**Theorem 1.** *SC-M\* is a complete algorithm.*

**Proof of Theorem 1.** SC-M\* inherits the sub-dimensional expansion from M\* (i.e., it changes the *Gsch* only when one of the soft-collision sets *Cp* changes). The algorithm applies A\* in the updated search graph. Due to the merging operation applied to collision set *Cp*, as shown in Equation (3), *Cp* for each vertex will change finite times (at most *m* times, which is the number of agents). Because A\* is complete, applying A\* to a given *Gsch* takes finite time to return a result. Therefore, SC-M\* is complete.

#### 4.3.2. Cost-Suboptimality

Different from M\*, which is optimal, SC-M\* is suboptimal because Equations (9) and (10) only include the immediate conflicting agents to the soft-collision set; the agents that softly interfere with the conflicting agents in the upstream path are excluded. Those excluded agents also contribute to the announced collision (i.e., making the collision score above the threshold). Because of this, SC-M\* cannot guarantee the *inclusion property*, which is the basis to ensure the optimality in M\* [6]: *The optimal path for some subset of agents costs no more than the optimal joint path for the entire set of agents*. Without the inclusion property, SC-M\* may not guarantee cost optimality.

Figure 3 provides a counterexample of the inclusion property of SC-M\* in the soft-collision MAPP context defined in this paper. Let *π* <sup>Ω</sup>(*vk*, *vf*) be the joint path constructed by combining the optimal path for a subset Ω ∈ *I* of agents with the individually optimal paths for the agents in *I*\Ω. The inclusion property is defined as follows: If the configuration graph contains an optimal path *π*∗(*vk*, *vf*), then ∀Ω ⊂ *I*, *g*(*π* <sup>Ω</sup>(*vk*, *vf*)) ≤ *g*(*π*∗(*vk*, *vf*)). See *Lemma 6* in [6].

In the soft-collision context, this inclusion property does not always hold. In Figure 3, we have a three-agent MAPP problem (*I* = {*r*1,*r*2,*r*3}) in the soft-collision context. Agents *r*1, *r*2, and *r*3 attempt to move from the vertexes *a*, *f* , and *h* to the vertexes *e*, *g*, and *i*, respectively. The individually optimal paths (shortest distance) are *a* → *b* → *c* → *d* → *e* with distance 4 for *r*1, *f* → *c* → *d* → *g* with distance 3 for *r*2, and *h* → *b* → *c* → *i* with distance 3 for *r*3. The total cost of the joint individually optimal path is 10. However, assuming that the agents can only tolerate a dissatisfying experience with distance 1, *r*1 will announce a collision at vertex *d* because of the interference on the edge *b* → *c* and *c* → *d* from agents *r*3 and *r*2, respectively.

If we choose Ω = {*r*1,*r*2} ∈ *I*, as can be seen in Figure 3, the only solution would be that *r*1 takes a detour through the vertex *x* to avoid the collision on the edge *c* → *d*, resulting in a cost of 5 for *r*1, and the total *g*(*π* <sup>Ω</sup>(*vk*, *vf*)) is 11 (3 for *r*2, 5 for *r*1 and 3 for *r*3). On the other hand, by searching through all three dimensions, a better solution would be that *r*3 detours through the vertex *y*, and *r*1 is free from collision because the interference on the edge *b* → *c* disappears. The total cost of this joint path is 10.5, and we have *g*(*π* <sup>Ω</sup>(*vk*, *vf*)) = 11 > *g*(*π*∗(*vk*, *vf*)) =10.5, which is contradictory to the inclusion property.

**Figure 3.** Counterexample of the inclusion property of soft-collision M\* (SC-M\*) in the soft-collision context. Agents *r*1,*r*2, and *r*3 have the planning O-D demands (*a*,*e*),(*f* , *g*), and (*h*, *i*), respectively. Vertexes in the system are labeled as *a*, *b*, *c*, etc.

The reason for this phenomenon is that, in the hard-collision context, only the immediate conflicting agent *r*2 contributes to the collision of *r*1 at vertex *d*. However, in the soft-collision context, both *r*2 and *r*3 contribute to the collision of *r*1 at vertex *d*, and thus, the inclusion property does not apply. Without this inclusion property, which is the basis of the optimality of M\*, the optimality of SC-M\* cannot be guaranteed.

However, we notice that suboptimal methods have long been used successfully to solve many interesting MAPP problems [15,25,26]. Given the fact that we show in the next section that SC-M\* is superior to other alternative SC-based MAPP solvers (e.g., SC-A\* and SC-CBS) in terms of scalability, run time, and path cost, we demonstrate that the proposed method, which is adjusted to MAPP in the soft-collision context, is a powerful tool in practice.

#### **5. Experiments and Results**

We evaluated SC-M\* in simulation on a grid public mass transit network with an Intel Core i7-6700 CPU at 3.4 GHz with 16 GB RAM. As shown in Figure 4, the grid transit environment has 20 × 20 stops. There are 20 bidirectional horizontal lines. Likewise, 20 bidirectional vertical lines are deployed in the environment. At each stop, agents can switch lines. The yellow areas are covered by some resources, such as the on-vehicle free Wi-Fi in our experiments. Agents traversing those areas can enjoy high-quality on-vehicle Wi-Fi connections. A fully covered edge has a Wi-Fi resource value of 100, and the Wi-Fi value of an edge is proportional to the length of coverage. Each agent wants to move from its source (square) to its destination (circle) with the lowest cost (i.e., a linear combination of distance cost and Wi-Fi cost) as well as bounded collision score. The second resource is the space on the edge, which is fixed at 5. The satisfying values are *ε*<sup>1</sup> = 20 and *ε*<sup>2</sup> = 1 for Wi-Fi and space resources, respectively.

We randomly generated a source–destination pair for each agent. Each trial was given a 1000-s run-time limitation to find a solution. For each configuration (including the number of agents, collision threshold *T*, and offset parameter *δ*), we ran 20 random trials to calculate the average metrics (i.e., the success rate and run time). The success rate is the number of trials ending with a solution divided by the number of trials. The run time is the average over trials ending with a solution or a no-solution declaration. If all trials under a certain configuration exceeded 1000 s, we used ">1000" to represent the run time of the corresponding configuration. We used the standard A\* as the coupled planner and policy generator in the SC-M\* framework and compared our results to the baselines.

**Figure 4.** Grid system with 20 × 20 stops and 40 bidirectional lines. Square and circle of the same color correspond to the source and destination of an agent, respectively.
