TY - JOUR

T1 - SETH-based Lower Bounds for Subset Sum and Bicriteria Path

AU - Abboud, Amir

AU - Bringmann, Karl

AU - Hermelin, Danny

AU - Shabtay, Dvir

N1 - Publisher Copyright:
© 2022 Copyright held by the owner/author(s). Publication rights licensed to ACM.

PY - 2022/1/22

Y1 - 2022/1/22

N2 - 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).

AB - 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).

KW - Strong Exponential Time Hypothesis

KW - Subset sum

KW - bicriteria shortest path

KW - fine-grained complexity

UR - http://www.scopus.com/inward/record.url?scp=85125689407&partnerID=8YFLogxK

U2 - 10.1145/3450524

DO - 10.1145/3450524

M3 - Article

AN - SCOPUS:85125689407

SN - 1549-6325

VL - 18

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 1

M1 - 6

ER -