$ P $-nearness for random TSP instances of size $50$ and $100$. The left column contains the edges with shortest distance, the center column has the optimal tour for the instances, and the right column contains the edges with the highest $ P $-nearness values for each city. For each city, we plotted the three edges with the highest nearness values.[]{data-label="fig:distance_vs_procrustes"}](Distance_vs_Procrustes){width="95.00000%"}

Thus, this is another example that demonstrates that dynamical systems theory can be used to analyze NP-hard problems and construct improved heuristics.

We now show how networks of Duffing oscillators can be used to construct a new algorithm for the iconic MAX-CUT problem.

Novel algorithm construction: network of Duffing oscillators for the MAX-CUT problem {#sec:maxsat_quantum}
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[Overview]{} MAX-CUT [@Cit:karp] is a well-known NP-hard problem that arises in graph theory. Simply stated, the goal is to compute a subset $S$ of the vertex set in a graph $\mathcal{G}$, such that the number of edges between $S$ and the rest of the graph are maximized. The best known approximation ratio of $0.878$ can be achieved in polynomial time using semi-definite programming [@Cit:Goemans]. The problem naturally arises in VLSI design and statistical physics, and has been extensively studied.

In [@Cit:quantum_net], the authors 