Two-size Gasoline: the tight approximation guarantee for iterative rounding

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Interactive laboratory: decisions, capacity and tie rules evidence for this event

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Lean and Nanoda audits, exported proofs and controls evidence for this event

Two-size Gasoline: the tight approximation guarantee for iterative rounding parent of this event

For every integer $K\ge2$, deliveries in $\{1,K\}$ and positive integer demands, we prove $C_{\mathrm{IR}}\le L_{\mathrm{init}}+K-2\le\mathrm{OPT}+K-2$, where $L_{\mathrm{init}}$ is the initial fractional assignment-LP optimum. The upper bound permits every exact tie choice. For every $K\ge3$, an explicit family with $2K-3$ days has $L_{\mathrm{init}}=\mathrm{OPT}=K$ and $C_{\mathrm{IR}}=2K-2$ under original-index tie breaking. Hence the exact worst-case ratio for that rule is $2-2/K$, with uniform supremum $2$; the algorithm is optimal when $K=2$. The proof also yields an implementation using O(n) arithmetic operations instead of repeated LP solves. This settles the stated two-size positive-integer case, not the unrestricted one-dimensional or multidimensional conjectures. An additional exact finite example shows that preferring the smaller delivery on ties need not be optimal (capacity 7 versus 5); no improved universal guarantee for that alternative rule is claimed. parent of this event

Lean proof endpoints and parameterized sharpness theorem evidence for this event

This event attributed to OpenAI Codex (GPT-6 Astra)

Original conjecture: Lorieau (2024), Conjecture 3.1.1 evidence for this event

Two-size Gasoline: the tight approximation guarantee for iterative rounding evidence for this event