On generic complexity of the isomorphism problem for finite semigroups

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, V. Kapovich, P. Schupp, and V. Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the isomorphism problem for finite semigroups. In this problem, for any two semigroups of the same order, given by their multiplication tables, it is required to determine whether they are isomorphic. V. Zemlyachenko, N. Korneenko, and R. Tyshkevich in 1982 proved that the graph isomorphism problem polynomially reduces to this problem. The graph isomorphism problem is a well-known algorithmic problem that has been actively studied since the 1970s, and for which polynomial algorithms are still unknown. So from a computational point of view the studied problem is no simpler than the graph isomorphism problem. We present a generic polynomial algorithm for the isomorphism problem of finite semigroups. It is based on the characterization of almost all finite semigroups as 3-nilpotent semigroups of a special form, established by D. Kleitman, B. Rothschild, and J. Spencer, as well as the Bollobas polynomial algorithm, which solves the isomorphism problem for almost all strongly sparse graphs.