ST-MR-IA & MT-MR-IA

Below we discuss two instances of MRTA problems: the ST-MR-IA problem which is the one studied in this dissertation, and then as a comparison the MT-MR-IA problem which is related to the previous one and more general as robots might be shared among different tasks. Note that both problems involve allocating groups or bundles of robots to tasks (i.e. Multi-Robot tasks); these require bundling mechanisms which we discuss in Section 2.1.2.

ST-MR-IAconsiders Single-Task robots supporting Multi-Robot tasks. This problem is referred to as the disjoint coalition formation in the multi-agent community and it has been formally studied in [109] and [115]. Following [40], ST-MR-IA can be modeled as a Set Partitioning Problem (SPP) [9] in which is given a set of robots R and a family F of acceptable subsets of R and a utility function u : F → R. The objective is to find

2.1 Multi-Robot Task Allocation & Taxonomy 26

a maximum-utility family X of elements in F such that X is a partition of R. Note that F represents the set of all feasible robot bundle-task pairs and the utility u represents the utility estimate for each pair. It is clear that for each different type of task we could evaluate utility in different ways, therefore using different estimation functions for example based on the distance of the robot or the energy required to accomplish the task. Finally note that casting the ST-MR-IA problem as an instance of SPP does not imply that all robots should be allocated to at least one task or that all the tasks should be assigned at least a bundle of robots, in fact the family F includes all the subsets of R where for some the utility might be zero.

If we restrict our attention to linear utilities, i.e. the utility of robots contributing to the same task adds up linearly with the others, then we can model ST-MR-IA as a Bipartite Weighted Semi-Matching problem [83]. Given a weighted bipartite graph G = (R ∪ T, E) where R is the set of robots, T the set of tasks and E is a set of edges. An edge (Ri, Tj, eij) indicates that robots Ri is capable of serving task Tj with

weight eij (which in our case represents utility). Then we seek a max-weight semi-

matching, i.e. a subset of edges of maximum combined weight under the restriction that no two chosen edges share an endpoint in R. This is similar to the formal models and algorithms presented for sensor selection problems in [106]. Clearly, the difference with SPP is that, in such a problem, the utility of a group of robots is calculated as the sum of the individual utilities of each robot assigned to that task, i.e. P

∀Ri→Tjeij.

The model we present in Chapter 3 adopts a similar approach but adds also budget constraints to each task and, in addition, it considers a utility demand and a threshold on that to be surpassed in order for the task to be satisfied.

SPP and Bipartite Weighted Semi-Matching mainly consider a static setting, where the set of tasks is known at once. For many real-life MRTA problems and also sensor net- work deployments, however, tasks arrive at different points in time, and each time a task arrives, we must assign robots or sensors to it (dynamic settings). In some environ- ments this happens knowing what is going to happen next, such as when we know the

plan of a particular mission (Time-Extended assignment); in others instead we have to take allocation desicions without knowledge of future tasks or energy constraints (In- stantaneous Assignment). So we could have a dynamic setting where we can achieve a time-extended allocation or, alternatively, one where we can only perform instantan- eous allocation due to the lack of information about the future, which is our case in Chapter 4 and 5. Note that instead in Chapter 3 we focus on the static setting, where we also assume instantaneous assignment.

Let us consider the dynamic settings, in the case in which these decisions are irrevoc- able, meaning that once a sensor is assigned, it cannot be reassigned to any future task. Authors studied different versions of this problem and in [58] such problems are re- ferred to as online assignment problems. These problems could be modeled in general by the online versions of bipartite matching problems (like the one discussed previ- ously) where nodes in T arrive over time. The performance of an online algorithm is evaluated by analyzing its competitive ratio, defined as the ratio between its worst per- formance and the performance of the best offline algorithm. In [58], the authors study weighted versions of the online exact matching problem (i.e. where each task is alloc- ated exactly one robot). In particular, they study Online Min-Matching (minimizing edge weight) and Online Max-Matching (maximizing edge weight), in metric spaces, where edge weights are computed based on the metric distance between nodes from the two sets, e.g. tasks and robots/sensors. We need to match n robots to m tasks such that the total distance between them is minimized. Note that the difference between these problems and our dynamic MSTA problem in Chapter 4 is that in general we consider non-linear utilities which could be proportional or not to the distance of the sensor from the task. In addition, we allow for reallocation of sensors to newly arrived more important tasks.

In the case in which a robot can be shared among different tasks, the problem becomes the MT-MR-IA instance. Of course each robot will have a limited maximum number of tasks it is able to serve; in the case of sensors this might be due to sensory limitations.

2.1 Multi-Robot Task Allocation & Taxonomy 28

For example, a camera could be able to detect up to a certain number of suspicious objects in a certain area, given that the camera might miss out-of-focus objects (as in [54]). MT-MR-IA has also been studied by the multi-agent community and it is referred to as the overlapping coalition formation problem [115]. Such a problem can be cast as a variant of the Set Covering Problem (SCP) [122] as follows: Given a set of robots R forming the network, a family F of acceptable subsets of R representing possible overlapping coalitions, and a utility function u : F → R as an estimate of the utility of assigning a subset of robots to a task, the objective is to find the maximum- utility family X ∈ F such that X is a cover of R. Note that also in this case casting the MT-MR-IA problem as an instance of SCP does not enforce that all robots should be allocated to at least one task and vice versa; in fact by definition the family F includes also subsets of R for which the utility is zero (representing unassigned robots or unsupported tasks). If full coverage of the tasks is required there could be several ways to include it in the problem formulation, but in our case we would allow for tasks being dropped due to the features of our environment in which sensing resources are scarce and in high demand. Note that we focus on Single-Task sensors which is the most restrictive scenario where sensors cannot be shared and need to either satisfy one task or the other. As discussed in Chapter 6 we propose to explore Multi-Task sensor problems as part of possible future work.