1 Introduction
Intersections are the most common merging points in urban traffic scenarios Xu et al. (2021). According to the Federal Highway Administration, more than 2.8 million intersectionrelated crashes occur in the US each year, accounting for 44% of all crashes Azimi et al. (2014). The rapid development of the V2X technology provides an opportunity to improve traffic mobility and safety in intersection management ContrerasCastillo et al. (2017). Through vehicletovehicle (V2V) and vehicletoinfrastructure (V2I) communication, a centralized controller is deployed at the intersection to coordinate the connected and automated vehicles (CAVs) to pass through the intersection; this guarantees conflictfree vehicle cooperation. Hence, CAVs have immense potential in improving traffic safety and mobility at intersections Zheng et al. (2020); Chen et al. (2021a).
In a fully autonomous intersection scenario, a hierarchical framework is frequently used to realize the CAV coordination Xu et al. (2017). First, after collecting the information of the CAVs in realtime, the centralized controller schedules the arrival time of the CAVs to improve the traffic efficiency. Then, a distributed controller is applied to the CAV to implement the determined arrival plan. Multiple methods have been proposed to address the vehicle control problem, e.g., fuzzy logic Milanés et al. (2010); Onieva et al. (2012), model predictive control Zheng et al. (2017); He et al. (2015), and optimal control Malikopoulos et al. (2018); Zhang et al. (2017). Moreover, the virtual platoon method was proposed in Xu et al. (2018), which projects the CAVs from different lanes into a virtual lane. Thus, the typical vehicle platoon analysis methods Zheng et al. (2015a) can be used in the control problem of the CAVs on different lanes as if they are traveling in the same lane.
The routes of incoming CAVs have complicated conflict relationships at intersections. Therefore, obtaining a highefficiency conflictfree arrival plan is a priority in CAV scheduling. Researchers have also pointed out that the passing order of the vehicles is the key factor influencing traffic mobility at intersections Li and Wang (2006); Meng et al. (2017). As incoming CAVs approach the intersection from different directions, the optimal passing order changes constantly and the solution space increases exponentially, preventing the determination of the optimal passing order using the bruteforce method. This problem is typically solved using a reservationbased method Dresner and Stone (2004). The most straightforward reservationbased solution is the firstinfirstout (FIFO) strategy, in which vehicles that enter the intersection first are scheduled to leave it first Xu et al. (2017); Malikopoulos et al. (2018); Dresner and Stone (2008). However, the FIFO passing order is not likely to be the optimal solution in most cases. Batchbased strategy is an improved version of the FIFO strategy, and it processes the vehicles in batches according to their direction to reduce the traffic delay Tachet et al. (2016); however, its performance has not been fully optimized. Another widely used strategy is the optimizationbased method. After formulating the scheduling problem as an optimization problem, various methods have been proposed to find the optimal passing order, e.g., spanning tree Li and Wang (2006), Monte Carlo tree search Xu et al. (2019), and dynamic programming Yan et al. (2009). However, the solution space exponentially increases with the number of the vehicles, making the use of optimizationbased methods unfeasible.
Apart from these specific methods, some researchers have also noticed that finding the optimal passing order of multiple CAVs is a discrete problem rather than a continuous problem, and thus graph theory is considered a promising methodology to describe and solve this problem. Studies have modeled the problem using graph theory by employing the conflict graph analysis to optimize the trafficsignal phase plan Pandit et al. (2013); Dey and Pal (2013). However, the scheduling of multiple CAVs is more difficult than scheduling a limited number of traffic signal phases because the number of incoming CAVs is much larger than the number of traffic signal phases. Some methods have been proposed to address this difficulty. Apart from the depthfirst spanning tree (DFST) algorithm proposed in Xu et al. (2018), Petri net Lin et al. (2019) and conflict duration graph Deng et al. (2020) are also used in modeling the scheduling problem. However, the discussion on the optimality of the scheduling problem of multiple CAVs is inadequate.
To tackle the problem of optimality, mathematical modeling, i.e., the computational reduction of the intersection scheduling problem, is an essential research aspect. Some researchers have focused on reducing this problem into typical algorithmic problems, including the jobshop scheduling problem Ahn and Del Vecchio (2017), abstractionbased verification problem Ahn and Colombo (2019), and polling system problem Miculescu and Karaman (2019). However, most of these methods mainly focus on the feasibility of a conflictfree passing order solution. In practice, traffic efficiency is another critical topic that has not been fully considered in the computational reduction of this problem.
In this study, we mainly focused on finding the optimal CAV passing order at unsignalized intersections, i.e., the CAV scheduling problem. Most of the existing studies have focused on obtaining the passing order through specific methods without considering the optimality and computational complexity Xu et al. (2018); Li and Wang (2006); Meng et al. (2017); Dresner and Stone (2008); Xu et al. (2019). Some have focused on the feasibility of a conflictfree solution but neglected traffic efficiency optimization Tachet et al. (2016); Ahn and Del Vecchio (2017); Ahn and Colombo (2019); Miculescu and Karaman (2019); Lin et al. (2019); Deng et al. (2020). To address this problem, we introduce a graphbased conflictfree cooperation method to model the conflict relationship of the vehicles. Based on this method, we propose two novel methods to obtain the local or global optimal vehicle passing order. The contributions of our study are as follows:

An improved DFST method is proposed based on Xu et al. (2018) to improve traffic efficiency. Instead of considering all conflict types as one united conflict set, we categorized different conflict types to further reduce the overall depth of the spanning tree, i.e., the evacuation time of vehicles. The computational complexity of the improved algorithm remains low, and the passing order solution generated by it is proved to be the local optimum.

