# Dubins Traveling Salesman Problem with Neighborhoods: A Graph-Based Approach

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## Abstract

**:**

## 1. Introduction

#### 1.1. Relevant Literature

#### 1.2. Contributions

#### 1.3. Organization

## 2. Problem Statement

**Problem 2.1**(DTSPN).

**Problem 2.2**(Sampled DTSPN).

## 3. DTSPN Intersecting Regions Algorithm

**Figure 1.**Example DTSPN with the corresponding “GTSP with intersecting node sets”. (

**a**) Example instance of DTSPN with three circular regions ${\mathcal{R}}_{1},{\mathcal{R}}_{2},\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{\mathcal{R}}_{3}$ and samples ${S}_{1},{S}_{2},\dots ,{S}_{8}$. The circuit through samples ${S}_{2},\phantom{\rule{4.pt}{0ex}}\text{and},{S}_{8}$ is the optimal tour; (

**b**) Problem (P0): A GTSP with intersecting node sets representation of the DTSPN example. Note: only an essential subset of arcs is shown for clarity of illustration.

#### 3.1. Stage 1

**Figure 2.**Example of Problem (P1) and Problem (P2) from Stage 1 of transformation. (

**a**) Problem (P1): Any arcs that do not enter at least one new node set $\{(3,5)$ and $(6,8)\}$ have been removed from the graph in Problem (P0); (

**b**) Problem (P2): A large finite cost α is added to each edge. Here ${\widehat{c}}_{i,j}={c}_{i,j}^{2}$, where ${c}_{i,j}^{2}$ is defined in Equation (4).

**Figure 3.**Example of Problem (P3) and Problem (P4) from Stage 1 and Stage 2 of transformation. (

**a**) Problem (P3): Nodes ${S}_{2}$ and ${S}_{3}$ from (P2) lie in multiple node sets. These nodes are duplicated and the spawned nodes ${S}_{{2}^{\prime}}$ and ${S}_{{3}^{\prime}}$ are placed in node set ${\mathcal{V}}_{2}$. Zero cost arcs (dashed arrows) are added connecting ${S}_{2}$ to ${S}_{{2}^{\prime}}$ and ${S}_{3}$ to ${S}_{{3}^{\prime}}$; (

**b**) Problem (P4): The intra-set arc $(5,{3}^{{}^{\prime}})$ from Problem (P3) is removed.

**Theorem 3.1**(Noon and Bean [17]). Given a GTSP in the form of $(P0)$, we can transform the problem to a problem of the form of $(P3)$. Given an optimal solution to $(P3)$ with cost less than $(m+1)\alpha $, we can construct an optimal solution to $(P0)$. If an optimal solution to $(P3)$ has a cost greater than or equal to $(m+1)\alpha $, the problem $(P0)$ is infeasible.

#### 3.2. Stage 2

- $\mathcal{M}(k)\subseteq \mathcal{M}(i)\cup \mathcal{M}(j)$,
- $\mathcal{M}(l)\subseteq \mathcal{M}(i)\cup \mathcal{M}(j)$,
- if $\mathcal{M}(l)=\mathcal{M}(i)$ then $\mathcal{M}(k)$ must also equal $\mathcal{M}(i)$,
- if $\mathcal{M}(k)=\mathcal{M}(j)$ then $\mathcal{M}(l)$ must also equal $\mathcal{M}(j)$.

**Theorem 3.2**(Noon and Bean [17]). Given an optimal solution, ${y}^{*}$, to $(P4)$, we can construct the optimal solution, ${x}^{*}$, to $(P3)$.

#### 3.3. Stage 3

**Theorem 3.3**(Noon and Bean [17]). Given a canonical GTSP in the form of $(P4)$ with n node sets, we can transform the problem into a standard TSP in the form of $(P6)$. Given an optimal solution ${y}^{*}$ to $(P6)$ with ${c}^{6}{y}^{*}<(n+1)\beta $, we can construct an optimal solution ${x}^{*}$ to $(P4)$.

