Reduction of the Graph Isomorphism Problem to Equality Checking of n variables Polynomials and the Algorithms that Use the Reduction
The graph isomorphism problem is considered. We assign modified characteristic polynomials for graphs and reduce the graph isomorphism problem to the following one. It is required to find out, is there such an enumeration of the graphs vertices that the polynomials of the graphs are equal. We present algorithms for the graph isomorphism problem that use the reduction. We prove the propositions that justify the possibility of a numerical realization of the algorithms for the general case of the graph isomorphism problem. The algorithms perform equality checking of graphs modified $n$ variables characteristic polynomials. We show that probability of obtaining a wrong solution of the graph isomorphism problem using recursive modification of the algorithm is negligible if the algorithm parameter is sufficiently large. In the course of its implementation, the algorithm checks the equality of the graphs modified characteristic polynomials in predefined points. For $n$ vertices graph, the polynomial has $2^n$ coefficients so its value in some point cannot be evaluated directly for large enough $n$ We show that we may check the equality of the polynomials in predefined points without direct evaluation of the polynomials values in these points. We prove that, for the graphs on $n$ vertices, it is required $O(n^4) elementary machine operations and it is requred machine numbers with mantissa's length $O(n^2) to check equality of the graphs polynomials values in predefined points. In general, it needs an exponential time to solve the $GI$ instance using the presented approach, but in practice, it is efficient even for compuationally hard instances of the graph isomorphism problem.
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