I was wondering if, given a min cost flow problem and an integer n, there is an efficient algorithm/package or mathematical method, to obtain the set of the n-best basic solutions of the min cost flow problem (instead of just the best).
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Not so easy. There were some special LP solvers that could do that (see: Ralph E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, 1986), but currently available LP solvers can't.
There is a way to encode a basis using binary variables:
b[i] = 1 if variable x[i] = basic
0 nonbasic
Using this, we can use "no good cuts" or "solution pool" technology to get the k best bases. See: https://yetanothermathprogrammingconsultant.blogspot.com/2016/01/finding-all-optimal-lp-solutions.html. Note that not all solution-pools can do the k-best. (Cplex can't, Gurobi can.) The "no-good" cuts work with any mip solver.
Update: a more recent reference is Craig A. Piercy, Ralph E. Steuer, Reducing wall-clock time for the computation of all efficient extreme points in multiple objective linear programming, European Journal of Operational Research, 2019, https://doi.org/10.1016/j.ejor.2019.02.042
Erwin Kalvelagen
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