  Submodularity is an important property of set functions and has been
extensively studied in the literature. It models set functions that exhibit a
diminishing returns property, where the marginal value of adding an element to
a set decreases as the set expands. This notion has been generalized to
considering sequence functions, where the order of adding elements plays a
crucial role and determines the function value; the generalized notion is
called sequence (or string) submodularity. In this paper, we study a new
problem of robust sequence submodular maximization with cardinality
constraints. The robustness is against the removal of a subset of elements in
the selected sequence (e.g., due to malfunctions or adversarial attacks).
Compared to robust submodular maximization for set function, new challenges
arise when sequence functions are concerned. Specifically, there are multiple
definitions of submodularity for sequence functions, which exhibit subtle yet
critical differences. Another challenge comes from two directions of
monotonicity: forward monotonicity and backward monotonicity, both of which are
important to proving performance guarantees. To address these unique
challenges, we design two robust greedy algorithms: while one algorithm
achieves a constant approximation ratio but is robust only against the removal
of a subset of contiguous elements, the other is robust against the removal of
an arbitrary subset of the selected elements but requires a stronger assumption
and achieves an approximation ratio that depends on the number of the removed
elements. Finally, we generalize the analyses to considering sequence functions
under weaker assumptions based on approximate versions of sequence
submodularity and backward monotonicity
