Genetic Algorithm for Combinatorial Path Planning: The Subtour Problem

Figure 4

(a) Given the set of targets 𝑇={𝑡₁,…,𝑡₄} and the agent 𝑎, 𝑉={𝑣₁,…,𝑣₅}=𝑇∪𝑎, with 𝑣₅=𝑎, is the augmented vertex set. In (b), a subtour of length 2 (the agent visits the targets associated with the vertices 𝑣₂ and 𝑣₁) is shown, while in (c), the cheapest Hamiltonian path (the agent visits all the given targets) is depicted. (d) shows the 𝑘-TSP with 𝑘=3, and in (e), the optimal solution of the Traveling Salesman Problem is drawn.

(a) Targets and agent

(b) Subtour in 𝒫₂(𝐾₅)

(c) Hamiltonian path in 𝒫₄(𝐾₅)

(d) 𝑘-TSP in 𝒞₄(𝐾₅(𝑉))

(e) Classic TSP