Find cycles with specific weights in complete graph

Question

Assume I have an undirected edge-weighted complete graph $G$ of $N$ nodes (every node is connected to every other node, and each edge has an associated weight). Assume that each node has a unique identifier.
Let's say I then have an input, $c$ of three edges (e.g $c=[4,7,6]$). Does an algorithm exist that lets me search $G$ for instances of $c$, and returns the identifiers of the matching nodes?
The cycles it returns must be closed loops, such as $[A, D, B, \text{(then back to A)}]$, rather than $[D, A, B, A]$

Here is a poorly-drawn example: .

Answer 1

You can form for each weight a matrix with $1$s for edges of that weight and $0$s elsewhere, and then multiply the matrices. The location of a nonvanishing diagonal entry will tell you the first vertex of your cycle. Then use partial products to succesively find the remaining vertices - if $i$ is the first vertex $M$ is the product of the first $j$ matrices, and $N$ is the product of the last $k-j$ matrices, then the $j+1$st element of the cycle should be an $l$ such that $M_{il}\neq 0$ and $N_{li} \neq 0$.

This takes time $O( k \cdot n^{\omega+\epsilon})$ where $k$ is the length of the cycle, $n$ is the number of vertices, and $\omega$ is the matrix multiplication constant, as long as we store partial matrix products so we don't have to compute them.

A trivial lower bound is $n^2$ (you might have to check most of the edges to find the cycle) so this is pretty close, at least when $k$ is small.