## Abstract

Subset Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. An important open problem in this area is to base the hardness of one of these problems on the other. Our main result is a tight reduction from k-SAT to Subset Sum on dense instances, proving that Bellman’s 1962 pseudo-polynomial O(T)-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T^{1}−^{ε} · 2^{o}(n^{)} for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails. As a corollary, we prove a “Direct-OR” theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time (NT)^{1}−o^{(1)}. As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s, t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to Õ(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).

Original language | English |
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Pages | 41-57 |

Number of pages | 17 |

DOIs | |

State | Published - 1 Jan 2019 |

Event | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States Duration: 6 Jan 2019 → 9 Jan 2019 |

### Conference

Conference | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 |
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Country/Territory | United States |

City | San Diego |

Period | 6/01/19 → 9/01/19 |

## ASJC Scopus subject areas

- Software
- General Mathematics