## Abstract

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 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_{j}U_{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.

Original language | English |
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Pages (from-to) | 29-40 |

Number of pages | 12 |

Journal | Journal of Computer and System Sciences |

Volume | 127 |

DOIs | |

State | Published - 1 Aug 2022 |

## Keywords

- Lower bounds
- Parallel machine problems
- SETH
- Scheduling
- Single machine problems
- Subset sum

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics