# 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∗(1.728n) time algorithm whereas the dynamic programming O∗(2n) 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 Ω(n1.08) lower bound for 3-SUM on n integers, k an Ω(n ck −ε) lower bound for k-SUM on n integers and k-Clique on n-node graphs, for any integer k ≥ 3, where ck ≤ log2 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 nΩ(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.