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

C. S. Elder, Guillaume Marçais, Carl Kingsford

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.

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

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 and a natural LP relaxation with -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 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.
Paper Structure (22 sections, 8 theorems, 9 equations, 4 figures, 2 algorithms)

This paper contains 22 sections, 8 theorems, 9 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. This work: a triangle-based assignment viewpoint
  3. Preliminaries
  4. Notation and multisets
  5. Intervals, refinements, and rulers
  6. Multi-matchings and realizability
  7. An MILP formulation
  8. Noisy variants and the (r,R) model
  9. A Triangle-Based ILP and LP Relaxation
  10. Two-partitions
  11. A triangle-based ILP
  12. Triangle-based LP relaxation and interpretation
  13. Structural consequences and formulation reductions
  14. Two-partition recovery under noise and rounding
  15. Experiments
  16. ...and 7 more sections

Key Result

Lemma 3

A matrix $\rho\in\mathbb{R}^{\left[n\right]\times\left[n\right]}$ is a ruler if and only if it satisfies $\rho_{ik}=\rho_{ij}+\rho_{jk}$ for all $i,j,k\in\left[n\right]$.

Figures (4)

  • Figure 1: Synthetic exact instances. LP integrality score on (left) linear Turnpike instances and (right) circular Beltway instances. Curves are stratified by the distribution of $x^\star$.
  • Figure 2: Noisy $(r,R)$ phase diagram for two-partition recovery. Top: recovery rate; bottom-left: false-positive rate (spurious two-partitions); bottom-right: false-negative rate (missing true two-partitions). The dashed line is the sufficient condition of Theorem \ref{['thm:two-partition-recovery']}.
  • Figure 3: Comparison with regression-based baselines. Panels show median labeling error (top), permutation distance (middle), and coordinate MAE (bottom) for the triangle ILP/LP, MM, GD, and the sorting-network Ext-ILP baseline.
  • Figure 4: Partial digest experiments on linear and circular genomes. Normalized interval-labeling error (left) and fragment recovery rate (right).

Theorems & Definitions (21)

  • Example 1
  • Definition 2
  • Lemma 3: Triangle equalities characterize rulers
  • proof
  • Remark 4
  • Definition 5
  • Lemma 6
  • proof
  • Example 7: One triangle constraint
  • Proposition 8
  • ...and 11 more