Symmetries: From Proofs To Algorithms And Back

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 ${\pi }^{\mathrm{\prime }}$ of the same input, such that swapping two items converts $\pi$ to ${\pi }^{\mathrm{\prime }}$, $f(\pi )=f({\pi }^{\mathrm{\prime }})$, 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.