About the connection of one problem of simple pursuit with the travelling salesman problem

Abstract

The article considers the traveling salesman problem on plane with a finite set of detour points M1, M2,…, Mn. A heuristic method is proposed for finding an approximate solution to this problem, in which at the first step the convex hull of the set of points M1, M2,…, Mn is constructed. This method can be used to solve the simple pursuit problem with one pursuer and several evaders, when the players make a simple move. The goal of the pursuer is to minimize the time of catching the last evader, and the evaders try to maximize this time. Players are supposed to use piecewise-constant strategies, i.e. players can change their direction of movement (the pursuer can also change the order of pursuit), only at certain moments of the correction t0, t1,…,tn,…. It is also assumed that at each moment of correction t, the pursuer chooses the pursuit order based on minimizing path the length of detour path E1(t), E2(t),…, En(t). If we assume, in addition, that the pursuit should occur along a closed loop, then at each moment of the correction t the pursuer solves the traveling salesman problem with detour points E1(t), E2(t),…, En(t), P(t).