  We call an objective function or algorithm symmetric with respect to an input
if after swapping two parts of the input in any algorithm, the solution of the
algorithm and the output remain the same. More formally, for a permutation
$\pi$ of an indexed input, and another permutation $\pi'$ of the same input,
such that swapping two items converts $\pi$ to $\pi'$, $f(\pi)=f(\pi')$, where
$f$ is the objective function.
  After reviewing samples of the algorithms that exploit symmetry, we give
several new ones, for finding lower-bounds, beating adversaries in online
algorithms, designing parallel algorithms and data summarization. We show how
to use the symmetry between the sampled points to get a lower/upper bound on
the solution. This mostly depends on the equivalence class of the parts of the
input that when swapped, do not change the solution or its cost.
