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 $1-1/e-\epsilon$ approximation for monotone functions and a $1/e-\epsilon$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/\epsilon^3)$, which is an exponential speedup over the existing algorithms.
We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-\epsilon$ approximation in $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective (Mahoney et al., 2016).
Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.
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Fast First-Order Methods for Monotone Strongly DR-Submodular Maximization
Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e. g., computing the stability number of a graph and mean-field inference for probabilistic log-submodular models, the DR-submodular function has the additional property of being strongly concave along non-negative directions that could be utilized for obtaining faster convergence rates. In this paper, we first introduce and characterize the class of strongly DR-submodular functions and show how such a property implies strong concavity along non-negative directions. Then, we study L-smooth monotone strongly DR-submodular functions that have bounded curvature, and we show how to exploit such additional structure to obtain algorithms with improved approximation guarantees and faster convergence rates for the maximization problem. In particular, we propose the SDRFW algorithm that matches the provably optimal approximation ratio after only iterations, where c ∈ [0,1] and μ ≥ 0 are the curvature and the strong DR-submodularity parameter. Furthermore, we study the Projected Gradient Ascent (PGA) method for this problem and provide a refined analysis of the algorithm with an improved approximation ratio (compared to ½ in prior works) and a linear convergence rate. Given that both algorithms require knowledge of the smoothness parameter L, we provide a novel characterization of L for DR-submodular functions showing that in many cases, computing L could be formulated as a convex optimization problem, i. e., a geometric program, that could be solved efficiently. Experimental results illustrate and validate the efficiency and effectiveness of our algorithms.
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- Award ID(s):
- 2023166
- NSF-PAR ID:
- 10443284
- Editor(s):
- Berry, Jonathan; Shmoys, David; Cowen, Lenore; Naumann, Uwe
- Date Published:
- Journal Name:
- Proceedings of SIAM Conference on Applied and Computational Discrete Algorithms
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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