1. Introduction
Multiagent path finding (MAPF) is the problem of planning collisionfree paths for multiple agents in known environments from their given start vertices to their given goal vertices. MAPF is important, for example, for aircrafttowing vehicles Morris et al. (2016), warehouse and office robots Wurman et al. (2008); Veloso et al. (2015), and game characters Ma et al. (2017). The objective is to minimize the sum of the arrival times of the agents or the makespan. The MAPF problem is NPhard to solve optimally Yu and LaValle (2013b) and even to approximate within a small constant factor for makespan minimization Ma et al. (2016). It can be solved with reductions to other wellstudied combinatorial problems Surynek (2015); Yu and LaValle (2013a); Erdem et al. (2013); Nguyen et al. (2017) and dedicated optimal Standley (2010); Standley and Korf (2011); Goldenberg et al. (2014); Sharon et al. (2013); Wagner and Choset (2015); Sharon et al. (2015); Boyarski et al. (2015); Felner et al. (2018), boundedsuboptimal Barer et al. (2014); Cohen et al. (2016), and suboptimal MAPF algorithms Silver (2005); Sturtevant and Buro (2006); Wang and Botea (2011); Luna and Bekris (2011); de Wilde et al. (2013); Wagner and Choset (2011), as described in several surveys Ma et al. (2016); Felner et al. (2017).
The MAPF problem has recently been generalized in different directions Ma and Koenig (2016); Hönig et al. (2016a); Ma et al. (2016); Hönig et al. (2016b); Ma et al. (2017a, b); Ma et al. (2017) but none of them capture an important characteristic of many applications, namely the ability to meet deadlines. We thus formalize the multiagent path finding problem with deadlines (MAPFDL problem). The objective is to maximize the number of agents that can reach their given goal vertices from their given start vertices within a given deadline, without colliding with each other. In previously studied MAPF problems, all agents have to be routed from their start vertices to their goal vertices, and the objective is with regard to resources such as fuel (sum of arrival times) or time (makespan). In the MAPFDL problem, on the other hand, the resources are the agents themselves. We first show that the MAPFDL problem is NPhard to solve optimally. We then present an optimal MAPFDL algorithm based on a reduction of the MAPFDL problem to a flow problem and a subsequent compact integer linear programming formulation of the resulting reduced abstracted multicommodity flow network.
2. MAPFDL Problem
We formalize the MAPFDL problem as follows: We are given a deadline, denoted by a time step , an undirected graph , and agents . Each agent has a start vertex and a goal vertex . In each time step, each agent either moves to an adjacent vertex or stays at the same vertex. Let be the vertex occupied by agent at time step . Call an agent successful iff it occupies its goal vertex at the deadline , that is, . A plan consists of a path assigned to each successful agent . Unsuccessful agents are removed at time step zero and thus have no paths assigned to them.^{1}^{1}1Depending on the application, the unsuccessful agents can be removed at time step zero, wait at their start vertices, or move out of the way of the successful agents. We choose the first option in this paper. If the unsuccessful agents are not removed, they can obstruct other agents. However, our proof of NPhardness does not depend on this assumption, and our MAPFDL algorithm can be adapted to other assumptions. A solution is a plan that satisfies the following conditions: (1) For all successful agents , [each successful agent starts at its start vertex]. (2) For all successful agents and all time steps , or [each successful agent always either moves to an adjacent vertex or does not move]. (3) For all pairs of different successful agents and and all time steps , [two successful agents never occupy the same vertex simultaneously]. (4) For all pairs of different successful agents and and all time steps , or [two successful agents never traverse the same edge simultaneously in opposite directions]. Define a collision between two different successful agents and to be either a vertex collision (, , , ) iff (corresponding to Condition 3) or an edge collision (, , , , ) iff and (corresponding to Condition 4). The objective is to maximize the number of successful agents .
Theorem 1.
It is NPhard to compute a MAPFDL solution with the maximum number of successful agents.
