Fine-Grained Reductions and Quantum Speedups for Dynamic Programming
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.
Fine-Grained Complexity
Set-Cover
3-SUM
k-Clique
k-SUM
Dynamic Programming
Quantum Algorithms
Theory of computation~Problems, reductions and completeness
8:1-8:13
Track A: Algorithms, Complexity and Games
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.
We thank Karl Bringmann and the anonymous reviewers for helpful feedback.
Amir
Abboud
Amir Abboud
IBM Almaden Research Center, San Jose, California, USA
10.4230/LIPIcs.ICALP.2019.8
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Amir Abboud
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