39 research outputs found
Practical Parallel Algorithms for Non-Monotone Submodular Maximization
Submodular maximization has found extensive applications in various domains
within the field of artificial intelligence, including but not limited to
machine learning, computer vision, and natural language processing. With the
increasing size of datasets in these domains, there is a pressing need to
develop efficient and parallelizable algorithms for submodular maximization.
One measure of the parallelizability of a submodular maximization algorithm is
its adaptive complexity, which indicates the number of sequential rounds where
a polynomial number of queries to the objective function can be executed in
parallel. In this paper, we study the problem of non-monotone submodular
maximization subject to a knapsack constraint, and propose the first
combinatorial algorithm achieving an -approximation under
adaptive complexity, which is \textit{optimal} up to a
factor of . Moreover, we also propose the first
algorithm with both provable approximation ratio and sublinear adaptive
complexity for the problem of non-monotone submodular maximization subject to a
-system constraint. As a by-product, we show that our two algorithms can
also be applied to the special case of submodular maximization subject to a
cardinality constraint, and achieve performance bounds comparable with those of
state-of-the-art algorithms. Finally, the effectiveness of our approach is
demonstrated by extensive experiments on real-world applications.Comment: Part of the contribution appears in AAAI-202
Submodular Maximization with Matroid and Packing Constraints in Parallel
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
approximation for monotone functions and a
approximation for non-monotone functions, which nearly matches the best
guarantees known in the fully adaptive setting. The number of rounds of
adaptivity is , 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
approximation using parallel rounds, which is again an exponential speedup
in parallel time over the existing algorithms. For monotone functions, we
obtain a approximation in
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.
Our results apply more generally to the problem of maximizing a diminishing
returns submodular (DR-submodular) function
Almost Optimal Streaming Algorithms for Coverage Problems
Maximum coverage and minimum set cover problems --collectively called
coverage problems-- have been studied extensively in streaming models. However,
previous research not only achieve sub-optimal approximation factors and space
complexities, but also study a restricted set arrival model which makes an
explicit or implicit assumption on oracle access to the sets, ignoring the
complexity of reading and storing the whole set at once. In this paper, we
address the above shortcomings, and present algorithms with improved
approximation factor and improved space complexity, and prove that our results
are almost tight. Moreover, unlike most of previous work, our results hold on a
more general edge arrival model. More specifically, we present (almost) optimal
approximation algorithms for maximum coverage and minimum set cover problems in
the streaming model with an (almost) optimal space complexity of
, i.e., the space is {\em independent of the size of the sets or
the size of the ground set of elements}. These results not only improve over
the best known algorithms for the set arrival model, but also are the first
such algorithms for the more powerful {\em edge arrival} model. In order to
achieve the above results, we introduce a new general sketching technique for
coverage functions: This sketching scheme can be applied to convert an
-approximation algorithm for a coverage problem to a
(1-\eps)\alpha-approximation algorithm for the same problem in streaming, or
RAM models. We show the significance of our sketching technique by ruling out
the possibility of solving coverage problems via accessing (as a black box) a
(1 \pm \eps)-approximate oracle (e.g., a sketch function) that estimates the
coverage function on any subfamily of the sets
Scalable Distributed Algorithms for Size-Constrained Submodular Maximization in the MapReduce and Adaptive Complexity Models
Distributed maximization of a submodular function in the MapReduce model has
received much attention, culminating in two frameworks that allow a centralized
algorithm to be run in the MR setting without loss of approximation, as long as
the centralized algorithm satisfies a certain consistency property - which had
only been shown to be satisfied by the standard greedy and continous greedy
algorithms. A separate line of work has studied parallelizability of submodular
maximization in the adaptive complexity model, where each thread may have
access to the entire ground set. For the size-constrained maximization of a
monotone and submodular function, we show that several sublinearly adaptive
algorithms satisfy the consistency property required to work in the MR setting,
which yields highly practical parallelizable and distributed algorithms. Also,
we develop the first linear-time distributed algorithm for this problem with
constant MR rounds. Finally, we provide a method to increase the maximum
cardinality constraint for MR algorithms at the cost of additional MR rounds.Comment: 45 pages, 6 figure
Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints
We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contributions are two single-pass (semi-)streaming algorithms that use O?(k)?poly(1/?) memory, where k is the size constraint. At the end of the stream, both our algorithms post-process their data structures using any offline algorithm for submodular maximization, and obtain a solution whose approximation guarantee is ?/(1+?)-?, where ? is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to 1/2-? approximation (which is nearly optimal). If we post-process with the algorithm of [Niv Buchbinder and Moran Feldman, 2019], that achieves the state-of-the-art offline approximation guarantee of ? = 0.385, we obtain 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715 due to [Feldman et al., 2018]. One of our algorithms is combinatorial and enjoys fast update and overall running times. Our other algorithm is based on the multilinear extension, enjoys an improved space complexity, and can be made deterministic in some settings of interest