The proof of the theorem reduces the 3,3SAT problem Tovey (1984), an NPcomplete version of the Boolean satisfiability problem, to the MAPFDL problem. The reduction is similar to the one used for proving the NPhardness of approximating the optimal makespan for the MAPF problem Ma et al. (2016). It constructs a MAPFDL instance with deadline that has a solution where all agents are successful iff the given 3,3SAT instance is satisfiable.
3. Optimal MAPFDL Algorithm
Our optimal MAPFDL algorithm first reduces the MAPFDL problem to the maximum (integer) multicommodity flow problem, which is similar to the reductions of the MAPF and TAPF problems to multicommodity flow problems Yu and LaValle (2013a); Ma and Koenig (2016): Given a MAPFDL instance with deadline , we construct a multicommodity flow network with vertices and directed edges with unit capacity. The vertices represent vertex at the end of time step and the beginning of time step , while the vertices are intermediate vertices. For each agent , we set a supply of one at (start) vertex and a demand of one at (goal) vertex , both for commodity type (corresponding to agent ). For each time step, we construct the gadgets shown in Figure 1(b) to prevent vertex and edge collisions. The objective is to maximize the total amount of integral flow received in all vertices , which can be achieved via a standard integer linear programming (ILP) formulation.
Theorem 2.
There is a onetoone correspondence between all solutions of a MAPFDL instance with the maximum number of successful agents and all maximum integral flows on the corresponding flow network.
The proof of the theorem is similar to the one for the reduction of the MAPF problem to the multicommodity flow problem Yu and LaValle (2013a).
Figure 1(a) shows a MAPFDL instance with deadline . Agents and have start vertices and and goal vertices and , respectively. The number of successful agents is at most because only agent can reach its goal vertex in two time steps. Figure 1(c) shows the corresponding flow network with a maximum flow (in color) that corresponds to a solution with unsuccessful agent and successful agent with path , , .
Abstracted Flow Network and Compact ILP Formulation We construct a compact ILP formulation based on an abstraction of the flow network and additional linear constraints to prevent vertex and edge collisions. We obtain the abstracted flow network by (1) contracting each and replacing and with a single vertex for all and (and with ); and (2) replacing the gadget for each and each with a pair of edges and . Figure 2 (left) shows an example. Then, we use the standard ILP formulation of this abstracted network augmented with the constraints shown in red:
where the 0/1 variable represents the amount of flow of commodity type on edge and the sets and contain all outgoing and incoming, respectively, edges of vertex . The top red constraints prevent vertex collisions of the form , and the bottom red constraints prevent edge collisions of the forms and .
Reduced Abstracted Flow Network We can remove all vertices and edges from the abstracted flow network that are not on some path from at least one start vertex to the corresponding goal vertex in the abstracted flow network. This can be done by performing one complete forward breadthfirst search from each start vertex and one complete backward breadthfirst search from each goal vertex and then keeping only those vertices and edges that are part of the search trees associated with at least one start vertex and the corresponding goal vertex. Figure 2 (right) shows an example. We then use the compact ILP formulation of the resulting reduced abstracted flow network.
Experimental Evaluation We tested our optimal MAPFDL algorithm on a 2.50 GHz Intel Core i52450M laptop with 6 GB RAM, using CPLEX V12.7.1 IBM (2011) as the ILP solver. We randomly generated MAPFDL instances with different numbers of agents (ranging from 10 to 100 in increments of 10) on 4neighbor 2D grids with deadline
50. We blocked all grid cells independently at random with 20% probability each. We generated 50 MAPFDL instances for each number of agents. We placed the start and goal vertices of each agent randomly at distance 48, 49, or 50. The following table shows the percentage of instances that could be solved within a runtime limit of 60 seconds per instance.
References
 (1)
 Barer et al. (2014) M. Barer, G. Sharon, R. Stern, and A. Felner. 2014. Suboptimal Variants of the ConflictBased Search Algorithm for the MultiAgent Pathfinding Problem. In SoCS. 19–27.
