**1. Introduction**

Multi-agent path planning (MAPP) involves finding the set of least-cost paths for a set of agents co-existing in a given *graph* such that each of the agents is free from collision, where a collision is defined as at least two agents moving to the same location at the same time. MAPP attracts increasing attention due to its practical applications in multi-robot systems for surveillance automation, video gaming, traffic control, and many other domains [1–4]. This problem is, however, difficult to solve because the configuration space grows exponentially with the number of agents in the system, incurring extremely heavy computational efforts. It is an NP-hard problem to find optimal solutions for MAPP in its general form [5].

Approaches to solving MAPP problems fold into three main categories: *coupled, decoupled* and *intermediate* [6]. *Coupled* approaches search the *joint configuration space* of the multi-agent system, which is the *Tensor product* of the free configuration spaces of all the individual agents. A popular coupled planner is the A\* algorithm [7] that directly searches the whole joint configuration space, making such an approach computationally infeasible when the number of agents is large. Enhanced variants of A\*, such as operator decomposition (OD), enhanced partial expansion A\* (EPEA\*), and iterative deepening A\* (IDA\*), can—to some extent—mitigate the exponential growth in the number of neighbors by improving the admissible heuristics [8–11]. Coupled approaches are optimal and complete, but usually at high computational cost. *Decoupled* approaches plan for each agent separately and then adjust the path to avoid collisions. Algorithms in this category are generally faster because they perform a graph search and collision-avoidance adjustment in low-dimensional spaces. However, optimality and completeness are not guaranteed [3,12].

*Intermediate* approaches lie between coupled and decoupled ones because they dynamically couple agents and grow the search space during the planning. In this way, the search space is initially small and grows when necessary. A few intermediate MAPP algorithms can guarantee optimality and completeness. State-of-the-art examples include Conflict-Based Search (CBS) [6,13]. CBS is a two-level algorithm. At the high level, conflicts are added into a *conflict tree* (CT). At the low level, solutions consistent with the constraints given by the CT are found and updated to agents. CBS behaves poorly when a set of agents is strongly coupled. Meta-agent CBS (MA-CBS) is then proposed by merging strongly coupled agents into a meta-agent to handle the strongly coupled scenarios.

The M\* algorithm is a state-of-the-art coupled approach. It starts with decoupled planning and applies a strategy called *sub-dimensional expansion* to dynamically increase the dimensionality of the search space in regions in which agent collisions occur. In this way, an efficient graph search with a strict collision-free constraint can be achieved, while minimizing the explored portion of the joint configuration space. M\* identifies which subsets of agents can be safely decoupled and hence plans for multi-agents in a lower-dimensional space. Compared to CBS and its variant MA-CBS, M\* and its variants, e.g., recursive M\* (rM\*), have much more fine-grained control over some technical details, such as the management of conflict sets for better scalability. The fine-grained nature of M\* allows it to be integrated into MA-CBS to take advantage of both [14]. Recent work extended both M\* and CBS algorithms to handle the imperfect path execution due to unmodeled environments and delays [15,16].

Most fundamental MAPP approaches assume *hard collisions*, which means that solutions in which agents share resources (nodes or edges) are rejected. In many real world scenarios, some degree of resource sharing between agents is acceptable, so the hard-collision constraint needlessly over-constrains the solution space. This paper relaxes the hard collisions constraint by allowing some sharing of resources, including space and various services on edges/nodes, by agents. Such sharing reduces the quality of the path, i.e., the satisfaction level of the agent using it, but as long as the quality reduction for each path is below a settable threshold, the solution is acceptable. We call this concept *soft collisions*. Hard collisions are still supported by having a very strict threshold, i.e., a penalty for sharing is very high. The reduction in satisfaction level experienced by an agent caused by soft collisions on resources in its path is quantified using a *collision score*. In this paper, we develop a generalized version of the M\* algorithm, called *soft-collision M\** (SC-M\*), for solving the MAPP problem in the soft-collision context. Note that we that we are not simply replacing hard with soft collisions, but instead introducing soft collisions as a generalization that allows modeling different types of collisions.

SC-M\* extends M\* by taking the perspective of soft collision on common resources. Specifically, SC-M\* tracks the collision score of each agent and places agents whose collision scores exceed certain thresholds into a *soft-collision set* for *sub-dimensional expansion*, a technique that limits the search space

while maintaining the optimality of the algorithm with respect to the objective. In this way, SC-M\* achieves improved scalability to handle a larger number of agents while limiting the probability of collisions on resources to a bound.

In this paper, we show that SC-M\* has advanced flexibility and scalability for efficiently solving the MAPP problem in the soft-collision context where common resources are considered, and can handle complex environments (e.g., with multiple types of agents requesting multiple types of resources). We theoretically prove that SC-M\* is *complete* and *suboptimal* under the soft-collision constraints on resources. Experimental results demonstrate the advantages/trade-offs of SC-M\* in terms of path cost, success rate and run time against baseline SC-based MAPP planners, such as SC-A\* and SC-CBS.

The rest of the paper is organized as follows. Section 2 discusses the motivation of soft collisions. Section 3 gives technical briefing of the M\* algorithm. Section 4 presents our proposed SC-M\* approach. Section 5 evaluates SC-M\* in a grid public transit network. Finally, Section 6 concludes our work.
