The minimum distance superset problem: formulations and algorithms

Leonardo Fontoura (), Rafael Martinelli (), Marcus Poggi () and Thibaut Vidal ()
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Leonardo Fontoura: Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)
Rafael Martinelli: Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)
Marcus Poggi: Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)
Thibaut Vidal: Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)

Journal of Global Optimization, 2018, vol. 72, issue 1, No 3, 27-53

Abstract: 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.

Keywords: Partial digest problem; Minimum distance superset; Turnpike problem; Combinatorial optimization; Integer programming (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-017-0579-9

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