Based on the concise and rigid graphical analysis, we first reduce the CAV scheduling problem to the minimum clique cover (MCC) problem. Unlike most of the existing research Malikopoulos et al. (2018); Xu et al. (2018); Dresner and Stone (2008)
, our reduction of the problem is proposed without the assumption of the FIFO principle. Hence, the global optimal passing order solution is obtained by solving the MCC problem. Moreover, we propose a heuristic method to solve the problem of numerous vehicles with low computation burden.

A distributed feedback control method is designed to apply the scheduling results of the proposed algorithms. In particular, the predecessor–leader following topology is used as the communication topology to lower the communication bandwidth requirement. Traffic simulations are conducted for various numbers of vehicle inputs and traffic volumes. The simulation results reveal that the proposed algorithms significantly improve both traffic efficiency and fuel economy.
The remainder of this paper is organized as follows. Section 2 introduces the scenario addressed in this study. Section 3 presents the conflict analysis, improved DFST (iDFST) method, MCC method, and distributed control. The simulation results are presented in Section 4, and Section 5 concludes the paper.
2 Problem Statement
Fig. 1(a) depicts one of the most congested intersections near the Tsinghua campus. In this scenario, because the number of approaching lanes is higher than the number of departure lanes, traffic congestion frequently occurs during rush hour. In this study, we simplified the traffic scenario by separating the vehicles into different lanes based on their destination. A hierarchical framework was established to realize the cooperation of the CAVs Xu et al. (2017). A central cloud coordinator was deployed at the intersection to guide the CAVs in driving through the intersection. After collecting the positions and velocities of all the CAVs, the upper coordinator schedules the passing order of the approaching CAVs, while the lower distributed vehicle controller controls the vehicle according to the determined passing order. The scheduling method is presented in Sections 3.1, 3.2, 3.3, and the distributed control is presented in Section 3.4. First, the following assumptions were devised to facilitate the design of the strategy.
Assumption 1
Assumption 2
An ideal communication condition without communication delay or packet loss is under consideration. CAVs transmit their velocity and position to the central cloud coordinator through wireless communication, e.g., V2I communication Gerla et al. (2014).
Assumption 3
The CAVs are capable of fully autonomous driving and are assumed to have perfect steering performance. Therefore, we only focused on the longitudinal control of the vehicle.
The incoming CAVs are indexed from to according to their arrival sequence at the control zone of length in Fig. 2, i.e., only the CAVs in the control zone are under coordination. For each CAV , the CAV dynamic model is described in the secondorder form as follows:
(1) 
where is the state space of CAV , and , , and denote the remaining distance to the stopping line, velocity, and control input of CAV at time . We consider homogeneous vehicle platoons, and thus a continuoustime linear timeinvariant system, , is used.
There also exist velocity and acceleration constraints of the vehicle.
(2)  
where is the maximum velocity limit; and are the deceleration and acceleration limits, respectively.
As mentioned earlier, we aim to propose a cooperation method to improve both traffic safety and efficiency, i.e., obtain the collisionfree optimal CAV passing order. Assign as the time step when vehicle enters the control zone, and as the time step when it arrives at the intersection. Several evaluation indexes have been proposed to measure the scheduling performance. In this study, we chose the evacuation time as the priority optimization target, as shown in Definition 1.
Definition 1 (Evacuation Time)
The evacuation time of CAVs is defined as the time when the last CAV reaches the stopping line; it is expressed as
(3) 
Considering incoming CAVs, represents the arrival time of the last CAV at the stopping line. For CAVs, smaller evacuation time means these CAVs pass through the intersection in shorter time. Thus, it shows the overall traffic efficiency performance, i.e.,the overall benefits of the CAVs.
In addition, the vehicle travels through the control zone in time, whereas it travels through it under the free driving condition in time. Accordingly, the average travel time delay (ATTD) is chosen to measure the average traffic efficiency of the vehicles, as described in Definition 2.
Definition 2 (Average Travel Time Delay)
The ATTD is designed to evaluate the average traffic efficiency of CAVs. It is expressed as
(4) 
where is the length of the control zone and represents the average travel delay of the CAVs. Since the travel time of every CAV is considered in , it denotes the individual benefits of CAVs, which is the secondary optimization target of traffic efficiency.
In this study, we mainly focused on finding the optimal CAV passing order to minimize the evacuation time, as defined in Definition 1. In addition, the passing order should also satisfy the conflictfree relationship between the CAVs. Section 3 presents the specific methods used to guarantee safety and improve traffic efficiency. Moreover, the ATTD defined in Definition 2 is introduced to evaluate the algorithm performance. The simulation results and algorithm performance are presented in Section 4.
3 Methodology
In Xu et al. (2018), a method using a virtual platoon was proposed, which projects the CAVs from different lanes onto a virtual lane. In this way, the CAVs from different lanes can drive through the intersection as if they were in the same lane. The most evident advantage is that the idling of the vehicles at the stopping line is avoided to improve the traffic efficiency. A DFST method is then introduced to find the CAV passing order.
