Scheduling lower bounds via AND subset sum

Given N instances (X₁,t₁),…,(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 O˜((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 O˜(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.