We study the problem of ranking with submodular valuations. An instance of this problem consists of a ground set @math , and a collection of @math monotone submodular set functions @math , where each @math . An additional ingredient of the input is a weight vector @math . The objective is to find a linear ordering of the ground set elements that minimizes the weighted cover time of the functions. The cover time of a function is the minimal number of elements in the prefix of the linear ordering that form a set whose corresponding function value is greater than a unit threshold value. Our main contribution is an @math -approximation algorithm for the problem, where @math is the smallest non-zero marginal value that any function may gain from some element. Our algorithm orders the elements using an adaptive residual updates scheme, which may be of independent interest. We also prove that the problem is @math -hard to approximate, unless P = NP. This implies that the outcome of our algorithm is optimal up to constant factors.
Submodular functions arise naturally in operations research and combinatorial optimization. One of the most extensively studied questions is how to minimize a submodular function. A series of results demonstrate that this task can be performed efficiently, either by the ellipsoid algorithm @cite_31 or through strongly polynomial time combinatorial algorithms @cite_18 @cite_5 @cite_11 @cite_24 @cite_13 @cite_17 . Recently, there has been a surge of interest in understanding the limits of tractability of minimization problems in which the classic linear objective function was replaced by a submodular one (see, e.g., @cite_26 @cite_32 @cite_20 @cite_4 ). Notably, these submodular problems are commonly considerably harder to approximate than their linear counterparts. For example, the minimum spanning tree problem, which is polynomial time solvable with linear cost functions is @math -hard to approximate with submodular cost functions @cite_20 , and the sparsest cut problem, which admits an @math -approximation algorithm when the cost is linear @cite_3 becomes @math -hard to approximate with submodular costs @cite_26 . Our work extends the tools and techniques in this line of research. In particular, our results establish a computational separation of logarithmic order between the submodular settings and the linear setting, which admits a constant factor approximation @cite_33 .
@cite_18 @cite_26 @cite_4 @cite_33 @cite_17 @cite_32 @cite_3 @cite_24 @cite_5 @cite_31 @cite_13 @cite_20 @cite_11
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The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed undirected graphs. In this paper we present the first known approximation algorithms for the problem of maximizing a nondecreasing submodular set function subject to multiple linear constraints. Given a d-dimensional budget vector [EQUATION], for some d ≥ 1, and an oracle for a non-decreasing submodular set function f over a universe U, where each element e ∈ U is associated with a d-dimensional cost vector, we seek a subset of elements S ⊆ U whose total cost is at most [EQUATION], such that f(S) is maximized. We develop a framework for maximizing submodular functions subject to d linear constraints that yields a (1 - e)(1 - e−1)-approximation to the optimum for any e > 0, where d > 1 is some constant. Our study is motivated by a variant of the classical maximum coverage problem that we call maximum coverage with multiple packing constraints. We use our framework to obtain the same approximation ratio for this problem. To the best of our knowledge, this is the first time the theoretical bound of 1 - e−1 is (almost) matched for both of these problems. We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions.The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle.The approximation guarantees for most of our algorithms are of the order of radic(n ln n). We show that this is the inherent difficulty of the problems by proving matching lower bounds.We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions. Submodular optimization has found many applications in machine learning and beyond. We carry out the first systematic investigation of inference in probabilistic models defined through submodular functions, generalizing regular pairwise MRFs and Determinantal Point Processes. In particular, we present L-FIELD, a variational approach to general log-submodular and log-supermodular distributions based on sub- and supergradients. We obtain both lower and upper bounds on the log-partition function, which enables us to compute probability intervals for marginals, conditionals and marginal likelihoods. We also obtain fully factorized approximate posteriors, at the same computational cost as ordinary submodular optimization. Our framework results in convex problems for optimizing over differentials of submodular functions, which we show how to optimally solve. We provide theoretical guarantees of the approximation quality with respect to the curvature of the function. We further establish natural relations between our variational approach and the classical mean-field method. Lastly, we empirically demonstrate the accuracy of our inference scheme on several submodular models. In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor @math approximation to the global maxima. We also study stochastic gradient methods and show that after @math iterations these methods reach solutions which achieve in expectation objective values exceeding @math . An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor @math approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort.