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Register a new userA Nearly-linear Time Algorithm for Submodular Maximization with a Knapsack Constraint
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Added by
Alina Ene
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, $1 - 1/e - epsilon$ approximation, using $(1/epsilon)^{O(1/epsilon^4)} n log^2{n}$ function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to $Omega(n^2)$ running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.
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We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a emph{general matroid constraint} with a $1 - 1/e - epsilon$ approximation that achieves a fast running time provided we have a fast data structure for maintaining a maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the emph{first algorithms} for these classes of matroids that achieve a nearly-optimal, $1 - 1/e - epsilon$ approximation, using a nearly-linear number of function evaluations and arithmetic operations.
Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a $5.83$ approximation and runs in $O(n log n)$ time, i.e., at least a factor $n$ faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a $9$-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.
In this paper, we study the tradeoff between the approximation guarantee and adaptivity for the problem of maximizing a monotone submodular function subject to a cardinality constraint. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomially-many parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. Previous algorithms achieving a nearly-optimal $1 - 1/e - epsilon$ approximation require $Omega(n)$ rounds of adaptivity. In this work, we give the first algorithm that achieves a $1 - 1/e - epsilon$ approximation using $O(ln{n} / epsilon^2)$ rounds of adaptivity. The number of function evaluations and additional running time of the algorithm are $O(n mathrm{poly}(log{n}, 1/epsilon))$.
The growing need to deal with massive instances motivates the design of algorithms balancing the quality of the solution with applicability. For the latter, an important measure is the emph{adaptive complexity}, capturing the number of sequential rounds of parallel computation needed. In this work we obtain the first emph{constant factor} approximation algorithm for non-monotone submodular maximization subject to a knapsack constraint with emph{near-optimal} $O(log n)$ adaptive complexity. Low adaptivity by itself, however, is not enough: one needs to account for the total number of function evaluations (or value queries) as well. Our algorithm asks $tilde{O}(n^2)$ value queries, but can be modified to run with only $tilde{O}(n)$ instead, while retaining a low adaptive complexity of $O(log^2n)$. Besides the above improvement in adaptivity, this is also the first emph{combinatorial} approach with sublinear adaptive complexity for the problem and yields algorithms comparable to the state-of-the-art even for the special cases of cardinality constraints or monotone objectives. Finally, we showcase our algorithms applicability on real-world datasets.
We study the problem of maximizing a monotone $k$-submodular function $f$ under a knapsack constraint, where a $k$-submodular function is a natural generalization of a submodular function to $k$ dimensions. We present a deterministic $(frac12-frac{1}{2e})$-approximation algorithm that evaluates $f$ $O(n^5k^4)$ times.
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