1 research outputs found
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
of an indexed input, and another permutation of the same input,
such that swapping two items converts to , , where
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.Comment: The definition of symmetry discussed here is too general, and the
examples in this paper has mislead people to believe this is a desirable or
useful property. A counter-example is SAT, where there is a symmetry between
each variable and its complement, and it is equivalent to the original
proble