We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearlyoptimal, 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 nearlyoptimal 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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MultiAgent Maximization of a Monotone Submodular Function via Maximum Consensus
This paper studies distributed submodular optimization subject to partition matroid. We work in the value oracle model where the only access of the agents to the utility function is through a black box that returns the utility function value. The agents are communicating over a connected undirected graph and have access only to their own strategy set. As known in the literature, submodular maximization subject to matroid constraints is NPhard. Hence, our objective is to propose a polynomialtime distributed algorithm to obtain a suboptimal solution with guarantees on the optimality bound. Our proposed algorithm is based on a distributed stochastic gradient ascent scheme built on the multilinearextension of the submodular set function. We use a maximum consensus protocol to minimize the inconsistency of the shared information over the network caused by delay in the flow of information while solving for the fractional solution of the multilinear extension model. Furthermore, we propose a distributed framework of finding a set solution using the fractional solution. We show that our distributed algorithm results in a strategy set that when the team objective function is evaluated at worst case the objective function value is in 1−1/e−O(1/T) of the optimal solution in the value oracle model where T is the number of communication instances of the agents. An example demonstrates our results.
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 Award ID(s):
 1724331
 NSFPAR ID:
 10328432
 Date Published:
 Journal Name:
 IEEE Conference on Decision and Control
 Page Range / eLocation ID:
 1238 to 1243
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 nearlyoptimal, $1  1/e  \epsilon$ 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 nearlyoptimal 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.more » « less

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