Limited memory influence diagrams are graph-based models that describe decision problems with limited information, as in the case of teams and agents with imperfect recall. Solving a (limited memory) influence diagram is an NP-hard problem, often approached through local search. In this paper we investigate algorithms for k-neighborhood local search. We show that finding a k-neighbor that improves on the current solution is W-hard and hence unlikely to be polynomial-time tractable. We then develop fast schema to perform approximate k-local search; experiments show that our methods improve on current local search algorithms both with respect to time and to accuracy.