Optimizing Murty´s ranked assignment method

We describe an implementation of an algorithm due to Murty for determining a ranked set of solutions to assignment problems. The intended use of the algorithm is in the context of multitarget tracking, where it has been shown that real-time multitarget tracking is feasible for some problems, but many other uses of the algorithm are also possible. The following three optimizations are discussed: (1) inheriting dual variables and partial solutions during partitioning, (2) sorting subproblems by lower cost bounds before solving, and (3) partitioning in an optimized order. When used to find the 100 best solutions to random 100×100 assignment problems, these optimizations produce a speedup of over a factor of 20, finding all 100 solutions in about 0.6 s. For a random cost matrix, the average time complexity for finding k solutions to random N×N problems appears to be nearly linear in both k and N, for sufficiently large k.