Section 3.1 first introduces the graphbased intersection conflict analysis method. Then, Section 3.2 presents the optimization of the DFST method to obtain the local optimal passing order solution, termed as the iDFST method. In Section 3.3, we introduce the MCC method to find the global optimal passing order solution. Finally, the distributed controller design is presented in Section 3.4 to undertake the scheduling plan.
3.1 Conflict Analysis
In Fig. 1
, the CAVs from different lanes form multiple conflict points. Despite the complicated conflict scenarios, they can be classified into the following conflict modes. Without loss of generality, several CAVs are selected in Fig.
2 to illustrate the conflict relationship. First, we introduce the route conflicts Roess et al. (2004), where CAVs have trajectory intersections on their paths.
Crossing Conflict: Vehicles from different lanes have the potential to collide while crossing the conflict points, indicated by the 24 red circles. For example, CAV and CAV have a crossing conflict point.

Diverging Conflict: Lane changing and overtaking are not permitted, as explained in Assumption 1. Thus, vehicles on the same lane cannot pass the intersection simultaneously, as indicated by the 14 orange squares. For example, CAV and CAV have a diverging conflict point.

Converging Conflict: Vehicles from different lanes cannot drive into the same lane simultaneously, as shown by the 6 green arrows. For example, CAV and CAV have a converging conflict point.
Apart from the abovementioned three kinds of conflicts that describe the route conflict of the CAVs, we claim that there exists another conflict type caused by the speed and acceleration limitation of the CAVs. For instance in Fig. 2, CAV arrives at the control zone when CAV is about to reach the stopping line with the designed virtual platoon velocity. In this case, CAV cannot catch up with CAV at the stopping line even if they have no conflict relationship, because of the limitations on the vehicle speed and acceleration. Hence, we further introduce the fourth type of CAV conflict, namely, the reachability conflict.

Reachability Conflict: Even if the CAVs have no route conflict, they cannot pass the intersection simultaneously because of the acceleration and velocity constraints. CAV and CAV have a reachability conflict.
Expanding equation (1) and (2), we obtain the judging condition as
(5) 
where is the distance of the preceding CAV from the stopping line, and is the designed virtual platoon velocity. In other words, the CAVs that are very close to the intersection should not be considered in the scheduling of new incoming CAVs. Note that most of the existing researches assume that the CAVs can reach the stopping line under all circumstances, i.e., the reachability conflict is ignored.
We define different conflict sets to describe the conflict relationship of the CAVs. For each CAV , the crossing set is defined as , diverging set as , converging set as , and reachability set as . Note that because the conflict sets are determined when the CAV reaches the control zone, the CAV indexes in the conflict sets are smaller than those of the ones on the control zone border, i.e., the elements in the conflict sets satisfy (6).
(6) 
Example 1 (Algorithms for CAVs)
Consider the example shown in Fig. 2. CAVs and are approaching from the east direction and have two separate destinations. Three CAVs are traveling straight from the west with on one lane and , on another. CAV is approaching from the south on the middle lane and CAV is on the right lane. Note that CAV is too far from CAVs and to catch up with them at the stopping line.
Note that in the virtual platoon, there exists a virtual leading vehicle traveling with a constant velocity in front of the CAVs that are closest to the intersection. Thus, the leading CAVs in the real world maintain a typical platooning behavior. Accordingly, CAV is added to the diverging set of the CAV closest to the intersection on each lane; therefore, in Example 1, . The remaining conflict sets are shown in Table 1.
1  2  3  4  5  6  7  
3.2 Improved DFST Method
Based on the conflict set analysis in Section 3.1, we further define the conflict directed graph (CDG) to represent the conflict relationship between the CAVs.
Definition 3 (Conflict Directed Graph)
The CDG is denoted as . If there are CAVs in the control zone, we have the node set (node denotes the virtual leading vehicle). The unidirectional edge set is defined as , and bidirectional edge set is defined as . The edge set is the union of these two sets as .
The CDG of the case scenario in Fig. 2 is drawn in Fig. 3(a). The nodes in the CDG represent the CAVs in the control zone. The black unidirectional edges represent the diverging conflicts and reachability conflicts. The existence of a unidirectional edge implies that CAV is not allowed to overtake CAV because of Assumption 1 or CAV is not able to catch up to CAV because it satisfies (5). Thus, CAV cannot reach the intersection earlier than CAV . The red bidirectional edges denote the crossing conflict and converging conflict, which means that the arrival sequence of the CAVs and ’s can be exchanged. Note that in the DFST method, the crossing conflict , diverging conflict , and converging conflict are treated as one union unidirectionaledge conflict. In the following paragraph, we will interpret the separation of these conflict sets to improve the performance of the original DFST algorithm.
It is straightforward that the CDG describes all the conflict relationships of the CAVs. Previous researches have proved that the CDG in the DFST method possesses a DFST, as shown in Lemma 1. Because this conclusion is drawn with only unidirectional edges in the CDG, it can be applied to our iDFST method.
Lemma 1 (Xu et al. (2018))
A CDG has a DFST with root node , i.e., the virtual leading vehicle.
Because the CDG describes all the conflict relationships of the CAVs, and the edges specifically describe the conflict type, a feasible passing order solution can be obtained by building a spanning tree from the CDG. Similar to the general graph theory, the depth of each node in the spanning tree is calculated based on its distance to the root node . Specifically, the depth of the nodes represents the passing order of the CAVs, i.e., the CAVs at the same depth shall pass the intersection simultaneously. Xu et al. (2018) has proved that if the spanning tree is built by the DFST method, the CAVs of the same depth have a conflictfree attribute, i.e., a feasible spanning tree, which has been proved in Lemma 2.
