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We present an empirical study of the relationship between map connectivity and the empirical hardness of the multi-agent pathfinding (MAPF) problem. By analyzing the second smallest eigenvalue (commonly known as lambda2) of the normalized Laplacian matrix of different maps, our initial study indicates that maps with smaller lambda2 tend to create more challenging instances when agents are generated uniformly randomly. Additionally, we introduce a map generator based on Quality Diversity (QD) that is capable of producing maps with specified lambda2 ranges, offering a possible way for generating challenging MAPF instances. Despite the absence of a strict monotonic correlation with lambda2 and the empirical hardness of MAPF, this study serves as a valuable initial investigation for gaining a deeper understanding of what makes a MAPF instance hard to solve. 

The A* algorithm is commonly used to solve NP-hard combinatorial optimization problems. When provided with a completely informed heuristic function, A* can solve such problems in time complexity that is polynomial in the solution cost and branching factor. In light of this fact, we examine a line of recent publications that propose fitting deep neural networks to the completely informed heuristic function. We assert that these works suffer from inherent scalability limitations since --- under the assumption of NP P/poly --- such approaches result in either (a) network sizes that scale super-polynomially in the instance sizes or (b) the accuracy of the fitted deep neural networks scales inversely with the instance sizes. Complementing our theoretical claims, we provide experimental results for three representative NP-hard search problems. The results suggest that fitting deep neural networks to informed heuristic functions requires network sizes that grow quickly with the problem instance size. We conclude by suggesting that the research community should focus on scalable methods for integrating heuristic search with machine learning, as opposed to methods relying on informed heuristic estimation. 

Conventional Multi-Agent Path Finding (MAPF) problems aim to compute an ensemble of collision-free paths for multiple agents from their respective starting locations to pre-allocated destinations. This work considers a generalized version of MAPF called Multi-Agent Combinatorial Path Finding (MCPF) where agents must collectively visit a large number of intermediate target locations along their paths before arriving at destinations. This problem involves not only planning collision-free paths for multiple agents but also assigning targets and specifying the visiting order for each agent (i.e., target sequencing). To solve the problem, we leverage Conflict-Based Search (CBS) for MAPF and propose a novel approach called Conflict-Based Steiner Search (CBSS). CBSS interleaves (1) the collision resolution strategy in CBS to bypass the curse of dimensionality in MAPF and (2) multiple traveling salesman algorithms to handle the combinatorics in target sequencing, to compute optimal or bounded sub-optimal paths for agents while visiting all the targets. We also develop two variants of CBSS that trade off runtime against solution optimality. Our test results verify the advantage of CBSS over the baselines in terms of computing cheaper paths and improving success rates within a runtime limit for up to 20 agents and 50 targets. Finally, we run both Gazebo simulation and physical robot tests to validate that the planned paths are executable. 

Multi-Agent Path Finding (MAPF), i.e., finding collision-free paths for multiple robots, is important for many applications where small runtimes are necessary, including the kind of automated warehouses operated by Amazon. CBS is a lead- ing two-level search algorithm for solving MAPF optimally. ECBS is a bounded-suboptimal variant of CBS that uses focal search to speed up CBS by sacrificing optimality and instead guaranteeing that the costs of its solutions are within a given factor of optimal. In this paper, we study how to decrease its runtime even further using inadmissible heuristics. Motivated by Explicit Estimation Search (EES), we propose Explicit Estimation CBS (EECBS), a new bounded-suboptimal variant of CBS, that uses online learning to obtain inadmissible estimates of the cost of the solution of each high-level node and uses EES to choose which high-level node to expand next. We also investigate recent improvements of CBS and adapt them to EECBS. We find that EECBS with the improvements runs significantly faster than the state-of-the-art bounded-suboptimal MAPF algorithms ECBS, BCP-7, and eMDD-SAT on a variety of MAPF instances. We hope that the scalability of EECBS enables additional applications for bounded-suboptimal MAPF algorithms.