  This article surveys the burgeoning area at the intersection of dynamical
systems theory and algorithms for NP-hard problems. Traditionally,
computational complexity and the analysis of non-deterministic polynomial-time
(NP)-hard problems have fallen under the purview of computer science and
discrete optimization. However, over the past few years, dynamical systems
theory has increasingly been used to construct new algorithms and shed light on
the hardness of problem instances. We survey a range of examples that
illustrate the use of dynamical systems theory in the context of computational
complexity analysis and novel algorithm construction. In particular, we
summarize a) a novel approach for clustering graphs using the wave equation
partial differential equation, b) invariant manifold computations for the
traveling salesman problem, c) novel approaches for building quantum networks
of Duffing oscillators to solve the MAX-CUT problem, d) applications of the
Koopman operator for analyzing optimization algorithms, and e) the use of
dynamical systems theory to analyze computational complexity.