Lemma 2 (Xu et al. (2018))
Consider a virtual platoon characterized by the spanning tree , which is built from the CDG . The trajectories of two CAVs at the same depth in have no conflict relationship with each other.
Hence, it is evident that for a group of CAVs in the control zone, a passing order solution is related to the DFST. It can be inferred that the depth of the spanning tree is related to the traffic mobility at the intersection. Evacuation time represents the overall time cost for all the CAVs to pass the intersection, i.e., the arrival time of the last CAV. It also represents the largest depth of the spanning tree nodes, denoted as . Thus, minimizing the evacuation time is equivalent to finding the smallest possible in all the feasible spanning trees. Formally, we describe this problem as
(7) 
We propose the iDFST method in Algorithm 1 and 2 to generate a DFST. For each node , we first find all its parent nodes in in Lines 5 and 6 of Algorithm 1. Note that both the original DFST method and iDFST method are based on the FIFO principle. Even though there are bidirectional edges in the CDG, the passing order is decided according to the arrival sequence of the CAVs. Thus, the largerindex CAVs should not be considered in the passingorder optimization process of the smallerindex CAVs. This also implies that only the smallerindex CAVs are selected as the parent node candidates, corresponding to the conflict definition in (6).
Then, in Line 7 of Algorithm 1, i.e., Algorithm 2, we treat the conflict types differently. If the parent node is in the diverging or reachability conflict set of CAV , i.e., , cannot surpass because of the overtaking restriction or acceleration limitation. In this case, we find the largest depth of the unidirectional edge parents, because the target depth of CAV should not surpass that of any of these CAVs. In other cases, the parent node is in the crossing conflict set or converging set of CAV . It means CAVs and cannot arrive at the intersection simultaneously, but their arrival order can be exchanged. Either CAV or CAV can pass the intersection first; therefore, we find the union depth set of the crossing conflict parents . To find the optimal parent , we should select the proper depth of CAV . Because the depth of the parent is , the depth of CAV is . As mentioned before, should be neither smaller than nor have an intersection with the set . Considering traffic efficiency, the depth should be as small as possible, as shown in Line 5 in Algorithm 2. The application of the iDFST method in Example 1 is shown in Table 2 and the output iDFST is plotted in Fig. 4(b).
Candidate Node  1  2  3  4  5  6  7 
Selected Parent Node  0  0  1  3  4  5  2 
The main improvement of the iDFST method is in Algorithm 2. In Xu et al. (2018), the conflict parent nodes are treated as one union conflict set . Therefore, the largest depth of all the parent nodes are found in the same way. For instance, in the DFST method, the parent nodes of CAV are the nodes , which causes the largest depth of the parent nodes to be ; therefore, the DFST method has , as shown in Fig. 4(a). On the contrary, in the iDFST method, the conflict nodes are separated into two sets . By distinguishing whether the parent nodes can be surpassed or not, the smaller , i.e., the locally optimal solution is obtained. As shown in Fig. 4(b), CAV is ranked in the iDFST method. Therefore, the total depth of the CAVs is in the DFST method whereas the iDFST method optimizes it to .
We conclude that the DFST method focuses on the feasibility of the passing order problem. The proposed iDFST method additionally considers the optimality, i.e., finds the smallest depth for each CAV, but the solution is still found CAV by CAV, i.e., based on the FIFO principle. In other words, the iDFST solution is locally optimal in arranging each CAV. Because each CAV needs to check all the conflict relationships of the parent nodes, the computational complexity is .
3.3 Minimum Clique Covering Method
As mentioned before, we aim to minimize the overall depth of the spanning tree, which corresponds to the evacuation time of all the CAVs. If we reconsider the traffic scenario in Fig. 1, a maximum number of six CAVs can pass the intersection simultaneously. In other words, because there is a limited number of CAVs in the control zone, the overall depth of the spanning tree can be minimized by maximizing the CAV groups that can drive through the intersection at the same time. From this point of view, another method to describe the conflict relationships of the CAV is to describe their coexistence relationships.
Definition 4 (Coexisting Undirected Graph)
The CUG is defined as the complement graph of the CDG , excluding node . Thus, , where , .
In Example 1, the CDG is drawn in Fig. 3(a) and the CUG in Fig. 3(b). Because the CDG edge implies that two CAVs have conflicts and the CUG is the complement graph of the CDG, the CUG edge implies that the two CAVs are conflictfree, i.e., they can pass through the intersection simultaneously.
Recall that we aim to minimize the overall depth of the spanning tree. Because the total number of CAVs, i.e., the node number , is a constant value , minimizing the overall depth of the spanning tree is equivalent to widening its average width. Thus, solving the optimal passing solution is equivalent to finding the minimum number of groups of the combinations of coexisting CAVs in the CUG. Note that this conclusion corresponds to common sense. The maximum groups of the coexisting CAVs represent the maximizing of the time utility of the intersection, which minimizes the overall evacuation time.
In graph theory, the clique is suitable for describing the coexistence relationship of the CAVs. The definition of the clique is shown in Definition 5, and the cliques of sizes are plotted in Fig. 5.
