Predicting Minimum Free Energy Structures of Multi-Stranded Nucleic Acid Complexes Is APX-Hard

Authors Anne Condon , Monir Hajiaghayi, Chris Thachuk

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Author Details

Anne Condon
  • The University of British Columbia, Vancouver, Canada
Monir Hajiaghayi
  • The University of British Columbia, Vancouver, Canada
Chris Thachuk
  • The University of Washington, Seattle, WA, USA


We thank Erik Winfree for helpful discussions and proposing the problem and we also thank DNA 27 reviewers for their feedback.

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Anne Condon, Monir Hajiaghayi, and Chris Thachuk. Predicting Minimum Free Energy Structures of Multi-Stranded Nucleic Acid Complexes Is APX-Hard. In 27th International Conference on DNA Computing and Molecular Programming (DNA 27). Leibniz International Proceedings in Informatics (LIPIcs), Volume 205, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Given multiple nucleic acid strands, what is the minimum free energy (MFE) secondary structure that they can form? As interacting nucleic acid strands are the basis for DNA computing and molecular programming, e.g., in DNA self-assembly and DNA strand displacement systems, determining the MFE structure is an important step in the design and verification of these systems. Efficient dynamic programming algorithms are well known for predicting the MFE pseudoknot-free secondary structure of a single nucleic acid strand. In contrast, we prove that for a simple energy model, the problem of predicting the MFE pseudoknot-free secondary structure formed from multiple interacting nucleic acid strands is NP-hard and also APX-hard. The latter result implies that there does not exist a polynomial time approximation scheme for this problem, unless 𝖯 = NP, and it suggests that heuristic methods should be investigated.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Applied computing → Chemistry
  • Nucleic Acid Secondary Structure Prediction
  • APX-Hardness
  • NP-Hardness


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