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 $\max_{|A|=k}\min_{i\in\{1,\dots,m\}}f_i(A)$. Krause et al. (2008) showed that when the number of functions $m$ grows as the cardinality $k$ i.e., $m=\Omega(k)$, the problem is inapproximable (unless $P=NP$). For the more general case of matroid constraint, Chekuri et al. (2010) gave a randomized $(1-1/e)-\epsilon$ approximation for constant $m$. The runtime (number of queries to function oracle) scales exponentially as $n^{m/\epsilon^3}$. We give the first polynomial time asymptotically constant factor approximations for $m=o(\frac{k}{\log^3 k})$: $(i)$ A randomized $(1-1/e)$ algorithm based on Chekuri et al. (2010). $(ii)$ A faster and more practical $\tilde{O}(n/\delta^3)$ time, randomized $(1-1/e)^2-\delta$ 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 $m$.