Definition 5 (CliqueLuce and Perry (1949))
A clique in an undirected graph is a subset of the nodes, such that every two distinct nodes are adjacent. This is equivalent to the condition that the subgraph of induced by is a complete graph.
The clique is the aforementioned group. Because the edges are selected from the CUG , the CAVs in one clique can pass through the intersection simultaneously in a conflictfree manner. In Fig. 3(b), is a clique, but it is not the maximum clique because cliques of larger sizes exist, e.g., . The maximum clique of a graph is a clique with the maximum number of nodes. Particularly, the number of nodes in the maximum clique in is called the clique number, denoted as . In Fig. 3(b), , which corresponds to the traffic scenario in Example 1, where a maximum number of CAVs can pass through the intersection simultaneously. Note that in the overall intersection scenario Fig. 1, as long as there is an adequate number of CAVs on the coexisting lanes.
Recall that our target is to find the minimum number of groups in the CUG, i.e., the minimum number of cliques covering all the nodes in the CUG . Therefore, we define the MCC problem as follows.
Definition 6 (Minimum Clique Cover (MCC) Karp (1972))
A clique cover of a graph is a partition of into disjoint subsets , such that for , the subgraph induced by is a clique, i.e., a complete graph. The MCC number of is the minimum number of subsets in a clique cover of , denoted as .
The MCC number of the CUG represents the minimum number of cliques covering the CUG. Because the cliques in the CUG represent the CAVs that can pass through the intersection simultaneously, these CAVs in the same clique can be scheduled to drive through the intersection at the same time, i.e., be scheduled at the same depth of the spanning tree. Note that the MCC does not lead to the maximum clique and vice versa. For example, considering the CUG in Fig. 3(b), the MCC number and the corresponding cliques are listed in Table 3. The maximum cliques and appear in solution and solution but not in the subsequent solutions. We conclude that for an arbitrary intersection scenario, the coexistence relationship of the incoming CAVs is perfectly depicted in the CUG . The clique number of the CUG represents the maximum number of CAVs that can simultaneously pass through the intersection, whereas the MCC number represents the possible minimum passing order solution.
[width=8em]SolutionsSubsets  
1  
2  
3  
4  
5  
6 
Note that in the CUG Fig. 3(b), which means the spanning tree generated by the MCC solutions are in . It also means that the theoretical evacuation times of these solutions are the same, i.e., the theoretical values of are the same. Thus, the evaluation index of in (3) corresponds to , which is the global optimal passing order considering the evacuation time. In addition, we further consider the ATTD among these solutions as a secondary index. The definition of in (4) can be rewritten in graphical terms as
(8)  
where is the spanning tree depth of the CAV nodes in , as defined in Section 3.2.
In this circumstance, it is evident that the subsets should be arranged in the descending order to decrease the average of CAVs in (8). Because of the same reason, we prefer to choose the solutions with the maximum cliques when they have the same . For example, in Table 3, we tend to choose the Solution , and the corresponding optimized spanning tree is scheduled as , as shown in Fig. 4(c).
The CUG only contains the coexistence information of the CAVs, whereas the passing order of the CAVs on the same lane should be strictly according to their relative positions because of Assumption 1. The MCC method can generate solutions that cannot be directly executed, e.g., Solution in Table 3. The spanning tree is formed as . However, this solution is not feasible because CAV is in front of CAV . We prove that this can be solved by exchanging the unfeasible sequence, as shown in Proposition 1.
Proposition 1
If the MCC method leads to an unfeasible solution, where the CAVs and have conflicting trajectories, exchanging the positions of and solves the conflict. The new solution is also an MCC solution.
Because the unfeasible solution is related to CAVs on the same lane, it is selfevident that exchanging the conflicted CAVs solves the conflict. Meanwhile, the CAVs on the same lane have identical edges in the CUG to those of the other CAVs. Therefore, permutation and combination of CAVs on the same lane result in similar MCC solutions, where only one is the feasible solution.
After introducing the MCC problem, we describe the formal complexity of the original SMALLESTDEPTH problem in Theorem 1.
Theorem 1
SMALLESTDEPTH problem is NPcomplete.
First, we prove that . Assume we have a CDG , a spanning tree , and an integral . The certificate we choose is . The verification algorithm consists of two procedures, i.e., whether the spanning tree is feasible and of the spanning tree is . Checking the edge connections of and the depth of each node of can be evidently accomplished in polynomial time.
Secondly, we prove that the SMALLESTDEPTH problem defined in (7) is NPhard by showing that it can be reduced to the MINIMUMCLIQUECOVER problem Karp (1972) defined in Definition 6, i.e., SMALLESTDEPTH MINIMUMCLIQUECOVER. As mentioned before, given a CDG , we define the corresponding CUG as , where , . The transformation of the CDG into CUG is easily achieved in polynomial time.
If the CDG has a feasible spanning tree with , we claim that the corresponding CUG has partitioned into disjoint cliques. Because is a feasible spanning tree, if are the nodes of same depth, they should not have a conflicting relationship, i.e., . Because the CUG is the complement graph of the CDG, it is evident that . Thus, the nodes of the same depth in the spanning tree are connected with each other, i.e., they form cliques in . The overall depth of the spanning tree corresponds to the clique cover number of .
On the contrary, we assume the the CUG has partitioned into disjoint cliques. From Proposition 1, we know that disjoint cliques lead to a feasible passing order, that is, the CDG has a feasible spanning tree with .
