Knowing All Optimal Solutions Does Not Help for TSP Reoptimization

Reoptimization of the Maximum Weighted P k -Free Subgraph Problem under Vertex Insertion

“…We can easily prove that max weighted P k -free subgraph has a PTAS for any fixed k. Moreover, we can adapt the proofs given in the paper to produce results depending on parameter M as done in [7]. Finally, in [11], the authors prove that the knowledge of all optimal solutions for free doesn't help TSP reoptimization. It is interesting to handle this question for max weighted P k -free subgraph.…”

On One Approach to TSP Structural Stability

“…Step 2(b) may be replaced by the application of some heuristic, but in this case the obtained solution would not necessarily be optimal. The conditions (15) and (22) are not polynomial and thus the analog of Algorithm 1 for the Theorem 2 does not make any practical sense. Namely, every addition of a new element to the initial data not only requires the solutions of NPhard problems for finding D ( , ) ( ) and D ( , ) ( ), but may also change the "old" values of D ( , ) ( ), so they should be recomputed.…”

“…The application of Theorem 2 allows reducing it to ≈ 2 ( 2 2 + | |) = 50 4 2 50 + 50 2 100 10 ≈ 3 ⋅ 10 23 . This example allows recommending the condition (15) for postoptimal analysis of the TSP solutions in multidimensional feature spaces.…”

“…Even if the optimal solution of the initial instance is known for free, the reoptimization remains NP-hard [14]. Moreover, the situation is not improved if all optimal solutions are known [15].…”

“…with an optimal tour as a result. Necessary and sufficient conditions(15) and(22)allow constructing ASA efficiently for a large finite . Firstly, we introduce formally the edge ASA…”