SETH-based Lower Bounds for Subset Sum and Bicriteria Path

• Published in: ACM transactions on algorithms Vol. 18; no. 1; pp. 1 - 22
• Main Authors: Abboud, Amir, Bringmann, Karl, Hermelin, Danny, Shabtay, Dvir
• Format: Journal Article
• Language: English
• Published: New York, NY ACM 22.01.2022
• Subjects: Graph algorithms analysis; Mathematical optimization; Parameterized complexity and exact algorithms; Problems, reductions and completeness; Theory of computation; bicriteria shortest path; fine-grained complexity; Strong Exponential Time Hypothesis; Subset sum
• ISSN: 1549-6325; 1549-6333
• DOI: 10.1145/3450524

Subset Sumand 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 T1-ε · 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 (N T)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).