License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.8
URN: urn:nbn:de:0030-drops-105846
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10584/
Abboud, Amir
Fine-Grained Reductions and Quantum Speedups for Dynamic Programming
Abstract
This paper points at a connection between certain (classical) fine-grained reductions and the question: Do quantum algorithms offer an advantage for problems whose (classical) best solution is via dynamic programming?
A remarkable recent result of Ambainis et al. [SODA 2019] indicates that the answer is positive for some fundamental problems such as Set-Cover and Travelling Salesman. They design a quantum O^*(1.728^n) time algorithm whereas the dynamic programming O^*(2^n) time algorithms are conjectured to be classically optimal. In this paper, fine-grained reductions are extracted from their algorithms giving the first lower bounds for problems in P that are based on the intriguing Set-Cover Conjecture (SeCoCo) of Cygan et al. [CCC 2010].
In particular, the SeCoCo implies:
- a super-linear Omega(n^{1.08}) lower bound for 3-SUM on n integers,
- an Omega(n^{k/(c_k)-epsilon}) lower bound for k-SUM on n integers and k-Clique on n-node graphs, for any integer k >= 3, where c_k <= log_2{k}+1.4427.
While far from being tight, these lower bounds are significantly stronger than what is known to follow from the Strong Exponential Time Hypothesis (SETH); the well-known n^{Omega(k)} ETH-based lower bounds for k-Clique and k-SUM are vacuous when k is constant.
Going in the opposite direction, this paper observes that some "sequential" problems with previously known fine-grained reductions to a "parallelizable" core also enjoy quantum speedups over their classical dynamic programming solutions. Examples include RNA Folding and Least-Weight Subsequence.
BibTeX - Entry
@InProceedings{abboud:LIPIcs:2019:10584,
author = {Amir Abboud},
title = {{Fine-Grained Reductions and Quantum Speedups for Dynamic Programming}},
booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages = {8:1--8:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-109-2},
ISSN = {1868-8969},
year = {2019},
volume = {132},
editor = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10584},
URN = {urn:nbn:de:0030-drops-105846},
doi = {10.4230/LIPIcs.ICALP.2019.8},
annote = {Keywords: Fine-Grained Complexity, Set-Cover, 3-SUM, k-Clique, k-SUM, Dynamic Programming, Quantum Algorithms}
}
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Keywords: |
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Fine-Grained Complexity, Set-Cover, 3-SUM, k-Clique, k-SUM, Dynamic Programming, Quantum Algorithms |
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Collection: |
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46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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04.07.2019 |
