We also consider a generalized problem, where a $k$-uniform hypergraph is given, along with an extra matroid or a $k'$-matchoid constraint imposed on the edges, with the same goal of finding a $b$-matching that maximizes a submodular function. When the extra constraint…
We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For $k$-unif…
Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to $k$ matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in $k$ is needed to obtain an $o(k / \log k)$-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obt…
In this paper we study submodular maximization under a matroid constraint in the adaptive complexity model. This model was recently introduced in the context of submodular optimization in [BS18a] to quantify the information theoretic complexity of black-box optimization in a parallel computation model. Informally, the adaptivity of an algorithm is the number of sequential rounds it makes when each round can execute polynomially-many function evaluations in parallel. Since submodular optimization is regularly applied on large datasets we seek algorithms with low adaptivity to enable speedups vi…
Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making $\mathrm{poly}(N)$ queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where $N$ denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least $N$ rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with $\mathrm{poly}(N)…
We consider the problem of maximizing a nonnegative (possibly non-monotone) submodular set function with or without constraints. Feige et al. [FOCS'07] showed a 2/5-approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Constant-factor approximation was also given for submodular maximization subject to a matroid independence constraint (a factor of 0.309 Vondrak [FOCS'09]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number is at least 2 (a 1/4-ap…
The classical problem of maximizing a submodular function under a matroid constraint is considered. Defining a new measure for the increments made by the greedy algorithm at each step, called the discriminant, improved approximation ratio guarantees are derived for the greedy algorithm. At each step, discriminant measures the multiplicative gap in the incremental valuation between the item chosen by the greedy algorithm and the largest potential incremental valuation for eligible items not selected by it. The new guarantee subsumes all the previous known results for the greedy algorithm, inclu…
We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-ε$ approximation for monotone functions and a $1/e-ε$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/ε^3)$, which is an…
Submodularity is one of the most important properties in combinatorial optimization, and $k$-submodularity is a generalization of submodularity. Maximization of a $k$-submodular function requires an exponential number of value oracle queries, and approximation algorithms have been studied. For unconstrained $k$-submodular maximization, Iwata et al. gave randomized $k/(2k-1)$-approximation algorithm for monotone functions, and randomized $1/2$-approximation algorithm for nonmonotone functions. In this paper, we present improved randomized algorithms for nonmonotone functions. Our algorithm give…
During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular matroid secretary problem (SMSP). So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up …
A recent line of research focuses on the study of the stochastic multi-armed bandits problem (MAB), in the case where temporal correlations of specific structure are imposed between the player's actions and the reward distributions of the arms (Kleinberg and Immorlica [FOCS18], Basu et al. [NeurIPS19]). As opposed to the standard MAB setting, where the optimal solution in hindsight can be trivially characterized, these correlations lead to (sub-)optimal solutions that exhibit interesting dynamical patterns -- a phenomenon that yields new challenges both from an algorithmic as well as a learnin…
We present an optimal, combinatorial 1-1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related non-oblivious potential function, which is also monotone submodular. Our algorithm runs in randomized time O(n^8u), where n is the rank of the given matroid and u is the size of its g…
The Open University of Israel（以色列开放大学）；Yale Institute for Network Science, Yale University（耶鲁大学网络科学研究所）；University of Haifa（海法大学）
In this paper, we propose a novel framework that converts streaming algorithms for monotone submodular maximization into streaming algorithms for non-monotone submodular maximization. This reduction readily leads to the currently tightest deterministic approximation ratio for submodular maximization subject to a $k$-matchoid constraint. Moreover, we propose the first streaming algorithm for monotone submodular maximization subject to $k$-extendible and $k$-set system constraints. Together with our proposed reduction, we obtain $O(k\log k)$ and $O(k^2\log k)$ approximation ratio for submodular …
We present a polynomial time algorithm for online maximization of $k$-submodular maximization. For online (nonmonotone) $k$-submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone $k$-submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as $k$ tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.
