Graph isomorphism problem

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level. At the same time, isomorphism for many special classes of graphs can be solved in polynomial time, and in practice graph isomorphism can often be solved efficiently. This problem is a special case of the subgraph isomorphism problem, which asks whether a given graph G contains a subgraph that is isomorphic to another given graph H; this problem is known to be NP-complete. It is also known to be a special case of the non-abelian hidden subgroup problem over the symmetric group. In the area of image recognition it is known as the exact graph matching. The best currently accepted theoretical algorithm is due to Babai & Luks (1983), and is based on the earlier work by Luks (1982) combined with a subfactorial algorithm of V. N. Zemlyachenko (Zemlyachenko, Korneenko & Tyshkevich 1985). The algorithm has run time 2O(√n log n) for graphs with n vertices and relies on the classification of finite simple groups. Without the CFSG theorem, 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 (1983). 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 obtained by Babai & Codenotti (2008). In November 2015, Babai announced a quasipolynomial time algorithm for all graphs, that is, one with running time 2 O ( ( log ⁡ n ) c ) {displaystyle 2^{O((log n)^{c})}} for some fixed c > 0 {displaystyle c>0} . On January 4, 2017, Babai retracted the quasi-polynomial claim and stated a sub-exponential time bound instead after Harald Helfgott discovered a flaw in the proof. On January 9, 2017, Babai announced a correction (published in full on January 19) and restored the quasi-polynomial claim, with Helfgott confirming the fix. Helfgott further claims that one can take c = 3, so the running time is 2O((log n)3). The new proof has not been fully peer-reviewed yet. There are several competing practical algorithms for graph isomorphism, such as those due to McKay (1981), Schmidt & Druffel (1976), and Ullman (1976). While they seem to perform well on random graphs, a major drawback of these algorithms is their exponential time performance in the worst case. 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 & Luks (1983) obtained complexity bounds similar to that for graph isomorphism.