 Boyarski et al. (2015) E. Boyarski, A. Felner, R. Stern, G. Sharon, D. Tolpin, O. Betzalel, and S. E. Shimony. 2015. ICBS: Improved ConflictBased Search Algorithm for MultiAgent Pathfinding. In IJCAI. 740–746.
 Cohen et al. (2016) L. Cohen, T. Uras, T. K. S. Kumar, H. Xu, N. Ayanian, and S. Koenig. 2016. Improved Solvers for BoundedSuboptimal MultiAgent Path Finding. In IJCAI. 3067–3074.
 de Wilde et al. (2013) B. de Wilde, A. W. ter Mors, and C. Witteveen. 2013. Push and Rotate: Cooperative MultiAgent Path Planning. In AAMAS. 87–94.
 Erdem et al. (2013) E. Erdem, D. G. Kisa, U. Oztok, and P. Schueller. 2013. A General Formal Framework for Pathfinding Problems with Multiple Agents. In AAAI. 290–296.

Felner et al. (2018)
A. Felner, J. Li,
E. Boyarski, H. Ma, L.
Cohen, T. K. S. Kumar, and S. Koenig.
2018.
Adding Heuristics to ConflictBased Search for MultiAgent Path Finding.
ICAPS (2018).  Felner et al. (2017) A. Felner, R. Stern, S. E. Shimony, E. Boyarski, M. Goldenberg, G. Sharon, N. R. Sturtevant, G. Wagner, and P. Surynek. 2017. SearchBased Optimal Solvers for the MultiAgent Pathfinding Problem: Summary and Challenges. In SoCS. 29–37.

Goldenberg et al. (2014)
M. Goldenberg, A. Felner,
R. Stern, G. Sharon,
N. R. Sturtevant, R. C. Holte, and
J. Schaeffer. 2014.
Enhanced Partial Expansion A*.
Journal of Artificial Intelligence Research
50 (2014), 141–187.  Hönig et al. (2016a) W. Hönig, T. K. S. Kumar, L. Cohen, H. Ma, H. Xu, N. Ayanian, and S. Koenig. 2016a. MultiAgent Path Finding with Kinematic Constraints. In ICAPS. 477–485.
 Hönig et al. (2016b) W. Hönig, T. K. S. Kumar, H. Ma, N. Ayanian, and S. Koenig. 2016b. Formation Change for Robot Groups in Occluded Environments. In IROS. 4836–4842.
 IBM (2011) IBM. 2011. IBM ILOG CPLEX Optimization Studio CPLEX User’s Manual.
 Luna and Bekris (2011) R. Luna and K. E. Bekris. 2011. Push and Swap: Fast Cooperative PathFinding with Completeness Guarantees. In IJCAI. 294–300.
 Ma et al. (2017) H. Ma, W. Hönig, L. Cohen, T. Uras, H. Xu, T. K. S. Kumar, N. Ayanian, and S. Koenig. 2017. Overview: A Hierarchical Framework for Plan Generation and Execution in MultiRobot Systems. IEEE Intelligent Systems 32, 6 (2017), 6–12.
 Ma and Koenig (2016) H. Ma and S. Koenig. 2016. Optimal Target Assignment and Path Finding for Teams of Agents. In AAMAS. 1144–1152.
 Ma et al. (2016) H. Ma, S. Koenig, N. Ayanian, L. Cohen, W. Hönig, T. K. S. Kumar, T. Uras, H. Xu, C. Tovey, and G. Sharon. 2016. Overview: Generalizations of MultiAgent Path Finding to RealWorld Scenarios. In IJCAI16 Workshop on MultiAgent Path Finding.
 Ma et al. (2017a) H. Ma, T. K. S. Kumar, and S. Koenig. 2017a. MultiAgent Path Finding with Delay Probabilities. In AAAI. 3605–3612.
 Ma et al. (2017b) H. Ma, J. Li, T. K. S. Kumar, and S. Koenig. 2017b. Lifelong MultiAgent Path Finding for Online Pickup and Delivery Tasks. In AAMAS. 837–845.
