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We show that given $m$ proteins (or protein backbones, which are modeled as3D polygonal chains each of length O(n)) the problem of protein local structurealignment under the discrete Fr\'{e}chet distance is as hard as IndependentSet. So the problem does not admit any approximation of factor$n^{1-\epsilon}$. This is the strongest negative result regarding the proteinlocal structure alignment problem. On the other hand, if $m$ is a constant,then the problem can be solved in polygnomial time.

On the Complexity of Protein Local Structure Alignment Under the Discrete Fr\'echet Distance