Submodular maximization is a general optimization problem with a wide range of applications in machine learning (e.g., active learning, clustering, and feature selection). In large-scale optimization, the parallel running time of an algorithm is governed by its adaptivity, which measures the number of sequential rounds needed if the algorithm can execute polynomially-many independent oracle queries in parallel. While low adaptivity is ideal, it is not sufficient for an algorithm to be efficient in practice-there are many applications of distributed submodular optimization where the number of function evaluations becomes prohibitively expensive. Motivated by these applications, we study the adaptivity and query complexity of submodular maximization. In this paper, we give the first constant-factor approximation algorithm for maximizing a nonmonotone submodular function subject to a cardinality constraint k that runs in O(log(n)) adaptive rounds and makes O(n log(k)) oracle queries in expectation. In our empirical study, we use three real-world applications to compare our algorithm with several benchmarks for non-monotone submodular maximization. The results demonstrate that our algorithm finds competitive solutions using significantly fewer rounds and queries.

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Given a permutation π = π 1 π 2 · · · π n ∈ S n , we say an index i is a peak if π i−1 < π i > π i+1 . Let P (π) denote the set of peaks of π. Given any set S of positive integers, define P S (n) = {π ∈ S n : P (π) = S}. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers S and sufficiently large n, |P S (n)| = p S (n)2 n−|S|−1 for some polynomial p S (x) depending on S. They conjectured that the coefficients of p S (x) expanded in a binomial coefficient basis centered at max(S) are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of p S (x). Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at 0, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.

Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of 1 − 1 /e. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial 1 /2-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing 1 /2 barrier and achieves a competitive ratio of at least 0.5086. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013) that restricts beating a 1 /2 competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting.The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems. * This paper merges and refines the results in arXiv: 1704.05384v2, arXiv:1910.02569, and arXiv:1910. In particular, we fix a bug in arXiv:1910.03287 and have a smaller competitive ratio as a result. Appendix C discusses the connections between the primal-dual algorithm in this work and the original algorithm of Fahrbach and Zadimoghaddam.

We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an O(n 4 log 4 n) upper bound and an Ω(n 4 ) lower bound for the transience class of the n × n grid. Our methods naturally extend to n d -sized d-dimensional grids to give O(n 3d−2 log d+2 n) upper bounds and Ω(n 3d−2 ) lower bounds.