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Fine-Grained Reductions and Quantum Speedups for Dynamic Programming

Author Amir Abboud

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Fine-Grained Complexity
  • Set-Cover
  • 3-SUM
  • k-Clique
  • k-SUM
  • Dynamic Programming
  • Quantum Algorithms

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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.

Cite As Get BibTex

Amir Abboud. Fine-Grained Reductions and Quantum Speedups for Dynamic Programming. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.8

Author Details

Amir Abboud
  • IBM Almaden Research Center, San Jose, California, USA

Funding

We acknowledge the support of the Quantum Computing Sciences program of the U.S. Air Force, Office of Scientific Research, administered through Air Force Research Laboratory contract FA8750-18-C-0098.

Acknowledgements

We thank Karl Bringmann and the anonymous reviewers for helpful feedback.

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