A Proof of Goulimis' Conjecture for the Corrugator Trim Problem Class

This paper investigates the one-dimensional Cutting Stock Problem (CSP) with equality constraints, focusing on Goulimis' Conjecture, which posits that an optimal solution requires at most $m+2$ distinct cutting patterns for $m$ item types. We propose a novel structural class of instances, inspired by the Corrugator Trim Problem (CTP), that is formally defined by subset-cover properties. Using residual-instance arguments and mathematical induction, we prove that for instances satisfying CTP Property 1, an optimal solution requires at most $m+1$ patterns. Furthermore, we demonstrate that this framework extends to instances satisfying CTP Property 2, thereby confirming the $ m+2$-bound conjectured by Goulimis. Our results provide theoretical validation for pattern sparsity in corrugator-motivated operational environments and contribute to the broader understanding of integer conic bounds in cutting and packing problems.