State of The Art
The best current theoretical algorithm is due to Eugene Luks (1983), and is based on the earlier work by Luks (1981), Babai & Luks (1982), combined with a subfactorial algorithm due to Zemlyachenko (1982). The algorithm relies on the classification of finite simple groups. Without CFSG, a slightly weaker bound 2O(√n log2 n) was obtained first for strongly regular graphs by László Babai (1980), and then extended to general graphs by Babai & Luks (1982). Improvement of the exponent √n is a major open problem; for strongly regular graphs this was done by Spielman (1996). For hypergraphs of bounded rank, a subexponential upper bound matching the case of graphs, was recently obtained by Babai & Codenotti (2008).
On a side note, the graph isomorphism problem is computationally equivalent to the problem of computing the automorphism group of a graph, and is weaker than the permutation group isomorphism problem, and the permutation group intersection problem. For the latter two problems, Babai, Kantor and Luks (1983) obtained complexity bounds similar to that for the graph isomorphism.
There are several competing practical algorithms for graph isomorphism, due to McKay (1981), Schmidt & Druffel (1976), Ullman (1976), etc. While they seem to perform well on random graphs, a major drawback of these algorithms is their exponential time performance in the worst case.
Read more about this topic: Graph Isomorphism Problem
Famous quotes containing the words state of, state and/or art:
“The twelve Cells for Incorrigibles ... are also carved out of the solid rock hill. On the walls of one of the cells human “liberty” is clearly inscribed, with the “liberty” in significant quotation marks.”
—Administration in the State of Ariz, U.S. public relief program (1935-1943)
“In every particular state of the world, those nations which are strongest tend to prevail over the others; and in certain marked peculiarities the strongest tend to be the best.”
—Walter Bagehot (1826–1877)
“Since art is the expression of beauty and beauty can be understood only in the form of the material elements of the true idea it contains, art has become almost uniquely feminine. Beauty is woman, and also art is woman.”
—Rémy De Gourmont (1858–1915)