Geodesic
An exact algorithm with linear complexity for a problem of visiting megalopolises
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 309-317

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A problem of visiting megalopolises with a fixed number of “entrances” and precedence relations defined in a special way is studied. The problem is a natural generalization of the classical traveling salesman problem. For finding an optimal solution we give a dynamic programming scheme, which is equivalent to a method of finding a shortest path in an appropriate acyclic oriented weighted graph. We justify conditions under which the complexity of the algorithm depends on the number of megalopolises polynomially, in particular, linearly.
Keywords: traveling salesman problem, $np$-hard problem, dynamic programming, precedence relations. 
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A. G. Chentsov; M. Yu. Khachai; M. Yu. Khachai. An exact algorithm with linear complexity for a problem of visiting megalopolises. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 309-317. http://geodesic.mathdoc.fr/item/TIMM_2015_21_3_a30/