Revisiting the complexity of and algorithms for the graph traversal edit distance and its variants

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

Flow network tracking for spatiotemporal and periodic point matching: Applied to cardiac motion analysis

The accurate quantification of left ventricular (LV) deformation/strain shows significant promise for quantitatively assessing cardiac function for use in diagnosis and therapy planning. However, accurate estimation of the displacement of myocardial tissue and hence LV strain has been challenging due to a variety of issues, including those related to deriving tracking tokens from images and following tissue locations over the entire cardiac cycle. In this work, we propose a point matching scheme where correspondences are modeled as flow through a graphical network. Myocardial surface points are set up as nodes in the network and edges define neighborhood relationships temporally. The novelty lies in the constraints that are imposed on the matching scheme, which render the correspondences one-to-one through the entire cardiac cycle, and not just two consecutive frames. The constraints also encourage motion to be cyclic, which an important characteristic of LV motion. We validate our method by applying it to the estimation of quantitative LV displacement and strain estimation using 8 synthetic and 8 open-chested canine 4D echocardiographic image sequences, the latter with sonomicrometric crystals implanted on the LV wall. We were able to achieve excellent tracking accuracy on the synthetic dataset and observed a good correlation with crystal-based strains on the in-vivo data.