We have proven that the SMALLESTDEPTH problem can be reduced to the MCC problem, which implies that the MCC number of the CUG is the smallest possible of the spanning tree solutions, i.e., the global optimal solution. However, solving the NPhard problem is a difficult task in real deployment. In the simulation, for a small number of vehicles, we applied the bruteforce method to find the MCC number, i.e., the strictly global optimal solution. In the previous work, we have shown that when the traffic scenario is simple, i.e., a maximum number of two vehicles are allowed to pass the intersection simultaneously, the CAV scheduling problem is reduced to a maximum matching problem Chen et al. (2021b). In this case, the global optimal solution is obtained in time.
For a large number of vehicles and complicated intersection scenarios, the dimensional explosion makes the bruteforce method impossible to deploy. Therefore, we apply a practical approach to solve the problem heuristically, as shown in Line 3 to 7 of Algorithm 3. The MCC problem of is proved to be reduced to the graph coloring problem of Garey (1979), and there are numerous heuristic methods to solve the graph coloring problem. Recall that we intend to find the solutions with larger cliques. Thus, we first generate a node sequence by breadthfirst search (BFS). Then, we greedily assign node the smallest possible color, i.e., the clique index according to the node sequence , forming the subset cliques . Line 8 is the spanning process, which arranges the cliques into a spanning tree and has a constant calculation time.
The BFS sequence generation needs . Afterwards, the MCC is solved greedily in and the ranking of the spanning tree requires a constant time. Hence, the overall computational complexity of the heuristically solved MCC method, i.e., Algorithm 3 is , which is lower than of the DFST and iDFST methods. In the simulation, we will show that even if the heuristically solved MCC method is not the strictly globally optimal solution, it outperforms the other two methods.
3.4 Distributed Control
In the ideal communication condition, every CAV obtains driving information, e.g., position and velocity, from the other CAVs in realtime. However, the communication resource, e.g., the bandwidth, is always limited around an intersection. Therefore, we design a hierarchical framework to lower the communication burden. In the upper level, the central coordinator collects the information of the CAVs and assigns the scheduled arrival plans to them. In the lower level, a CAV executes its arrival time through distributed control. Similar to the virtual platoon analysis in Xu et al. (2018), we design the distributed control as follows.
3.4.1 Geometric Topology
As mentioned in Section 3, all the three methods, i.e., DFST, iDFST, and MCC, lead to a feasible spanning tree , as shown in Fig. 4, which forms the geometric topology of the virtual platoon. Because each node in represents a CAV and the edge represents the virtual preceding CAV , the CAV can always follow its parent node CAV . In other words, CAV and its parent CAV intend to maintain a constant desired carfollowing distance and the same velocity .
(9) 
where and denote the velocity and position of the CAVs and .
Lemma 2 has proven that the CAVs at the same depth have no conflict relationship. Therefore, if CAVs and are of the same depth,
(10) 
3.4.2 Communication Topology
The communication topology represents the information exchange route among the CAVs. Various communication topologies have been studied in the field of vehicle platoon control Zheng et al. (2015b). For simplicity, we use the predecessor–leader following (PLF) topology, where each CAV communicates with its preceding vehicle in the virtual platoon and the virtual leading vehicle. Similar to the topology analysis in Xu et al. (2018); Zheng et al. (2015a), we introduce as the communication graph of the spanning tree . The nodes still represent the CAVs; thus, . Meanwhile, the edges represent the information exchange among the CAVs. The communication topologies of the iDFST and MCC spanning trees are shown in Fig. 6. Specifically, the CAVs collect information from the following two types of CAVs.

Virtual leading vehicle: because the CAV obtains its target layer in the scheduled spanning tree, every CAV in the spanning tree communicates with the virtual leading vehicle .
(11) 
Virtual preceding vehicle: the second type is the virtual preceding vehicle in the spanning tree . This term is used to help the CAVs maintain a safe distance from the upper layer in the spanning tree. Note that the preceding vehicle in the spanning tree does not symbolize the preceding vehicle of the same lane in the simulation and vice versa. The CAVs and are the preceding vehicles of CAV in the iDFST and MCC spanning trees. respectively.
(12)
The adjacency matrix and pinning matrix are further introduced to represent the communication topology.
(13) 
where indicates that CAV receives information from CAV .
(14) 
where indicates that CAV receives information from the virtual leading vehicle , forming a diagonal matrix .
Further, a Laplacian matrix is also introduced as
(15) 
Note that we assume the information exchange is bidirectional; thus, . Hence, both , and are symmetric matrices.
The communication topology graph is designed based on the geometric topology graph of the spanning tree . Because the geometric topology represents the feasibility and optimality of an arrival plan, the communication topology represents the communication realization of the arrival plan. Xu et al. (2018) does not specify the communication topology. However, in this paper, we use PLF to simplify the communication topology and demonstrate the effectiveness of the algorithm. Topologies with greater complexity will be addressed in future research.
3.4.3 Controller Design
The control input of CAV is calculated according to the communication topology. The carfollowing distance in the virtual platooning is defined as in (9). Meanwhile, to align the CAVs at the same depth to arrive at the stopping line, (10) is indirectly guaranteed by the virtual leading vehicle control feedback. Hence, the virtual platoon controller is designed as follows.