 Ma et al. (2016) H. Ma, C. Tovey, G. Sharon, T. K. S. Kumar, and S. Koenig. 2016. MultiAgent Path Finding with Payload Transfers and the PackageExchange RobotRouting Problem. In AAAI. 3166–3173.
 Ma et al. (2017) H. Ma, J. Yang, L. Cohen, T. K. S. Kumar, and S. Koenig. 2017. Feasibility Study: Moving NonHomogeneous Teams in Congested Video Game Environments. In AIIDE. 270–272.
 Morris et al. (2016) R. Morris, C. Pasareanu, K. Luckow, W. Malik, H. Ma, S. Kumar, and S. Koenig. 2016. Planning, Scheduling and Monitoring for Airport Surface Operations. In AAAI16 Workshop on Planning for Hybrid Systems.
 Nguyen et al. (2017) V. Nguyen, P. Obermeier, T. C. Son, T. Schaub, and W. Yeoh. 2017. Generalized Target Assignment and Path Finding Using Answer Set Programming. In IJCAI. 1216–1223.
 Sharon et al. (2015) G. Sharon, R. Stern, A. Felner, and N. R. Sturtevant. 2015. ConflictBased Search for Optimal MultiAgent Pathfinding. Artificial Intelligence 219 (2015), 40–66.
 Sharon et al. (2013) G. Sharon, R. Stern, M. Goldenberg, and A. Felner. 2013. The Increasing Cost Tree Search for Optimal MultiAgent Pathfinding. Artificial Intelligence 195 (2013), 470–495.
 Silver (2005) D. Silver. 2005. Cooperative Pathfinding. In AIIDE. 117–122.
 Standley (2010) T. S. Standley. 2010. Finding Optimal Solutions to Cooperative Pathfinding Problems. In AAAI. 173–178.
 Standley and Korf (2011) T. S. Standley and R. E. Korf. 2011. Complete Algorithms for Cooperative Pathfinding Problems. In IJCAI. 668–673.
 Sturtevant and Buro (2006) N. R. Sturtevant and M. Buro. 2006. Improving Collaborative Pathfinding Using Map Abstraction. In AIIDE. 80–85.
 Surynek (2015) P. Surynek. 2015. Reduced TimeExpansion Graphs and Goal Decomposition for Solving Cooperative Path Finding SubOptimally. In IJCAI. 1916–1922.
 Tovey (1984) C. A. Tovey. 1984. A Simplified NPComplete Satisfiability Problem. Discrete Applied Mathematics 8 (1984), 85–90.
 Veloso et al. (2015) M. Veloso, J. Biswas, B. Coltin, and S. Rosenthal. 2015. CoBots: Robust Symbiotic Autonomous Mobile Service Robots. In IJCAI. 4423–4429.
 Wagner and Choset (2011) G. Wagner and H. Choset. 2011. M*: A Complete Multirobot Path Planning Algorithm with Performance Bounds. In IROS. 3260–3267.
 Wagner and Choset (2015) G. Wagner and H. Choset. 2015. Subdimensional Expansion for Multirobot Path Planning. Artificial Intelligence 219 (2015), 1–24.
 Wang and Botea (2011) K. Wang and A. Botea. 2011. MAPP: A Scalable MultiAgent Path Planning Algorithm with Tractability and Completeness Guarantees. Journal of Artificial Intelligence Research 42 (2011), 55–90.
 Wurman et al. (2008) P. R. Wurman, R. D’Andrea, and M. Mountz. 2008. Coordinating Hundreds of Cooperative, Autonomous Vehicles in Warehouses. AI Magazine 29, 1 (2008), 9–20.
 Yu and LaValle (2013a) J. Yu and S. M. LaValle. 2013a. Planning Optimal Paths for Multiple Robots on Graphs. In ICRA. 3612–3617.
 Yu and LaValle (2013b) J. Yu and S. M. LaValle. 2013b. Structure and Intractability of Optimal MultiRobot Path Planning on Graphs. In AAAI. 1444–1449.
Comments
There are no comments yet.