A greedy and distributable approach to the Lexicographic Bottleneck Assignment Problem with conditions on exactness

Solving the Lexicographic Bottleneck Assignment Problem (LexBAP) typically relies on centralised computation with order O(n⁴) complexity. We consider the Sequential Bottleneck Assignment Problem (SeqBAP), which yields a greedy solution to the LexBAP and discuss the relationship between the SeqBAP, the LexBAP, and the Bottleneck Assignment Problem (BAP). In particular, we reexamine tools used to analyse the structure of the BAP, and apply them to derive an O(n³) algorithm that solves the SeqBAP. We show that the set of solutions of the LexBAP is a subset of the solutions of the SeqBAP and analyse the conditions for which the solutions sets are identical. Furthermore, we provide a method to verify the satisfaction of these conditions. In cases where the conditions are satisfied, the proposed algorithm for solving the SeqBAP solves the LexBAP with computation that has lower complexity and can be distributed over a network of computing agents. The applicability of the approach is demonstrated with a case study where mobile robots are assigned to goal locations.