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• W4207048092 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)." @default.
• W4207048092 created "2022-01-26" @default.
• W4207048092 creator A5032224171 @default.
• W4207048092 creator A5044394587 @default.
• W4207048092 creator A5051832024 @default.
• W4207048092 creator A5052950579 @default.
• W4207048092 date "2022-01-23" @default.
• W4207048092 modified "2023-09-24" @default.
• W4207048092 title "SETH-based Lower Bounds for Subset Sum and Bicriteria Path" @default.
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