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

2026-03-18

AI Summary

This work addresses the problem of reconstructing a one-dimensional point set from unlabeled pairwise distance multiset measurements corrupted by bounded noise and rounding errors. The authors propose an integer linear programming (ILP) model grounded in triangle equality constraints, which relies solely on a bipartition set 𝒫_y, and introduce a two-stage coordinate estimation procedure that prioritizes assignment before regression. Theoretical analysis establishes exact recovery of the original combinatorial structure under a deterministic separation condition and provides deterministic reconstruction guarantees in the presence of bounded noise. Experimental results validate the integrality of solutions obtained via LP relaxation and demonstrate the method’s performance beyond the theoretical recovery regime.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

Abstract

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.

Problem

• Turnpike problem
• uncertain measurements
• pairwise distances
• bounded noise
• rounding

Innovation

• Turnpike problem
• triangle equality
• integer linear programming
• deterministic recovery
• combinatorial realizability