Zak's modified integer round-down conjecture for the skiving stock problem

Theorem 1: for every integer $c\ge0$ there is a rational instance with $5B$ items, sizes in $(53/270,11/54)$ for threshold $1$, $LP_{\mathrm{SSP}}=LP_{\mathrm{SSP,prop}}=B$ and $OPT_{\mathrm{SSP}}\le B-c-1$; $c=1$ refutes MIRDP with no rounding ambiguity. Theorem 2 and Corollary 3: fixed additive approximation is NP-hard. The result is conditional on OpenAI's bin-packing theorem (itself a Candidate here). Not shown: any growth rate of the gap in terms of instance size.

Theorem 1: for every integer $c\ge0$ there is a rational instance with $5B$ items, sizes in $(53/270,11/54)$ for threshold $1$, $LP_{\mathrm{SSP}}=LP_{\mathrm{SSP,prop}}=B$ and $OPT_{\mathrm{SSP}}\le B-c-1$; $c=1$ refutes MIRDP with no rounding ambiguity. Theorem 2 and Corollary 3: fixed additive approximation is NP-hard. The result is conditional on OpenAI's bin-packing theorem (itself a Candidate here). Not shown: any growth rate of the gap in terms of instance size.