Multi-agent path finding (MAPF) is the problem of planning collision-free paths for multiple agents in known environments from their given start vertices to their given goal vertices. MAPF is important, for example, for aircraft-towing 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 NP-hard 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 well-studied 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), bounded-suboptimal 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 multi-agent path finding problem with deadlines (MAPF-DL 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 MAPF-DL problem, on the other hand, the resources are the agents themselves. We first show that the MAPF-DL problem is NP-hard to solve optimally. We then present an optimal MAPF-DL algorithm based on a reduction of the MAPF-DL problem to a flow problem and a subsequent compact integer linear programming formulation of the resulting reduced abstracted multi-commodity flow network.
2. MAPF-DL Problem
We formalize the MAPF-DL 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.111Depending 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 NP-hardness does not depend on this assumption, and our MAPF-DL 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 .
It is NP-hard to compute a MAPF-DL solution with the maximum number of successful agents.
The proof of the theorem reduces the 3,3-SAT problem Tovey (1984), an NP-complete version of the Boolean satisfiability problem, to the MAPF-DL problem. The reduction is similar to the one used for proving the NP-hardness of approximating the optimal makespan for the MAPF problem Ma et al. (2016). It constructs a MAPF-DL instance with deadline that has a solution where all agents are successful iff the given 3,3-SAT instance is satisfiable.
3. Optimal MAPF-DL Algorithm
Our optimal MAPF-DL algorithm first reduces the MAPF-DL problem to the maximum (integer) multi-commodity flow problem, which is similar to the reductions of the MAPF and TAPF problems to multi-commodity flow problems Yu and LaValle (2013a); Ma and Koenig (2016): Given a MAPF-DL instance with deadline , we construct a multi-commodity 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.
There is a one-to-one correspondence between all solutions of a MAPF-DL 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 multi-commodity flow problem Yu and LaValle (2013a).
Figure 1(a) shows a MAPF-DL 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 breadth-first search from each start vertex and one complete backward breadth-first 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 MAPF-DL algorithm on a 2.50 GHz Intel Core i5-2450M laptop with 6 GB RAM, using CPLEX V12.7.1 IBM (2011) as the ILP solver. We randomly generated MAPF-DL instances with different numbers of agents (ranging from 10 to 100 in increments of 10) on 4-neighbor 2D grids with deadline
50. We blocked all grid cells independently at random with 20% probability each. We generated 50 MAPF-DL 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.
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