Affiliation:
1. IBM Almaden Research Center, USA
2. Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
3. Department of Industrial Engineering and Management, Ben-Gurion University, Beer Sheva, Israel
Abstract
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
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 (
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).
Funder
NSF
BSF
IBM Almaden Research Center and in Stanford University
European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme
People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme
REA
ISRAEL Science Foundation
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
7 articles.
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