Turnpike with Uncertain Measurements: Triangle-Equality ILP with a Deterministic Recovery Guarantee
This work addresses a fundamental combinatorial geometry problem with applications in fields like computational biology and signal processing, though it is incremental as it builds on existing Turnpike formulations with new noise-handling methods.
The paper tackles the Turnpike problem with uncertain measurements by reconstructing a one-dimensional point set from noisy, unlabeled pairwise distances, achieving a deterministic recovery guarantee under bounded noise and rounding conditions.
We study Turnpike 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 ILP based on the triangle equality whose constraint structure depends only on the two-partition set $\mathcal{P}_y=\{(r,s,t): y_r+y_s=y_t\}$ and a natural LP relaxation with $\{0,1\}$-coefficient constraints. Integral solutions certify realizability and output an explicit assignment matrix, enabling an assignment-first, regression-second pipeline for downstream coordinate estimation. Under bounded noise followed by rounding, we prove a deterministic separation condition under which $\mathcal{P}_y$ 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.