Firstly, a union set is defined to describe the information exchange of CAV , as follows:
(16) 
The distance and velocity errors are defined as
(17) 
where is the carfollowing distance error of CAV , and is the carfollowing velocity error considering all the CAVs in .
A linear feedback controller is designed as follows:
(18)  
where and are the feedback gains of the distance and velocity errors of CAV . Because we consider a homogeneous scenario, the same gains are set for all the CAVs.
As mentioned before, we consider a secondorder vehicle dynamics as shown in (1). We define the car following errors as the new vehicle state.
(19) 
The vehicle input remains the same, i.e., .
Therefore, the carfollowing vehicle dynamic model is
(20) 
The linear feedback controller is simplified to
(21) 
Defining , we have
(22) 
4 Simulation
4.1 Simulation Environment and Evaluation Index
Types  Parameter  Symbol  Value 
Simulation Parameters  Simulation step    0.1 s 
Initial vehicle speed    2 m/s  
Control zone length  900 m  
Controller Parameters  Feedback gain of distance error  0.1  
Feedback gain of velocity error  0.3  
Desired velocity of the virtual leading vehicle  10 m/s  
Desired carfollowing distance of the virtual platoon  30 m  
Maximum acceleration  5 m/s  
Minimum acceleration  6 m/s  
Maximum velocity  25 m/s  
Minimum velocity  0 m/s 
The traffic simulation was conducted in SUMO, which is widely used in traffic researchesLopez et al. (2018). The simulation was run on an Intel Core i77700 3.6 GHz processor. The intersection scenario and lane direction settings are the same as shown in Fig. 1, and the control zone length
. The CAV arrival is assumed to be a Poisson distributed flow, given by
(23) 
where represents the vehicle’s arrival at the control zone.
is the expected value as well as the variance of the Poisson distribution.
4.2 Case Study of Algorithm Comparison
We demonstrate a case study in Example 2 to show the effectiveness of the proposed algorithms. The benchmark algorithm is the DFST method proposed in Xu et al. (2018). Based on DFST, the first algorithm is the iDFST presented in Section 3.2. The MCC algorithm has been explained in Section 3.3, which applies a heuristic algorithm to solve the MCC problem.
Example 2 (A case study of CAVs)
The traffic scenario remains the same as shown in Fig 1(b). A total number of vehicles arrive at the intersection with Poisson distribution. The vehicle arrivals are generated randomly and remain identical among the three algorithms.
As mentioned before, CDG and CUG are employed to describe the conflict relationship of the vehicles. The CDG is plotted in Fig. 7(a) as explained in Definition 3. The red dashed edges are the crossing conflicts and converging conflicts, whereas the black solid edges are the diverging and reachability conflicts. Fig. 7(b) depicts the CUG defined in Definition 4, which is the complement graph of the CDG. The edges are the coexisting relationships, i.e., the conflictfree vehicles.
Although identical vehicle arrivals are set for the three algorithms, different spanning trees, i.e., passing order solutions, are obtained. The spanning trees are carried out by scheduling the vehicles in the same layer to pass the intersection simultaneously. Thus, the overall depth of the spanning tree represents the evacuation time of the solution. First, the spanning trees of the DFST and iDFST methods are shown in Fig 7(c) and Fig. 7(d), which are derived from Fig. 7(a). Both the DFST and iDFST methods are based on the FIFO principle, i.e., obtaining the spanning tree vehicle by vehicle. Because the DFST has limited optimization of vehicle scheduling, it is a feasible solution with . In contrast, iDFST separates the conflict types to find the optimal place for each vehicle and obtains a locally optimal solution. Hence, the overall depth of the iDFST method is . Note that the red nodes are the vehicles that are brought forward when compared with the DFST method. Fig 7(e) depicts the solution derived from Fig. 7(b) by using the MCC method. The nodes of the same color represent the vehicles in the same clique, i.e., the vehicles that are scheduled to pass the intersection at the same time. The same cliques and the corresponding colors are drawn in Fig. 7(b) to show the MCC result with . Moreover, because the cliques can be arranged in the descending order, the average travel time of the vehicles in the MCC method is also better than that in the former algorithms. A detailed discussion is presented in the following simulations.
4.3 Algorithm Validation with Small Number of Vehicles
Next, we focus on the optimality of the three algorithms in general cases. In this section, we introduce the MCCBruteForce algorithm. It enumerates all the possible combinations of the cliques, i.e., solves the MCC problem by using the brute force method. Although MCCBruteForce provides us the theoretical global optimal solution, the huge calculation burden makes it impossible to apply it to a large number of vehicles. Thus, is set as the number of input vehicles.
Because the vehicle scheduling problem is strongly related to the vehicle arrival time, conducting a small number of simulations is insufficient. Hence, repetitions of the simulation were conducted for each algorithm. It is worth mentioning that these instances of vehicle arrival input are randomly generated by Poisson distribution, as shown in (23), and they are identical across all four algorithms.
