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The graph traversal edit distance (GTED), introduced by Ebrahimpour Boroojenyet al.~(2018), is an elegant distance measure defined as the minimum editdistance between strings reconstructed from Eulerian trails in two edge-labeledgraphs. GTED can be used to infer evolutionary relationships between species bycomparing de Bruijn graphs directly without the computationally costly anderror-prone process of genome assembly. Ebrahimpour Boroojeny et al.~(2018)propose two ILP formulations for GTED and claim that GTED is polynomiallysolvable because the linear programming relaxation of one of the ILPs alwaysyields optimal integer solutions. The claim that GTED is polynomially solvableis contradictory to the complexity results of existing string-to-graph matchingproblems. We resolve this conflict in complexity results by proving that GTEDis NP-complete and showing that the ILPs proposed by Ebrahimpour Boroojeny etal. do not solve GTED but instead solve for a lower bound of GTED and are notsolvable in polynomial time. In addition, we provide the first two, correct ILPformulations of GTED and evaluate their empirical efficiency. These resultsprovide solid algorithmic foundations for comparing genome graphs and point tothe direction of heuristics. The source code to reproduce experimental resultsis available at https://github.com/Kingsford-Group/gtednewilp/.

Revisiting the Complexity of and Algorithms for the Graph Traversal Edit Distance and Its Variants