**2. Motivation**

In some planning problems, solutions in which agents share resources, i.e., they collide using the traditional MAPP problem definition, may be acceptable, at the cost of having a reduced level of agent satisfaction. Problems of this type have two properties in common: (1) Agents' satisfaction conditions are reduced when meeting at the same place; and (2) the extent of reduction in satisfaction depends on how long the dissatisfying situation lasts in terms of distance or time.

One motivating example of this type of problems involves mass transit systems, in which passengers have various preferences, even necessities, in terms of *common resources*, such as seat availability (necessary for seniors) and on-vehicle Wi-Fi supply (preferred by video viewers and game players during the trip). Passengers may interfere with one another on common resources in crowded situations. Individually optimal paths can cause serious interference, leading to low-quality experiences. Interference between passengers is soft because it is possible that they do not call for the same resource when they are on the same public vehicle. In addition, even when they call for the same resource and interfere, they are able to tolerate each other over a short time and distance. Intuitively, how likely a collision (intolerable interference) actually happens depends on: (1) whether the resource supply is less than the demands; and (2) how long the lack-of-supply condition lasts in terms of the time and distance that the passengers stay together. Passengers can be viewed as agents, moving through the transportation network. When planners plan for all the agents, sticking to eliminating any hard collision is neither necessary nor feasible. Thus, people are more interested in another problem: How can the resource received by all agents be maximized such that the probability of collision of each agent is less than a bound? This is an important topic of passenger-centered research [17–19].

In addition to public transit scenarios, other examples include: network traffic engineering, where multiple data streams can route through a router. Long streams will have a higher chance of being blocked when unexpected traffic spikes pop up, exceeding the link capacity [20]. How to maximize the throughput with a bounded chance of blocking is of great interest to researchers in the field of communications and computer networks.

Another example is planning for large-scale self-driving cars, where multiple cars can share the same lane, and the number of cars on a road will influence the chance of crashes among autonomous vehicles [21,22]. Scholars and engineers dealing with the fundamentals of autonomous vehicles in unstructured and dynamic environments aim to increase the road traffic while bounding the crash risk.

Military transportation also has the soft-collision property, in which transport aircrafts or vehicles are subject to higher risks to be detected and attacked by enemy troops when many of they move together due to path overlap for a long distance. Formally, as the transportation volume on a road increases, the degree of concealment decreases [23]. The dispatcher must bound the security risk when attempting to maximize the military transportation efficiency.

To support these application classes, we introduce the soft-collision property (related to common resources) to MAPP. SC-M\*, introduced in this paper, is the first attempt to generalize M\* to handle real MAPP problems in a soft-collision context, considering various common resources requested by agents. Specifically, SC-M\* changes M\*'s definition of a collision so it can represent soft collisions on resources and their impact on an agent's dissatisfaction level. We show the advantages of the SC-M\* against other SC-based MAPP solvers.
