SETH-based lower bounds for subset sum and bicriteria path

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 T1-e . 2°(n) for any e > 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-°(1). As an application of this corollary, we prove a tight SETH-based lower bound for the classical BICRITERIAs, 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 O(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).