SEtH-based lower bounds for subset sum and bicriteria path

Amir Abboud, Karl Bringmann, Dan Hermelin, Dvir Shabtay

Research output: Contribution to journalConference articlepeer-review

29 Scopus citations

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 T1ε · 2o(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 languageEnglish GB
Pages (from-to)41-57
Number of pages17
JournalProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
DOIs
StatePublished - 1 Jan 2019
Event30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States
Duration: 6 Jan 20199 Jan 2019

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

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