  The $k$-batch greedy strategy is an approximate algorithm to solve
optimization problems where the optimal solution is hard to obtain. Starting
with the empty set, the $k$-batch greedy strategy adds a batch of $k$ elements
to the current solution set with the largest gain in the objective function
while satisfying the constraints. In this paper, we bound the performance of
the $k$-batch greedy strategy with respect to the optimal strategy by defining
the total curvature $\alpha_k$. We show that when the objective function is
nondecreasing and submodular, the $k$-batch greedy strategy satisfies a
harmonic bound $1/(1+\alpha_k)$ for a general matroid constraint and an
exponential bound $\left(1-(1-{\alpha}_k/{t})^t\right)/{\alpha}_k$ for a
uniform matroid constraint, where $k$ divides the cardinality of the maximal
set in the general matroid, $t=K/k$ is an integer, and $K$ is the rank of the
uniform matroid. We also compare the performance of the $k$-batch greedy
strategy with that of the $k_1$-batch greedy strategy when $k_1$ divides $k$.
Specifically, we prove that when the objective function is nondecreasing and
submodular, the $k$-batch greedy strategy has better harmonic and exponential
bounds in terms of the total curvature. Finally, we illustrate our results by
considering a task-assignment problem.
