  Motivated by applications in online advertising, we consider a class of
maximization problems where the objective is a function of the sequence of
actions as well as the running duration of each action. For these problems, we
introduce the concepts of \emph{sequence-submodularity} and
\emph{sequence-monotonicity} which extend the notions of submodularity and
monotonicity from functions defined over sets to functions defined over
sequences. We establish that if the objective function is sequence-submodular
and sequence-non-decreasing, then there exists a greedy algorithm that achieves
$1-1/e$ of the optimal solution. We apply our algorithm and analysis to two
applications in online advertising: online ad allocation and query rewriting.
We first show that both problems can be formulated as maximizing non-decreasing
sequence-submodular functions. We then apply our framework to these two
problems, leading to simple greedy approaches with guaranteed performances. In
particular, for online ad allocation problem the performance of our algorithm
is $1-1/e$, which matches the best known existing performance, and for query
rewriting problem the performance of our algorithm is $1- 1/e^{1-1/e}$ which
improves upon the best known existing performance in the literature.
