Scheduling Lower Bounds via AND Subset Sum

Abstract

Given $N$ instances $(X_1,t_1),\dots ,(X_N,t_N)$ 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 $\tilde{O}((N\cdot t_{max}{)}^{1-\epsilon })$, 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 $\tilde{O}(n+P_{max}\cdot n^{1-\epsilon })$-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 $1||\sum w_j{U}_j$, 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.

14 pages, ICALP'20