as a function of m, the number of orders. As the number of orders increases, it may therefore become impractical to enumerate the possible cutting patterns. An alternative approach uses delayed column-generation. This method solves the cutting-stock problem by starting with just a few patterns. It generates additional patterns when they are needed. For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program. The knapsack problem has well-known methods to solve it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be much more efficient than the original approach, particularly as the size of the problem grows. The column generation approach as applied to the cutting stock problem was pioneered by Gilmore and Gomory in a series of papers published in the 1960s. Gilmore and Gomory showed that this approach is guaranteed to converge to the (fractional) optimal solution, without needing to enumerate all the possible patterns in advance. A limitation of the original Gilmore and Gomory method is that it does not handle integrality, so the solution may contain fractions, e.g. a particular pattern should be produced 3.67 times. Rounding to the nearest integer often does not work, in the sense that it may lead to a sub-optimal solution and/or under- or over-production of some of the orders (and possible infeasibility in the presence of two-sided demand constraints). This limitation is overcome in modern algorithms, which can solve to optimality (in the sense of finding solutions with minimum waste) very large instances of the problem (generally larger than encountered in practice). The cutting-stock problem is often highly degenerate, in that multiple solutions with the same amount of waste are possible. This degeneracy arises because it is possible to move items around, creating new patterns, without affecting the amount of waste. This gives rise to a whole collection of related problems which are concerned with some other criterion, such as the following: The minimum pattern count problem: to find a minimum-pattern-count solution amongst the minimum-waste solutions. This is a very hard problem, even when the waste is known. There is a conjecture that any equality-constrained one-dimensional instance with n sizes has at least one minimum waste solution with no more than n + 1 patterns. This conjecture was first refuted in April 2020 with an example with 9