Abstract
The partial digest problem consists in retrieving the positions of a set of points on the real line from their unlabeled pairwise distances. This problem is critical for DNA sequencing, as well as for phase retrieval in X-ray crystallography. When some of the distances are missing, this problem generalizes into a “minimum distance superset problem”, which aims to find a set of points of minimum cardinality such that the multiset of their pairwise distances is a superset of the input. We introduce a quadratic integer programming formulation for the minimum distance superset problem with a pseudo-polynomial number of variables, as well as a polynomial-size integer programming formulation. We investigate three types of solution approaches based on an available integer programming solver: (1) solving a linearization of the pseudo-polynomial-sized formulation, (2) solving the complete polynomial-sized formulation, or (3) performing a binary search over the number of points and solving a simpler feasibility or optimization problem at each step. As illustrated by our computational experiments, the polynomial formulation with binary search leads to the most promising results, allowing to optimally solve most instances with up to 25 distance values and 8 solution points.
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Acknowledgements
This research is partially supported by CNPq (grants number 310855/2013-6, 308498/2015-1, and 425962/2016-4), CAPES (grant number 1192880), and FAPERJ in Brazil. This support is gratefully acknowledged.
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Appendix: Detailed results
Appendix: Detailed results
In Tables 3, 4, 5 and 6, we present the trivial lower and upper bounds and, for each method, the lower and upper bounds obtained, the remaining gap and its running time in seconds. Furthermore, entries which did not reach the time limit of 3600 s and nevertheless have not obtained the optimal solution raised an out of memory exception. These instances are indicated with an “—”.
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Fontoura, L., Martinelli, R., Poggi, M. et al. The minimum distance superset problem: formulations and algorithms. J Glob Optim 72, 27–53 (2018). https://doi.org/10.1007/s10898-017-0579-9
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DOI: https://doi.org/10.1007/s10898-017-0579-9