Simplified problem formulations

The TSP is a canonical path planning problem, which is often used to describe infor- mative path planning formulations. This subsection introduces the TSP and variants relevant to active perception. These variants are relevant to several formulations considered in this thesis, particularly Section 4.2 and Section 3.5.

Travelling salesman problem

The TSP is the problem of finding the shortest path that visits all given cities and returns to the origin. More specifically, a TSP instance can be described as a graph with vertices corresponding to cities and edges corresponding to travel distances be- tween pairs of cities. The goal is to find a Hamiltonian cycle, i.e. a closed path that visits each vertex exactly once, that has minimum weight.

The TSP is studied in a wide range of fields, particularly operations research (Toth and Vigo, 2001). It is an NP-hard problem and significant attention has been devoted to finding efficient heuristic algorithms for the problem.

In its purest form, the TSP can be used to describe simple robotic coverage prob- lems (Galceran and Carreras, 2013). However, there are several variants and gener- alisations of the TSP that more closely relate to complex robotics tasks; we describe several of these below. In some cases, the TSP is used to help setup and motivate a robotics problem before developing a custom solution algorithm, while in other cases TSP solution algorithms are used to solve sub-problems of a more general robotics problem.

m-TSP

The m-TSP generalises the TSP to multiple agents, which requires assigning nodes to agents and finding a path for each agent. There are several variations of the m- TSP with different objective functions such as minimising the maximum-cost path,

or minimising the sum of path costs (Bektas, 2006; Lagoudakis et al., 2005). Many different approach have been proposed, such as exact algorithms, heuristics based on the standard TSP, neural networks and genetic algorithms. The focus in the literature is solely on centralised and offline algorithms.

Generalised TSP

In robotic coverage problems it is often desired to observe a set of points using range sensors. This does not require the robot to visit the points, but rather just requires the robot to be within observation range of the points. This problem naturally maps to the generalised travelling salesman problem (GTSP) (Noon and Bean, 1989) and related variants, where it is required to visit one city from every set of cities, for a collection of city sets.

The GTSP is the case where these city sets are discrete and finite. A well known solution to the GTSP transforms the problem into a standard TSP and then any TSP solver can be applied (Noon and Bean, 1989). Recently, specialised solvers have been proposed that are typically more efficient (Smith and Imeson, 2017). The GTSP can be thought of as a generalisation of the set cover problem (Vazirani, 2001; Hochbaum, 1997) where path constraints are imposed on the set-selection costs; however, counterexamples show that GTSP is fundamentally harder than set cover and greedy solutions can perform arbitrarily poorly (Best and Fitch, 2016).

Several variants describe the city-sets as continuous spatial regions. In some ways this problem is harder than the discrete-set case due to having to deal with the infinite search space; however, efficient approximation algorithms have been developed that exploit the spatial-structure of the problem. This problem is often referred to as the TSP with neighbourhoods for the case of circular regions (Dumitrescu and Mitchell, 2003), and has occasionally been extended for the case with polygonal regions (Faigl et al., 2013), which is closely related to the watchman route problem (Faigl, 2010). The GTSP has been formulated for robotics applications. For example, Mathew et al. (2013) formulate a mobile refuelling problem as a GTSP where the sets describe possible refuel points in time and space. In Chapter 4 we consider an active perception

formulation with continuous regions, but additionally has budget constraints, which more closely maps to the OP, described below.

Orienteering problem

The standard TSP requires finding a path that visits all vertices in shortest time. In many applications, particularly in informative path planning, a full coverage is not desirable, or even possible. Instead, the problem is to visit the maximum number of vertices in a given time. This time or distance constraint represents some form of budget that cannot exceeded, such as due to fuel constraints, or specifications given by an operator. In contrast to the TSP, these problems not only require determining the order to visit vertices, but also the selection of which vertices to visit. Also, typically some vertices may be more important than others and therefore a common objective is to maximise a weighted sum of the visited vertices.

The prize-collecting TSP (Balas, 1989) requires finding a selection of vertices to visit, where rewards are gained by visiting vertices and deducted for omitting vertices. This presents a trade-off between the reward-value of selecting a vertex versus the travel cost required to visit the vertex. This formulation is useful in some scenarios, however it does not enforce budget constraints, and it may be difficult to define the rewards correctly to reflect a desired trade-off (Faigl and Hollinger, 2018).

The orienteering problem (OP) (sometimes known as the selective TSP) is a distance- constrained variant of the TSP that appears in a wide range of contexts (Laporte and Martello, 1990; Vansteenwegen et al., 2011; Gunawan et al., 2016). This problem is also known to be NP-hard as there exists a transformation from the Hamiltonian circuit problem (Laporte and Martello, 1990). Similar to the standard TSP, there are many relevant variants to the OP; the most relevant variants to informative path planning and this thesis include the team-OP that extends the problem for multi- agent systems, generalised-OP (Geem et al., 2005) that defines the objectives as a function of discrete sets, and the orienteering problem with neighbourhoods (OPN) where rewards are collected by visiting continuous regions.

Chapter 4 considers a new OP variant that includes continuous polygonal goal regions (similar to the GTSP variants), and multiple agents (similar to the team-OP). While

there are existing techniques for the team-OP (Dang et al., 2013a; Archetti et al., 2007; Dang et al., 2013b), none of these address continuous polygonal goal regions. Also, while OP formulations have been applied to many problems (Gunawan et al., 2016; Vansteenwegen et al., 2011), the focus has mostly been on offline planning rather than online settings where goals are discovered over time, which is more applicable to robotics.