Optimal Feedback Strategy in The Game Variant of Generalized Travelling Salesman Problem

Abstract

The game problem of optimization of a tour in the generalized travelling salesman problem (GTSP) is considered, when it is required to visit all sets of a given system of sets M1,…,Mm in Rn and minimize the cost (additive or bottleneck) of a tour. The points Xi of visit to the sets Mi, i ∊ 1, m, are supposed to be an uncertain factors (opposing nature) or controls of real player-opponent trying to maximize the cost. The future behaviour of uncertainties is unknown. So, we have the game variant (both static and feedback) of GTSP. We state this combinatorial optimization problem in the class of feedback strategies. To choose a strategy forming the route does not mean an a priori choice of some permutation r of the integers 1,…,m (as, e.g., in TSP or GTSP), but is a choice of mechanism producing for every position arising during a motion its own number of a set to be visited next. For every current position a strategy may generate, depending on what uncertainties are realizing, a various continuations of a route. As is shown, the result of the game in the class of feedback strategies may be, in general, strictly better than one in the class of open-loop strategies r. The optimal (minimax) feedback strategy is designed, which provides the minimal guaranteed value of the cost, no matter how the uncertainties behave. A number of heuristics is given, and some applications are discussed.