Constrained Submodular Maximization: Beyond 1/e

Ene, Alina; Nguyêݱn, Huy L.

doi:10.1109/focs.2016.34

Cited by 50 publications

(51 citation statements)

References 16 publications

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“…First-order optimization methods have recently gained high popularity due to their applicability to large-scale problem instances arising from modern datasets, their relatively low computational complexity, and their potential for parallelizing computation [36]. Moreover, such methods have also been successfully applied in discrete optimization leading to faster numerical methods [19,35], graph algorithms [18,22,34], and submodular optimization methods [16].…”

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confidence: 99%

The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods

Diakonikolas¹,

Orecchia²

2019

SIAM J. Optim.

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We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the algorithm converges. We show that in continuous time enforcement of an invariant that this approximate duality gap decreases at a certain rate exactly recovers a wide range of first-order continuous-time methods. We characterize the discretization errors incurred by different discretization methods, and show how iteration-complexity-optimal methods for various classes of problems cancel out the discretization error. The techniques are illustrated on various classes of problems -including convex minimization for Lipschitz-continuous objectives, smooth convex minimization, composite minimization, smooth and strongly convex minimization, solving variational inequalities with monotone operators, and convex-concave saddle-point optimization -and naturally extend to other settings. 1 Width-independent algorithms enjoy poly-logarithmic dependence of their convergence times on the constraints matrix width -namely, the ratio between the constraint matrix maximum and minimum non-zero elements. By contrast, standard first-order methods incur (at best) linear dependence on the matrix width, which is not even considered to be polynomial-time convergence [27].

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confidence: 99%

The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods

Diakonikolas¹,

Orecchia²

2019

SIAM J. Optim.

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“…The last decade has seen a surge of work on submodular maximization problems. Arguably, the main factor that allowed this surge was the invention of the multilinear relaxation for submodular maximization problems as well as algorithms for (approximately) solving this relaxation [4,6,8,9,11]. The invention of the multilinear relaxation was so influential because it allowed algorithms for submodular maximization to use the technique of first solving a relaxed version of the problem, and then rounding the fractional solution obtained.…”

Section: Introductionmentioning

confidence: 99%

Guess Free Maximization of Submodular and Linear Sums

Feldman

2019

Lecture Notes in Computer Science

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We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. [16] described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of [16].

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“…A comparably recent result is an optimal 1 − e −1approximation to maximizing f (S) over a matroid constraint S ∈ I when f is monotone [18], a significant generalization of the cardinality constraint problem. Subsequent developments obtained a e −1 -approximation for nonnegative submodular functions subject to a matroid constraint [35] and then improved beyond e −1 [27,13]. The techniques underlying these results take a fractional point of view with one part continuous optimization and another part rounding, somewhat analogous to the use of LP's for approximating integer programs.…”

Section: Introductionmentioning

confidence: 99%

Parallelizing greedy for submodular set function maximization in matroids and beyond

Chekuri

Quanrud

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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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 *

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Constrained Submodular Maximization: Beyond 1/e

Cited by 50 publications

References 16 publications

The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods

The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods

Guess Free Maximization of Submodular and Linear Sums

Parallelizing greedy for submodular set function maximization in matroids and beyond

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