Ranking with Submodular Valuations

We study the problem of ranking with submodular valuations. An instance of this problem consists of a ground set $[m]$, and a collection of $n$ monotone submodular set functions $f^1, \ldots, f^n$, where each $f^i: 2^{[m]} \to R_+$. An additional ingredient of the input is a weight vector $w \in R_+^n$. The objective is to find a linear ordering of the ground set elements that minimizes the weighted cover time of the functions. The cover time of a function is the minimal number of elements in the prefix of the linear ordering that form a set whose corresponding function value is greater than a unit threshold value. Our main contribution is an $O(\ln(1 / \epsilon))$-approximation algorithm for the problem, where $\epsilon$ is the smallest non-zero marginal value that any function may gain from some element. Our algorithm orders the elements using an adaptive residual updates scheme, which may be of independent interest. We also prove that the problem is $\Omega(\ln(1 / \epsilon))$-hard to approximate, unless P = NP. This implies that the outcome of our algorithm is optimal up to constant factors.