# Sub-exponential for Subset Sum on the total bit length of the input

Was crawling the internet, and found this in a paper (1)

"[...], a variant of dynamic programming called dynamic dynamic programming has been shown to have a worst-case sub-exponential time complexity of $2^{O(\sqrt x)}$ when the total bit length $x$ of the input set is used as the complexity parameter."

As I understand this claim violates the Exponential Time Hypothesis, and there has been some answers in the forum clearly stating no such sub-exponential algorithm exists, would be interesting to know what the experts in the area think about the proposed algorithm in terms of worst case complexity.

Direct link to the paper (2) explaining the dynamic dynamic approach.

(1) Thomas E. O’Neil -An Empirical Study of Algorithms for the Subset Sum Problem

(2) Thomas E. O’Neil, Scott Kerlin - A Simple $2^{O(\sqrt x)}$ Algorithm for PARTITION and SUBSET SUM

• Sounds interesting. Can you clarify what your question is? Are you asking for a justification for the running time bound listed? Something else? Not sure what you mean by "the reason". Also, can you edit the question to give a full citation to the paper (title, authors, where published), so that the question still makes sense even if the link stops working, and so that others with a similar question about the paper can find this page by searching on the paper title? Thank you! – D.W. Oct 27 '17 at 4:04
• Let me do the changes – Jesus Salas Oct 27 '17 at 4:08
• If there exists an $o(n^2)$ reduction from SAT to Subset-Sum [and papers are correct], ETH fails. But does such reduction exist? – rus9384 Oct 27 '17 at 8:13
• Note: standard reduction from SAT is $\Theta(n^2)$. – rus9384 Oct 27 '17 at 8:26

This doesn't violate the exponential time hypothesis. The exponential time hypothesis says that k-SAT takes exponential time, i.e., $\Omega(2^{cn})$ for some constant $c>0$. So if you found a $O(2^{\sqrt{n}})$ time algorithm for k-SAT, that would violate the exponential time hypothesis -- but a a $O(2^{\sqrt{n}})$ time algorithm for subset sum does not violate the hypothesis.