We investigate the computational complexity of finding an element of
a permutation group~$H\subseteq S_n$ with a minimal distance to a
given~$\pi\in S_n$, for different metrics on~$S_n$. We assume
that~$H$ is given by a set of generators, such that the problem
cannot be solved in polynomial time ... more >>>

We develop the theory of holographic algorithms. We give
characterizations of algebraic varieties of realizable
symmetric generators and recognizers on the basis manifold,
and a polynomial time decision algorithm for the
simultaneous realizability problem.
Using the general machinery we are able to give
unexpected holographic algorithms for
some counting problems, ... more >>>

Suppose you ran a chess tournament, everybody played everybody, and you wanted to use the results to rank everybody. Unless you were really lucky, the results would not be acyclic, so you could not just sort the players by who beat whom. A natural objective is to find a ranking ... more >>>