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Turnpike with Uncertain Measurements: Triangle-Equality Integer Programming with a Deterministic Recovery Guarantee

Authors C. S. Elder , Guillaume Marçais , Carl Kingsford

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Discrete optimization
  • Applied computing → Bioinformatics
Keywords
  • Turnpike problem
  • partial digest
  • integer linear programming
  • LP relaxation
  • unlabeled distances
  • partitions
  • bounded noise

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Abstract

We study the Turnpike problem with uncertain measurements: reconstructing a one-dimensional point set from an unlabeled multiset of pairwise distances under bounded noise and rounding. We give a combinatorial characterization of realizability via a multi-matching that labels interval indices by distinct distance values while satisfying all triangle equalities. This yields an integer linear program (ILP) based on the triangle equality whose constraint structure depends only on the two-partition set P_y = {(r,s,t): y_r + y_s = y_t, (r ≠ s or μ_r ≥ 2)} and a natural linear-programming (LP) relaxation with {0,1}-coefficient constraints. Integral solutions certify realizability and output an explicit assignment matrix, enabling a modular assignment-first, regression-second pipeline for downstream coordinate estimation. Under bounded noise followed by rounding, we prove deterministic separation conditions under which distinct rounded values do not collide and P_y is recovered exactly, so the ILP/LP receives the same combinatorial input as in the noiseless case. Experiments illustrate integrality behavior and degradation outside the provable regime.

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C. S. Elder, Guillaume Marçais, and Carl Kingsford. Turnpike with Uncertain Measurements: Triangle-Equality Integer Programming with a Deterministic Recovery Guarantee. In 26th International Conference on Algorithms for Bioinformatics (WABI 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 390, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WABI.2026.20

Author Details

C. S. Elder
  • Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, Pittsburgh, PA, USA
Guillaume Marçais
  • Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, Pittsburgh, PA, USA
Carl Kingsford
  • Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, Pittsburgh, PA, USA

Funding

This work was supported in part by the US National Science Foundation [III-2232121] and the US National Institutes of Health [R01HG012470].

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