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Exact computation of graph edit distance (GED) can be solved in exponential time complexity only. A previously introduced approximation framework reduces the computation of GED to an instance of a linear sum assignment problem. Major benefit of this reduction is that an optimal assignment of nodes (including local structures) can be computed in polynomial time. Given this assignment an approximate value of GED can be immediately derived. Yet, since this approach considers local  rather than the global  structural properties of the graphs only, the GED derived from the optimal assignment is suboptimal. The contribution of the present paper is twofold. First, we give a formal proof that this approximation builds an upper bound of the true graph edit distance. Second, we show how the existing approximation framework can be reformulated such that a lower bound of the edit distance can be additionally derived. Both bounds are simultaneously computed in cubic time.

Computing Upper and Lower Bounds of Graph Edit Distance in Cubic Time