We consider the Unconstrained Submodular Maximization problem in which we are given a non-negative submodular function f : 2 N → R + , and the objective is to find a subset S ⊆ N maximizing f (S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include Max-Cut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem.

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In this paper, we propose the first continuous optimization algorithms that achieve a constant factor approximation guarantee for the problem of monotone continuous submodular maximization subject to a linear constraint. We first prove that a simple variant of the vanilla coordinate ascent, called Coordinate-Ascent+, achieves a ( e−1 2e−1 − ε)-approximation guarantee while performing O(n/ε) iterations, where the computational complexity of each iteration is roughly O(n/ √ ε+n log n) (here, n denotes the dimension of the optimization problem). We then propose Coordinate-Ascent++, that achieves the tight (1 − 1/e − ε)-approximation guarantee while performing the same number of iterations, but at a higher computational complexity of roughly O(n 3 /ε 2.5 + n 3 log n/ε 2 ) per iteration. However, the computation of each round of Coordinate-Ascent++ can be easily parallelized so that the computational cost per machine scales as O(n/ √ ε + n log n).complexity of each iteration is O(n B/ε + n log n). Here, n and B denote the dimension of the optimization problem and the ℓ 1 radius of the constraint set, respectively.• We then develop Coordinate-Ascent++, that achieves the tight (1 − 1/e − ε) approximation guarantee while performing O(n/ǫ) iterations, where the computational complexity of each iteration is O(n 3 √ B/ε 2.5 + n 3 log n/ε 2 ). Moreover, Coordinate-Ascent++ can be easily parallelized so that the computational complexity per machine in each round scales as O(n B/ǫ + n log n).Notably, to establish these results, we do not assume that the continuous submodular function satisfies the diminishing returns condition.

We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider.We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constantfactor algorithm for weighted stochastic probing with deadlines.

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