What exactly is Lawler's modification to Yen's algorithm and how does it work?
Yen's original paper 1, from 1971, only establishes an upper bound of $O(Kn^4)$ operations (see Table 1).
Lawler's original paper 2, from 1972, improves the time complexity upper bound to $O(Kn^3)$.
Wikipedia's analysis of Yen's algorithm, leading to the $O(Kn(m+n\log n))=O(Kn^3)$ upper bound, is based on the Fibonacci heap implementation of Dijkstra's algorithm, from 1984. [3] But even the $O(n^2)$ upper bound for Dijkstra's algorithm would be fine; the issue, as far as I can tell, is that Wikipedia is not considering the same problem.


Namely, Yen's paper, as well as Lawler's, consider graphs where negative weights are allowed. The Wikipedia seems to restrict itself to non-negative weights, so... one can use Dijkstra there.

See for instance this quote from Yen's paper (Section 5):



and this one for Lawler's paper (Section 5):





1 Finding the K Shortest Loopless Paths in a Network, Jin Y. Yen (1971).
Management Science 197117:11 , 712-716 

2 A Procedure for Computing the K Best Solutions to Discrete Optimization Problems and Its Application to the Shortest Path Problem, 
Eugene L. Lawler (1972). Management Science 197218:7 , 401-405 

[3]  Fibonacci heaps and their uses in improved network optimization algorithms. Fredman, Michael Lawrence; Tarjan, Robert E. (1984). 25th Annual Symposium on Foundations of Computer Science. IEEE. pp. 338–346. doi:10.1109/SFCS.1984.715934.