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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. One of the most intriguing open
problems 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 $T^{1-\varepsilon}\cdot 2^{o(n)}$ for any $\varepsilon>0$, unless the
Strong Exponential Time Hypothesis (SETH) fails. This is one of the strongest
known connections between any two of the core problems of fine-grained
complexity.
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 $\tilde{O}(L+m)$, in
contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).