Multi-Agent Path Finding on Strongly Connected Digraphs

Abstract

The pebble motion on trees (PMT) problem consists in finding a feasible sequence of moves that repositions a set of pebbles to assigned target vertices. This problem has been widely studied because, in many cases, the more general Multi-Agent path finding (MAPF) problem on graphs can be reduced to PMT. We propose a simple and easy to implement procedure, which finds solutions of length O(knc + n 2 ), where n is the number of nodes, k is the number of pebbles, and c the maximum length of corridors in the tree. This complexity result is more detailed than the current best known result O(n 3 ), which is equal to our result in the worst case, but does not capture the dependency on c and k.

An Algorithm with Improved Complexity for Pebble Motion/Multi-Agent Path Finding on Trees

“…These conditions are discussed in (Goraly & Hassin, 2010;Krontiris et al, 2013) for the case of undirected graphs and in (Ardizzoni et al, 2022;Botea & Surynek, 2015) for the case of directed graphs. Given a graph G, we can convert it into a tree (called biconnected component tree), adding a new type of vertex called trans-shipment (Ardizzoni et al, 2022;Goraly & Hassin, 2010;Krontiris et al, 2013). In particular, each biconnected component of the graph is converted into a star subgraph, whose internal vertex is a trans-shipment (see Figure 10).…”

“…Figures 11 and 12 show an example of this conversion. For more details on the conversion algorithm see (Ardizzoni et al, 2022;Goraly & Hassin, 2010;Krontiris et al, 2013). For this reason, we need to study a variant of the PMT problem, the pebble motion on trees with trans-shipment vertices (ts-PMT), which is a PMT problem on a tree such that the vertex set is partitioned in trans-shipment and regular vertices.…”

“…Instead, the complexity of checking MAPF feasibility depends on the specific graph class. For instance, it is polynomial on undirected graphs (Kornhauser et al, 1984), on strongly biconnected directed graphs (Botea & Surynek, 2015), and on strongly connected directed graphs (Ardizzoni et al, 2022). Instead, it is NP-hard in the general case of directed graphs (Nebel, 2020).…”

“…Instead, it is NP-hard in the general case of directed graphs (Nebel, 2020). Optimal and suboptimal algorithms have been proposed in the last forty years (Achá et al, 2022;Alotaibi & Al-Rawi, 2016;Ardizzoni et al, 2022;Auletta et al, 1999; Botea be a tree with n vertices and P a set of distinct pebbles such that |P | < n, numbered 1, . .…”

“…Although PMT is one of the simplest versions of MAPF, it is quite relevant. Indeed, various algorithms that solve MAPF on more general graphs are based on a reformulation of MAPF as a PMT, over a suitably defined tree (Ardizzoni et al, 2022;Goraly & Hassin, 2010;Krontiris et al, 2013). We will further discuss this in Section 6.…”