Given N instances (X_1,t_1),\ldots,(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  O((N \cdot t_{max})^{1-\epsilon}) time , for t_{max}=\max_i t_i and any \epsilon > 0, assuming the \forall \exists Strong Exponential Time Hypothesis (\forall \exists-SETH). We then use this result to exclude O(n+P_{max} \cdot n^{1-\epsilon})-time algorithms for several scheduling problems on n jobs with maximum processing time P_{max}, based on \forall \exists-SETH. These include classical problems such as 1||\sum w_jU_j, the problem of minimizing the total weight of tardy jobs on a single machine, and P_2||\sum U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.
Given N instances (X_1,t_1),\ldots,(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 \cdot t_{max})^{1-\epsilon}) , for t_{max}=\max_i t_i and any \epsilon > 0, assuming the \forall \exists Strong Exponential Time Hypothesis (\forall \exists-SETH). We then use this result to exclude O(n+P_{max} \cdot n^{1-\epsilon})-time algorithms for several scheduling problems on n jobs with maximum processing time P_{max}, based on \forall \exists-SETH. These include classical problems such as 1||\sum w_jU_j, the problem of minimizing the total weight of tardy jobs on a single machine, and P_2||\sum U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.
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