We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general $p$-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of $p$ arbitrary matroid constraints and $p$-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of $p+1+\varepsilon$ using $O(p/\varepsilon)$-passes and requires storing only $O(k)$ elements, where $k$ is the maximum …
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…
While greedy algorithms have long been observed to perform well on a wide variety of problems, up to now approximation ratios have only been known for their application to problems having submodular objective functions $f$. Since many practical problems have non-submodular $f$, there is a critical need to devise new techniques to bound the performance of greedy algorithms in the case of non-submodularity. Our primary contribution is the introduction of a novel technique for estimating the approximation ratio of the greedy algorithm for maximization of monotone non-decreasing functions based on…
We study the following problem: Given a variable of interest, we would like to find a best linear predictor for it by choosing a subset of $k$ relevant variables obeying a matroid constraint. This problem is a natural generalization of subset selection problems where it is necessary to spread observations amongst multiple different classes. We derive new, strengthened guarantees for this problem by improving the analysis of the residual random greedy algorithm and by developing a novel distorted local-search algorithm. To quantify our approximation guarantees, we refine the definition of weak …
Motivated by several applications, we consider the problem of randomly rounding a fractional solution in a matroid (base) polytope to an integral one. We consider the pipage rounding technique and also present a new technique, randomized swap rounding. Our main technical results are concentration bounds for functions of random variables arising from these rounding techniques. We prove Chernoff-type concentration bounds for linear functions of random variables arising from both techniques, and also a lower-tail exponential bound for monotone submodular functions of variables arising from random…
While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of $2$ for the \emph{unweighted} matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor $2$ achieved by the…
Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank $k$ (i.e., any solution has cardinality at most $k$), our main results are: 1) a single-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $0.3178$, and 2) a multi-pass streaming algorithm that uses $\…
We consider parallel, or low adaptivity, algorithms for submodular function maximization. This line of work was recently initiated by Balkanski and Singer and has already led to several interesting results on the cardinality constraint and explicit packing constraints. An important open problem is the classical setting of matroid constraint, which has been instrumental for developments in submodular function maximization. In this paper we develop a general strategy to parallelize the well-studied greedy algorithm and use it to obtain a randomized $\left(\frac{1}{2} - ε\right)$-approximation in…
Group Equality in Adaptive Submodular Maximization
Naveen Jindal School of Management, The University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金德尔管理学院）；Department of Computer Science and Engineering, The University of North Texas（北德克萨斯大学计算机科学与工程系）
Submodularity is one of the most important property of combinatorial optimization, and $k$-submodularity is a generalization of submodularity. Maximization of $k$-submodular function is NP-hard, and approximation algorithms are studied. For monotone $k$-submodular function, [Iwata, Tanigawa, and Yoshida 2016] gave $k/(2k-1)$-approximation algorithm. In this paper, we give a deterministic algorithm by derandomizing that algorithm. Derandomization scheme is from [Buchbinder and Feldman 2016]. Our algorithm is $k/(2k-1)$-approximation and polynomial-time algorithm.
We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of …
The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a $1/2$-approximation, and we also present an appropriate extension to the non-monotone setting. However, reaching a local optimum quickly is a non-trivial task. Moreover, we describe a fast and very general local search procedure that applies to a wide range of constra…
In this work we present the first practical $\left(\frac{1}{e}-ε\right)$-approximation algorithm to maximise a general non-negative submodular function subject to a matroid constraint. Our algorithm is based on combining the decreasing-threshold procedure of Badanidiyuru and Vondrak (SODA 2014) with a smoother version of the measured continuous greedy algorithm of Feldman et al. (FOCS 2011). This enables us to obtain an algorithm that requires $O(\frac{nr^2}{ε^4} \big(\frac{a+b}{a}\big)^2 \log^2({\frac{n}ε}))$ value oracle calls, where $n$ is the cardinality of the ground set, $r$ is the matro…
