Multi-Objective Maximization of Monotone Submodular Functions with Cardinality Constraint

We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, one formulation of which is max|A|=k mini∈{1,...,m} fi(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)− ǫ approximation for constant m. The runtime (number of queries to function oracle) scales exponentially as n 3 . We give the first polynomial time asymptotically constant factor approximations for m = o( k log k ): (i) A randomized (1 − 1/e) algorithm based on Chekuri et al. (2010). (ii) A faster and more practical Õ(n/δ) time, randomized (1 − 1/e) − 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.