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.

翻译：我们研究具有不确定测量的Turnpike问题：在有界噪声和舍入条件下，从未标记的多重集成对距离重建一维点集。通过对满足所有三角等式的区间索引用不同距离值进行标记的多重匹配，我们给出了可实现性的组合刻画。由此得到一个基于三角等式的整数线性规划，其约束结构仅依赖于二分集$\mathcal{P}_y=\{(r,s,t): y_r+y_s=y_t\}$以及具有$\{0,1\}$系数约束的自然线性规划松弛。整数解可证明可实现性并输出显式赋值矩阵，从而形成先赋值后回归的下游坐标估计流程。在有界噪声后接舍入的条件下，我们证明了确定性分离条件，该条件下$\mathcal{P}_y$被精确恢复，因此整数线性规划/线性规划接收与无噪声情形相同的组合输入。实验展示了可证明区域外的整数性行为和退化现象。