Monte-Carlo algorithms in graph isomorphism testing

Abstract. We present an O(V 4 log V ) coin flipping algorithm to test vertex-colored graphs with bounded color multiplicities for color-preserving isomorphism. We are also able to generate uniformly distributed random automorphisms of such graphs. A more general result finds generators for the intersection of cylindric subgroups of a direct product of groups in O(n7/2 log n) time, where n is the length of the input string. This result will be applied in another paper to find a polynomial time coin flipping algorithm to test isomorphism of graphs with bounded eigenvalue multiplicities. The most general result says that if AutX is accessible by a chain G0 ≥ · · · ≥ Gk = AutX of quickly recognizable groups such that the indices |Gi−1 : Gi| are small (but unknown), the order of |G0| is known and there is a fast way of generating uniformly distributed random members of G0 then a set of generators of AutX can be found by a fast algorithm. Applications of the main result improve the complexity of isomorphism testing for graphs with bounded valences to exp(n1/2+o(1)) and for distributive lattices to O(n6 log logn). ∗Apart from possible typos, this is a verbatim transcript of the author’s 1979 technical report, “Monte Carlo algorithms in graph isomorphism testing” (Université de Montréal, D.M.S. No. 79-10), with two footnotes added to correct a typo and a glaring omission in the original. We present an O(V 4 log V ) coin flipping algorithm to test vertex-colored graphs with bounded color multiplicities for color-preserving isomorphism. We are also able to generate uniformly distributed random automorphisms of such graphs. A more general result finds generators for the intersection of cylindric subgroups of a direct product of groups in O(n7/2 log n) time, where n is the length of the input string. This result will be applied in another paper to find a polynomial time coin flipping algorithm to test isomorphism of graphs with bounded eigenvalue multiplicities. The most general result says that if AutX is accessible by a chain G0 ≥ · · · ≥ Gk = AutX of quickly recognizable groups such that the indices |Gi−1 : Gi| are small (but unknown), the order of |G0| is known and there is a fast way of generating uniformly distributed random members of G0 then a set of generators of AutX can be found by a fast algorithm. Applications of the main result improve the complexity of isomorphism testing for graphs with bounded valences to exp(n1/2+o(1)) and for distributive lattices to O(n6 log logn). ∗Apart from possible typos, this is a verbatim transcript of the author’s 1979 technical report, “Monte Carlo algorithms in graph isomorphism testing” (Université de Montréal, D.M.S. No. 79-10), with two footnotes added to correct a typo and a glaring omission in the original.