Algorithm to find if exists a k-length acyclic path with negative weight on an weighted undirected graph
If $k$ is part of the input to the algorithm (as opposed to a constant chosen ahead of time), and if "acyclic path" means a path that does not visit any vertex more than once, then this problem is already at least as hard as Hamiltonian Path, which is NP-hard: to reduce the latter to this problem, just give every edge some negative weight (e.g., -1) and set $k=n−1$.