Approximation guarantee of the iterative rounding algorithm for the Gasoline problem

Ascertain the worst‑case approximation ratio of the iterative rounding algorithm that iteratively rounds the linear programming relaxation for the Gasoline problem and its canonical d‑dimensional generalization, by proving a formal approximation guarantee.

Background

A heuristic based on iterative rounding of the LP relaxation has been studied for the Gasoline problem, and was conjectured to be a 2‑approximation; this work provides counterexamples showing the ratio can exceed 2.

Despite these counterexamples, a definitive approximation guarantee for the iterative rounding algorithm—both in 1D and in its canonical d‑dimensional generalization—remains unknown.