Asymptotic behavior of the iterative rounding algorithm’s ratio on a constructed family of d‑dimensional Gasoline instances

Prove whether, for the constructed family of d‑dimensional Gasoline instances defined by structured sequences X and Y, the approximation factor of the iterative rounding algorithm scales linearly with input length and converges asymptotically to 4, 6, and 8 for dimensions d=2, 3, and 4, respectively.

Background

The paper presents a family of d‑dimensional instances extending Lorieau’s construction and reports computational evidence that both the algorithm’s returned value and the optimum scale linearly with instance length, suggesting limiting ratios of 4, 6, and 8 for d=2, 3, and 4.

However, a proof is lacking because the LP relaxation’s optimum changes at each rounding step, preventing reuse of prior proof techniques.

References

If this scaling held for larger $k$, the approximation-factors would approach $4,6,8$ for $d=2,3, 4$ respectively. Unfortunately, the proof-strategy employed in \citet{Lorieau} does not apply here, as the optimum value of the relaxed Linear Program changes at each step of the algorithm. Hence, we are unable to provide a proof that these trends hold asymptotically.

— The Art of Being Difficult: Combining Human and AI Strengths to Find Adversarial Instances for Heuristics  (2601.16849 - Nikoleit et al., 23 Jan 2026) in Subsubsection “Gasoline” (Section 3.4)