Isomorphism Testing for Graphs, Semigroups, and Finite Automata are Polynomially Equivalent Problems

Abstract

Two problems are polynomially equivalent if each is polynomially reducible to the other. The problems of testing either two graphs, two semigroups, or two finite automata for isomorphism are shown to be polynomially equivalent. For graphs the isomorphism problem may be restricted to regular graphs since we show that this is equivalent to the general case. Using the techniques of Hartmanis and Berman we then show that this equivalence is actually a polynomial isomorphism. It is conjectured that the isomorphism problem for groups is not in this equivalence class, but that it is an easier problem. If the conjecture is true then $P \ne NP$; if it is false then there exists a “subexponential” $O(n^{c_1 \log n + c_2 } )$ algorithm for graph isomorphism.