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Since all of these problems are in general -hard, much attention has been given to the regime where there is only a small number k<\/jats:italic> of job types, but possibly the number of jobs n<\/jats:italic> is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (\n \n $$Q|HM|C_{\\max }$$<\/jats:tex-math>\n \n \n \n Q<\/mml:mi>\n |<\/mml:mo>\n H<\/mml:mi>\n M<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>) is -hard already with 6 job types, and that the related Cutting Stock<\/jats:sc> problem is -hard already with 8 item types. For the more general unrelated machines model (\n \n $$R|HM|C_{\\max }$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n |<\/mml:mo>\n H<\/mml:mi>\n M<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>), we show that if the largest job size \n \n $$p_{\\max }$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> or the number of jobs n<\/jats:italic> is polynomially bounded in the instance size\u00a0|I<\/jats:italic>|, there are algorithms with complexity \n \n $$|I|^{{{\\,\\mathrm{\\textrm{poly}}\\,}}(k)}$$<\/jats:tex-math>\n \n \n \n |<\/mml:mo>\n I<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n \n \n poly<\/mml:mtext>\n \n <\/mml:mrow>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Our main result is that this is unlikely to be improved because \n \n $$Q||C_{\\max }$$<\/jats:tex-math>\n \n \n \n Q<\/mml:mi>\n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is \n \n $$\\mathsf {W[1]}$$<\/jats:tex-math>\n \n \n W<\/mml:mi>\n [<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-hard parameterized by k<\/jats:italic> already when n<\/jats:italic>, \n \n $$p_{\\max }$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and the numbers describing the machine speeds are polynomial in\u00a0|I<\/jats:italic>|; the same holds for \n \n $$R||C_{\\max }$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (without machine speeds) when the job sizes matrix has rank\u00a02. Our positive and negative results also extend to the objectives \n \n $$\\ell _2$$<\/jats:tex-math>\n \n \n \u2113<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-norm minimization of the load vector and, partially, sum of weighted completion times \n \n $$\\sum w_j C_j$$<\/jats:tex-math>\n \n \n \u2211<\/mml:mo>\n \n w<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \n C<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (\n \n $$P||C_{\\max }$$<\/jats:tex-math>\n \n \n \n P<\/mml:mi>\n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>) is fixed-parameter tractable parameterized by k<\/jats:italic>, extending our understanding of this fundamental problem. Together with our hardness results for \n \n $$Q||C_{\\max }$$<\/jats:tex-math>\n \n \n \n Q<\/mml:mi>\n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, this implies that the complexity of \n \n $$P|HM|C_{\\max }$$<\/jats:tex-math>\n \n \n \n P<\/mml:mi>\n |<\/mml:mo>\n H<\/mml:mi>\n M<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n C<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is the only remaining open case.<\/jats:p>","DOI":"10.1007\/s10951-024-00827-8","type":"journal-article","created":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T18:29:59Z","timestamp":1738261799000},"page":"139-156","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Complexity of scheduling few types of jobs on related and unrelated machines"],"prefix":"10.1007","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7846-0053","authenticated-orcid":false,"given":"Martin","family":"Kouteck\u00fd","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7398-718X","authenticated-orcid":false,"given":"Johannes","family":"Zink","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,30]]},"reference":[{"issue":"3","key":"827_CR1","doi-asserted-by":"publisher","first-page":"2152","DOI":"10.1137\/17M1162792","volume":"28","author":"I Aliev","year":"2018","unstructured":"Aliev, I., Loera, J. 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