Graph isomorphism problem

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 2^{O(√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 2^{O(√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 ${\displaystyle 2^{O((\log n)^{c})}}$ for some fixed ${\displaystyle c>0}$.[6][7][8] 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.[9][10] Helfgott further claims that one can take c = 3, so the running time is 2^{O((log n)3)}.[11][12] 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.[13]

The graph isomorphism problem is computationally equivalent to the problem of computing the automorphism group of a graph,[14][15] 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.

A number of important special cases of the graph isomorphism problem have efficient, polynomial-time solutions:

• Trees[16][17]
• Planar graphs[18] (In fact, planar graph isomorphism is in log space,[19] a class contained in P)
• Interval graphs[20]
• Permutation graphs[21]
• Circulant graphs[22]
• Bounded-parameter graphs
  • Graphs of bounded treewidth[23]
  • Graphs of bounded genus[24] (Planar graphs are graphs of genus 0.)
  • Graphs of bounded degree[25]
  • Graphs with bounded eigenvalue multiplicity[26]
  • k-Contractible graphs (a generalization of bounded degree and bounded genus)[27]
  • Color-preserving isomorphism of colored graphs with bounded color multiplicity (i.e., at most k vertices have the same color for a fixed k) is in class NC, which is a subclass of P[28]

Since the graph isomorphism problem is neither known to be NP-complete nor known to be tractable, researchers have sought to gain insight into the problem by defining a new class GI, the set of problems with a polynomial-time Turing reduction to the graph isomorphism problem.[29] If in fact the graph isomorphism problem is solvable in polynomial time, GI would equal P. On the other hand, if the problem is NP-complete, GI would equal NP and all problems in NP would be solvable in quasi-polynomial time.

As is common for complexity classes within the polynomial time hierarchy, a problem is called GI-hard if there is a polynomial-time Turing reduction from any problem in GI to that problem, i.e., a polynomial-time solution to a GI-hard problem would yield a polynomial-time solution to the graph isomorphism problem (and so all problems in GI). A problem ${\displaystyle X}$ is called complete for GI, or GI-complete, if it is both GI-hard and a polynomial-time solution to the GI problem would yield a polynomial-time solution to ${\displaystyle X}$.

The graph isomorphism problem is contained in both NP and co-AM. GI is contained in and low for Parity P, as well as contained in the potentially much smaller class SPP.[30] That it lies in Parity P means that the graph isomorphism problem is no harder than determining whether a polynomial-time nondeterministic Turing machine has an even or odd number of accepting paths. GI is also contained in and low for ZPP^NP.[31] This essentially means that an efficient Las Vegas algorithm with access to an NP oracle can solve graph isomorphism so easily that it gains no power from being given the ability to do so in constant time.

Manuel Blum and Sampath Kannan (1995) have shown a probabilistic checker for programs for graph isomorphism. Suppose P is a claimed polynomial-time procedure that checks if two graphs are isomorphic, but it is not trusted. To check if G and H are isomorphic:

• Ask P whether G and H are isomorphic.
  • If the answer is "yes':
    • Attempt to construct an isomorphism using P as subroutine. Mark a vertex u in G and v in H, and modify the graphs to make them distinctive (with a small local change). Ask P if the modified graphs are isomorphic. If no, change v to a different vertex. Continue searching.
    • Either the isomorphism will be found (and can be verified), or P will contradict itself.
  • If the answer is "no":
    • Perform the following 100 times. Choose randomly G or H, and randomly permute its vertices. Ask P if the graph is isomorphic to G and H. (As in AM protocol for graph nonisomorphism).
    • If any of the tests are failed, judge P as invalid program. Otherwise, answer "no".

This procedure is polynomial-time and gives the correct answer if P is a correct program for graph isomorphism. If P is not a correct program, but answers correctly on G and H, the checker will either give the correct answer, or detect invalid behaviour of P. If P is not a correct program, and answers incorrectly on G and H, the checker will detect invalid behaviour of P with high probability, or answer wrong with probability 2⁻¹⁰⁰.

Notably, P is used only as a blackbox.

Graphs are commonly used to encode structural information in many fields, including computer vision and pattern recognition, and graph matching, i.e., identification of similarities between graphs, is an important tools in these areas. In these areas graph isomorphism problem is known as the exact graph matching.[42]

In cheminformatics and in mathematical chemistry, graph isomorphism testing is used to identify a chemical compound within a chemical database.[43] Also, in organic mathematical chemistry graph isomorphism testing is useful for generation of molecular graphs and for computer synthesis.

Chemical database search is an example of graphical data mining, where the graph canonization approach is often used.[44] In particular, a number of identifiers for chemical substances, such as SMILES and InChI, designed to provide a standard and human-readable way to encode molecular information and to facilitate the search for such information in databases and on the web, use canonization step in their computation, which is essentially the canonization of the graph which represents the molecule.

In electronic design automation graph isomorphism is the basis of the Layout Versus Schematic (LVS) circuit design step, which is a verification whether the electric circuits represented by a circuit schematic and an integrated circuit layout are the same.[45]

• Graph automorphism problem
• Graph canonization