37 research outputs found

    Lazier Than Lazy Greedy

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    Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm (also known as accelerated greedy), both in theory and practice? In this paper, we develop the first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can achieve a (11/eε)(1-1/e-\varepsilon) approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data and independent of the cardinality constraint. We empirically demonstrate the effectiveness of our algorithm on submodular functions arising in data summarization, including training large-scale kernel methods, exemplar-based clustering, and sensor placement. We observe that STOCHASTIC-GREEDY practically achieves the same utility value as lazy greedy but runs much faster. More surprisingly, we observe that in many practical scenarios STOCHASTIC-GREEDY does not evaluate the whole fraction of data points even once and still achieves indistinguishable results compared to lazy greedy.Comment: In Proc. Conference on Artificial Intelligence (AAAI), 201

    Weakly Submodular Functions

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    Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the "standard greedy algorithm" achieves a constant approximation ratio where the constant (experimentally) converges to 5.95 as the cardinality constraint increases. For a general matroid constraint, a simple local search algorithm achieves a constant approximation ratio where the constant (analytically) converges to 10.22 as the rank of the matroid increases

    Streaming Algorithms for Submodular Function Maximization

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    We consider the problem of maximizing a nonnegative submodular set function f:2NR+f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+ subject to a pp-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a Ω(1p)\Omega(\frac{1}{p})-approximation using O(klogk)O(k \log k)-space, where kk is an upper bound on the cardinality of the desired set. The model assumes value oracle access to ff and membership oracles for the matroids defining the pp-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201
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