Types of Collisions

In order to have a valid solution for a MAPF problem, it is necessary that the single-agent plans of the ${\displaystyle k}$ agents do not collide one another. Given plan ${\displaystyle \pi _{i}}$, the expression ${\displaystyle \pi _{i}[x]}$ denotes the position of agent ${\displaystyle i}$ after having performed ${\displaystyle x}$ steps of plan ${\displaystyle \pi _{i}}$. It is possible to distinguish five different types of collisions between two plans ${\displaystyle \pi _{i}}$ and ${\displaystyle \pi _{j}}$.^[1]

• Vertex conflict: there is a vertex conflict between plans ${\displaystyle \pi _{i}}$ and ${\displaystyle \pi _{j}}$ when the two agents occupy the same location at the same time. Formally, a vertex conflict happens when ${\displaystyle \pi _{i}[x]=\pi _{j}[x]}$.
• Edge conflict: an edge conflict occurs whenever two agents cross the same edge in the same direction at the same time, that is ${\displaystyle \pi _{i}[x]=\pi _{j}[x]}$ and ${\displaystyle \pi _{i}[x+1]=\pi _{j}[x+1]}$. If vertex conflicts are not allowed, then edge conflicts cannot exist.
• Following conflict: a following conflict happens when at a certain time step an agent occupies a location that was occupied by another agent in the previous time step. Mathematically, a following conflict is described as ${\displaystyle \pi _{i}[x+1]=\pi _{j}[x]}$.
• Cycle conflict: a cycle conflict happens whenever a set of agents (at least three) move as if they are spinning in a cycle. It means that each agent takes the position that was previously occupied by the other agent one step-ahead in the cycle. If following conflicts are forbidden, then cycle conflicts cannot happen.
• Swapping conflict: a swapping conflict is the case in which two agents exchange their position, passing on the same edge at the same time in two different directions. It is expressed as ${\displaystyle \pi _{i}[x+1]=\pi _{j}[x]}$ and ${\displaystyle \pi _{j}[x+1]=\pi _{i}[x]}$.

When formalizing a MAPF problem, it is possible to decide which conflicts are allowed and which are forbidden. A unified standard about permitted and denied conflicts does not exist, however vertex and edge conflicts are usually not allowed.^[1]

Objective Functions

When computing single-agent plans, the aim is to maximize a user-defined objective function. There is not a standard objective function to adopt, however the most common are:^[1]

• flowtime: this measure is obtained by summing the time steps employed by each agent to reach their target location. Formally, it is equal to ${\displaystyle \sum _{i\in A}|\pi _{i}|}$, where the plans ${\displaystyle \pi _{i},i\in A}$ are single-agent plans without collisions;
• makespan: the number of time steps necessary so that all the agents complete theirs tasks, defined as ${\displaystyle \max _{i\in A}|\pi _{i}|}$, where ${\displaystyle \pi _{i}}$${\displaystyle ,i\in A}$ are part of a valid solution;
• maximization of reached targets given a deadline: the aim is to find a valid solution that maximizes the number of agents that reach their target given a time deadline.^[2]

Prioritized Planning

One possible approach to face the computational complexity is prioritized planning.^[8] It consists in decoupling the MAPF problem into ${\displaystyle k}$ single-agent pathfinding problems.

The first step is to assign to each agent a unique number ${\displaystyle \{1,2,...,k\}}$ that corresponds to the priority given to the agent. Then, following the priority order, for each agent a plan is computed to reach the target location. When planning, agents have to avoid collisions with paths of other agents with a higher priority that have already computed their plans.

Finding a solution for the MAPF problem in such setting corresponds to the shortest path problem in a time-expansion graph.^[9] A time-expansion graph is a graph that takes into account the passing of time. Each node is composed by two entries ${\displaystyle (v,t)}$, where ${\displaystyle v}$ is the node name and ${\displaystyle t}$ is the time step. Each node ${\displaystyle (v,t)}$ is linked to the nodes ${\displaystyle (u,t+1)}$ such that ${\displaystyle u}$ is adjacent to ${\displaystyle v}$ and ${\displaystyle u}$ is not occupied at time step ${\displaystyle t+1}$.

The drawback of prioritized planning is that, even if it is a sound approach (it returns valid solutions), it is neither optimal nor complete.^[10] This means that it is not assured that the algorithm will return a solution and, even in that case, the solution may not be optimal.

Optimal MAPF Solvers

It is possible to distinguish four different categories of optimal MAPF solvers:^[10]

• Extensions of A*: algorithms in this category employ modified versions of the A* approach.
• Increasing Cost Tree Search:^[11] a novel formalization of the MAPF problem is proposed, that comprehends an increasing search tree and the corresponding algorithm. The algorithm is composed by two levels and relies on the assumption that a valid solution for the MAPF problem is composed by a set of solutions for the single agents.
• Conflict-Based Search:^[12] this algorithm computes paths as when solving single-agent pathfinding problems, and then it adds constraints in an incremental way in order to avoid collisions.
• Constraints programming:^[13] with this kind of approach, MAPF problems are transformed into a set of constraints and then solved using specific constraint solvers such as SAT and Mixed Integer Programming (MIP) solvers.

Bounded Suboptimal MAPF Solvers

Bounded suboptimal algorithms offer a trade-off between the optimality and the cost of the solution. They are said to be bounded by a certain factor because they return solutions with a cost at most equal to the optimal solution cost times the factor. MAPF bounded suboptimal solvers can be divided following the same categorization presented for optimal MAPF solvers.^[10]

Anonymous MAPF

It is a version of MAPF in which there is a set of target locations but agents are not assigned a specific target.^[14] It does not matter whether the agent that reaches the target, the important thing is that targets are completed. A slight modification of this version is the one in which agents are divided into groups and each group has to perform a set of targets.^[15]

Multi-Agent Pick-up and Delivery hww

MAPF problem is not able to capture some aspects relative to real world applications. For example, in automated warehouses it happens that robots have to complete several tasks one after the other. For this reason, an extended MAPF version is proposed, called Multi-Agent Pick-up and Delivery (MAPD).^[16] In a MAPD setting, agents have to complete a stream of tasks, where each task is composed by a pick-up a location and a delivery location. When planning for the completion of a task, the path has to start from the current position of the robot and to end in the delivery position of the task, passing through the pick-up point. MAPD is considered a "lifelong" version of MAPF in which tasks arrive incrementally.^[16]

Beyond Classical MAPF

The assumptions that the agents are in a grid environment, that their speed is constant and that time is discrete are simplifying hypotheses. Many works take into account the kinematic constraints of agents,^[17] such as velocity and orientation, or go past the assumption that the weights of the arcs are all equal to 1.^[18] Other works focus on eliminating the discrete time assumptions and that the duration of actions is exactly equal to one time step.^[19] Another assumption that does not reflect reality is that agents occupy exactly one cell of the environment in which they are: some studies have been conducted to overcome this hypothesis.^[20] It is interesting to note that the shape and geometry of agents may introduce new types of conflicts, since agents may crash with one another even if they are not completely overlapped.