Multi-Objective Maximization of Monotone Submodular Functions with Cardinality Constraint
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We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, one formulation of which is _|A|=k_i∈{1,...,m}f_i(A). Krause et al. (2008) showed that when the number of functions m grows as the cardinality k i.e., m=Ω(k), the problem is inapproximable (unless P=NP). For the more general case of matroid constraint, Chekuri et al. (2010) gave a randomized (1-1/e)-ϵ approximation for constant m. The runtime (number of queries to function oracle) scales exponentially as n^m/ϵ^3. We give the first polynomial time asymptotically constant factor approximations for m=o(k/^3 k): (i) A randomized (1-1/e) algorithm based on Chekuri et al. (2010). (ii) A faster and more practical Õ(n/δ^3) time, randomized (1-1/e)^2-δ approximation based on Multiplicative-Weight-Updates. Finally, we characterize the variation in optimal solution value as a function of the cardinality k, leading to a derandomized approximation for constant m.
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