Scheduling Lower Bounds via AND Subset Sum

03/16/2020

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by   Amir Abboud, et al.

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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 Õ((N · t_max)^1-ϵ) time, 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, based on ∀∃-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 P_2||∑ U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.