Given N instances (X_1,t_1),…,(X_N,t_N) of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers X_i has a subset that sums up to the target integer t_i. We prove that this problem cannot be solved in time Õ((N ⋅ t_max)^{1-ε}), for t_max = max_i t_i and any ε > 0, assuming the ∀ ∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude Õ(n+P_max⋅n^{1-ε})-time algorithms for several scheduling problems on n jobs with maximum processing time P_max, assuming ∀∃-SETH. These include classical problems such as 1||∑ w_jU_j, the problem of minimizing the total weight of tardy jobs on a single machine, and P₂||∑ U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.
@InProceedings{abboud_et_al:LIPIcs.ICALP.2020.4, author = {Abboud, Amir and Bringmann, Karl and Hermelin, Danny and Shabtay, Dvir}, title = {{Scheduling Lower Bounds via AND Subset Sum}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {4:1--4:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.4}, URN = {urn:nbn:de:0030-drops-124119}, doi = {10.4230/LIPIcs.ICALP.2020.4}, annote = {Keywords: SETH, fine grained complexity, Subset Sum, scheduling} }
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