# Scheduling lower bounds via AND subset sum

## Abstract

Given N instances (X1,t1),…,(XN,tN) 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 Xi has a subset that sums up to the target integer ti. We prove that this problem cannot be solved in time O˜((N⋅tmax)1−ε), for tmax=maxiti and any ε>0, assuming the ∀∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude O˜(n+pmax⋅n1−ε)-time algorithms for several scheduling problems on n jobs with maximum processing time pmax, assuming ∀∃-SETH. These include classical problems such as 1||∑wjUj, the problem of minimizing the total weight of tardy jobs on a single machine, and P2||∑Uj, the problem of minimizing the number of tardy jobs on two identical parallel machines.