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Abstract: There are many folded states consistent with a given mountain–valley pattern of equidistant creases on a long paper strip. We would like to fold a paper strip such that the number of paper layers between each pair of hinged paper segments, called the crease width at the crease point, is minimized. This problem is called the stamp folding problem and there are two variants of this problem: minimization of the maximum crease width, and minimization of the total crease width. This optimization problem was recently introduced and investigated from the viewpoint of the counting problem. However, its computational complexity is not known. In this paper, we first show that the minimization problem of the maximum crease width is strongly NP-complete. Hence we cannot solve the problem in polynomial time unless P=NP. We next propose an algorithm that solves the minimization problem. The algorithm itself is a straightforward one, but its analysis is not trivial. We show that this algorithm runs in time where is the number of creases and is the total crease width. [Copyright &y& Elsevier]

Abstract: This paper studies the semi-online scheduling on two uniform machines under a grade of service (GoS) provision, where one machine is available for all jobs and the other one is only available for partial jobs. The objective is to minimize the makespan. We consider three variants, where the optimal makespan, the total size of jobs, and the largest job size are known in advance respectively, and design optimal algorithms for all of them. [Copyright &y& Elsevier]