Abstract
We prove that path puzzles with complete row and column information—or equivalently, 2D orthogonal discrete tomography with Hamiltonicity constraint—are strongly NP-complete, ASP-complete, and #P-complete. Along the way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional Matching and Numerical\(3\)-Dimensional Matching.
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Notes
Most sets of row and column constraints are ambiguous; constraining the output image makes the problem harder by preventing an easy image from being found instead.
Section 3.2.4 of [8] proves that 1-in-3-SAT is ASP-hard. Unfortunately, their problem definition allows negative clauses, while we need Positive 1-in-3-SAT.
In [6], Positive 1-in-3-SAT is called “1-Ex3MonoSat”.
If a Numerical\(4\)-Dimensional Matching instance has any elements \(\ge t\), it trivially has no solutions (as all elements are positive). Otherwise, we can convert it to an instance with this property by adding 2t to each element in W, X, Y, Z and changing the target sum from t to \({\hat{t}} = 9 t\). Then every element is strictly between 2t and 3t, and thus strictly between \({\hat{t}}/5 = 9t / 5\) and \({\hat{t}}/3 = 9t / 3\).
If a Numerical\(3\)-Dimensional Matching instance has any elements \(\ge t\), it trivially has no solutions (as all elements are positive). Otherwise, we can convert it to an instance with this property by adding t to each element in X, Y, Z and changing the target sum from t to \({\hat{t}} = 4 t\). Then every element is strictly between t and 2t, and thus strictly between \({\hat{t}}/4 = 4t / 4\) and \({\hat{t}}/2 = 4t / 2\).
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Acknowledgements
We thank Jayson Lynch for useful discussions and debugging help, and Quanquan Liu for help in constructing the figures for this paper. Most figures were produced using SVG Tiler (https://github.com/edemaine/svgtiler).
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Bosboom, J., Demaine, E.D., Demaine, M.L. et al. Path Puzzles: Discrete Tomography with a Path Constraint is Hard. Graphs and Combinatorics 36, 251–267 (2020). https://doi.org/10.1007/s00373-019-02092-5
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DOI: https://doi.org/10.1007/s00373-019-02092-5