The Travelling Salesman Problem on Permuted Monge Matrices

We consider traveling salesman problems (TSPs) with a permuted Monge matrix as cost matrix where the associated patching graph has a specially simple structure: a multistar, a multitree or a planar graph. In the case of multistars, we give a complete, concise and simplified presentation of Gaikov's theory. These results are then used for designing an O(m3 + mn) algorithm in the case of multitrees, where n is the number of cities and m is the number of subtours in an optimal assignment. Moreover we show that for planar patching graphs, the problem of finding an optimal subtour patching remains NP-complete.