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

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.

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.