We use the box plot to illustrate the maximum depth of the spanning tree, i.e., the overall traffic efficiency, as shown in Fig. 8
. The red lines are the median values of the corresponding simulation results, whereas the green triangles represent the average values. Compared with the DFST method, both the iDFST and MCC algorithms can reduce the maximum depth. The first quartile value, median value, and third quartile value are
, respectively. However, the average values of the results are not the same. The iDFST method attained an average value of , whereas the average value in the MCC method was and that of MCCBruteForce was . Recall that the vehicle arrivals were identical for these algorithms; therefore, it can be inferred that the MCCBruteForce method obtains the global optimal solution. Because of the heuristic MCC method introduced in Section 3.3, the MCC method cannot obtain the global optimal solution. Nevertheless, it achieves better result than the iDFST method, which obtains the locally optimal solution.4.4 Simulation Results for Various Numbers of Input Vehicles
For conducting the simulation on a larger scale, we extended the number of input vehicles from to and further compared the algorithms. was set as the input for the Poisson distribution of the vehicle arrival. We used the evacuation time and ATTD to evaluate the traffic efficiency. In terms of fuel consumption, the HBEFA3 model was used to evaluate the fuel consumption of the CAVs, which was integrated with SUMO Keller et al. (2010). As shown in Fig 9, the red lines in the box plot are the median value and the triangles are the average values. Ten repetitions of the simulation were conducted for each number of vehicles, and the vehicle arrivals were identical across the three algorithms.
In terms of the overall traffic efficiency evaluation, both the iDFST and MCC methods reduce the evacuation time significantly when compared with DFST. As shown in Fig 9(a), the average value of the evacuation time in DFST for the number of input vehicles of is , whereas those in iDFST and MCC are and , respectively. The two proposed algorithms save approximately of the evacuation time because of the improved scheduling method. Similar results are observed in the ATTD in Fig. 9(b), where iDFST and MCC save approximately of the ATTD for input vehicles. Although the MCC method prioritizes the overall efficiency, i.e., the evacuation time in the scheduling, the reordering of the cliques minimizes the ATTD (line 8 in the MCC algorithm). In other words, the MCC method heuristically finds the MCC groups to minimize the evacuation time. Then, it obtains a spanning tree from the MCC groups to minimize the ATTD. This also helps to interpret the outstanding fuel consumption performance of the MCC, as shown in Fig. 9(c). Better scheduling also helps to avoid the deceleration and idling of the CAVs in the control zone. iDFST and MCC saves and of the fuel consumption, respectively, for input vehicles.
4.5 Simulation Results at Various Traffic Volumes
The previous simulations were conducted with , i.e., a constant Poisson distribution. In the last simulation, we examined the influence of the traffic volume on the algorithm performance. Because the vehicle arrival follows the Poisson distribution as shown in (23), we varied the value to change the traffic volumes on each lane. As before, repetitions of the simulation were conducted for each number of vehicles, and the vehicle arrivals were identical across the three algorithms. The simulation results are plotted in Fig 10.
is the expected value of the Poisson distribution, which represents the average gap in the vehicle arrival time on each lane. As shown in Fig. 10(a), the evacuation times obtained with the three algorithms show limited fluctuations with increase in . The proposed two algorithms, i.e., iDFST and MCC, reduce the evacuation time obtained with DFST by approximately . Note that MCC has better performance when . Recall that in Table 4, we designed the car following distance of the virtual platoon as and the speed of the virtual leading vehicle as . Thus, the theoretical vehicle output of the intersection is bounded as per layer. In other words, if the average input vehicle gap , the accumulation and queuing of the vehicles is inevitable. Under this circumstance, MCC shows better performance than iDFST because of its better usage of the intersection space. Fig. 10(b) shows the ATTD performance of the algorithms. As the traffic volume increases, the vehicle is more likely to be stuck behind other preceding vehicles. Thus, the vehicle travel time is positively correlated with the traffic volume, i.e., the ATTD decreases as the vehicle arrival gap increases. Similar to the evacuation time, MCC shows better performance when . Because of the abovementioned reason for the MCC to show better utilization of the intersection space, idling behavior and fuel consumption performance are greatly improved. As shown in Fig. 10(c), iDFST saves at most and MCC saves of the fuel consumption when compared with DFST.
5 Conclusion
In this paper, a graphbased cooperation method was proposed to formulate the conflictfree scheduling problem at unsignalized intersections. Based on the graphical description of the conflict relationships among the CAVs, an iDFST method was introduced to obtain the local optimal solution. After concise and rigid graphical analysis, we reduced the CAV scheduling problem to the MCC problem, which yielded the global optimal solution. Because the MCC problem was proven to be an NPhard problem, a heuristic method was also proposed to solve this problem with low computation complexity in the case of large number of vehicles. Furthermore, a distributed control framework and communication topology were designed to realize the conflictfree cooperation of the vehicles. Traffic simulations proved the effectiveness of the proposed algorithms.
A future direction of our research is to permit the lanechanging behavior, i.e., the CAVs are allowed to change lanes while approaching the intersection. Several practical methods Cai et al. (2021); Chouhan et al. (2020) have been proposed to apply the lane changing behavior of multiple CAVs. It is interesting to combine these works to further extend the intersection scenario, which will be considered in our future research. Another interesting topic is the communication topology of the virtual platoon. In this study, we designed a PLF communication topology to realize the scheduling plan. The problem of identifying the best topology is yet to be addressed. Finally, field experiments are needed for further validation of the proposed algorithm.
Acknowledgments
This work was supported by the National Key Research and Development Program of China under Grant
2018YFE0204302, National Natural Science Foundation of China under Grant 52072212, Key Algorithms in Intelligent and Connected Cloud Control System for China Intelligent and Connected Vehicles Research Institute (CICV), and Tsinghua UniversityDidi Joint Research Center for Future Mobility.
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