**Figure 4.**Example of Problem (P5) and Problem (P6) from Stage 3 of transformation. (

**a**) Problem (P5): The clustered TSP is created by forming zero cost intra-set cycles and adjusting the originating node in each inter-set arc; (

**b**) Problem (P6): A large finite cost β is added to each inter-set edge. Here ${\overline{c}}_{i,j}={c}_{i,j}^{6}$, where ${c}_{i,j}^{6}$ is defined in Equation (5). The optimal tour is shown in red with a cost of ${\widehat{c}}_{8,2}+\beta +{\widehat{c}}_{2,8}$.

#### 3.4. Performance Comparison

**Theorem 3.4**(IRA Performance). Given $\rho >0$, the set of $n\ge 2$ possibly intersecting regions, R, and the set of m sample configurations, S, let ${T}_{IRA}$ and ${T}_{RCM}$ denote the tours produced by IRA and the RCM [15], respectively. Then the length of ${T}_{IRA}$ is no greater than that of ${T}_{RCM}$,

**Corollary 3.5**(IRA is Resolution Complete). Given $\rho >0$, the set of $n\ge 2$ possibly intersecting regions, R, and the set of m sample configurations, S drawn from a Halton quasi-random sequence [21] as in RCM, then IRA is Resolution Complete.

**Figure 5.**A comparison of IRA and RCM on an example DTSPN instance with three regions and three sample poses. (

**a**) Example Tour: IRA, Tour Length $=7.7$; (

**b**) Example Tour: RCM, Tour Length $=15.4$.

#### 3.5. Complexity of Intersecting Regions Algorithm

## 4. Numerical Results

**Figure 6.**Simulation results for 100 Monte Carlo trials where both IRA and RCM optimized over the same 50 sample poses. (

**a**) The color represents the average of the ratio of the tour length under IRA to the tour length under the RCM planning algorithm. Here the red regions indicate near parity in performance while the blue regions indicate that IRA produced tours that are approximately half the length of tours produced by the RCM algorithm; (

**b**) The color represents the average of the ratio of the size of the ATSP solved under IRA to the size of the ATSP solved under the RCM planning algorithm. Here the blue regions indicate near parity in size while the red regions indicate that IRA increased the size of the ATSP by as much as four times.

**Figure 7.**Simulation results for the where IRA optimized over 50 sample poses and RCM optimized over the same 50 samples plus an additional sample for each duplicated node in the IRA. These extra samples ensured that both algorithms solved the same size ATSP.

## 5. Demonstration

**Figure 8.**The configuration of sensors and UAV trajectory during the field demonstration at Camp Roberts, CA. (

**a**) Field Demonstration Description. The acoustic sensors visited by the data collecting UAV are shown as yellow dots; (

**b**) The blue lines represent the GPS logs of the path taken by data collecting UAV during the test. The desired path was sent to the autopilot via the square waypoints. The sensors and communication regions are represented by green and blue circles respectively.

#### 5.1. Modifications for the Demonstration

## 6. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Isaacs, J.T.; Hespanha, J.P. Dubins Traveling Salesman Problem with Neighborhoods: A Graph-Based Approach. *Algorithms* **2013**, *6*, 84-99.
https://doi.org/10.3390/a6010084

**AMA Style**

Isaacs JT, Hespanha JP. Dubins Traveling Salesman Problem with Neighborhoods: A Graph-Based Approach. *Algorithms*. 2013; 6(1):84-99.
https://doi.org/10.3390/a6010084

**Chicago/Turabian Style**

Isaacs, Jason T., and João P. Hespanha. 2013. "Dubins Traveling Salesman Problem with Neighborhoods: A Graph-Based Approach" *Algorithms* 6, no. 1: 84-99.
https://doi.org/10.3390/a6010084