DMS*: Minimizing Makespan for Multi-Agent Combinatorial Path Finding

TL;DR

This letter proposes a min-max variant of MCPF, denoted as MCPF-max, that minimizes the makespan of the agents, and develops two new techniques based on <monospace>MS*</monospace> to defer the expensive target sequencing during planning to expedite the overall computation.

Abstract

Multi-Agent Combinatorial Path Finding (MCPF) seeks collision-free paths for multiple agents from their initial to goal locations, while visiting a set of intermediate target locations in the middle of the paths. MCPF is challenging as it involves both planning collision-free paths for multiple agents and target sequencing, i.e., solving traveling salesman problems to assign targets to and find the visiting order for the agents. Recent work develops methods to address MCPF while minimizing the sum of individual arrival times at goals. Such a problem formulation may result in paths with different arrival times and lead to a long makespan, the maximum arrival time, among the agents. This paper proposes a min-max variant of MCPF, denoted as MCPF-max, that minimizes the makespan of the agents. While the existing methods (such as MS*) for MCPF can be adapted to solve MCPF-max, we further develop two new techniques based on MS* to defer the expensive target sequencing during planning to expedite the overall computation. We analyze the properties of the resulting algorithm Deferred MS* (DMS*), and test DMS* with up to 20 agents and 80 targets. We demonstrate the use of DMS* on differential-drive robots.

Figures (4)

• Figure 1: MCPF-max and MCPF-sum. MCPF-max seeks a set of collision-free paths while minimizing the maximum arrival time of the agents. The color of a target or goal indicates the assignment constraints, i.e., the subset of agents that are eligible to visit that target or goal.
• Figure 2: An illustration of DMS* and related concepts. (a) shows the workspace graph $G^W$. (b) shows the target graph $G^T$ where each edge in $G^T$ corresponds to a minimum cost path in $G^W$ between the respective vertices. (c) shows the workflow of DMS* in Alg. \ref{['dms:alg:dms']}. (d) shows DMS* first ignores any agent-agent conflict and solves a corresponding mHPP, which provides a joint sequence $\gamma(l_o)$ (d1). This joint sequence $\gamma(l_o)$ can be converted to a joint path $\pi(l_o)$ (d2,d3), whose corresponding makespan $f_{max}$ is 9. (e) The joint path $\pi(l_o)$ leads to a policy $\phi_{\gamma(l_o)}$ that maps a joint vertex to the next joint vertex along $\pi(l_o)$. Since conflicts are ignored in this policy, agents may run into conflict (e2). When a conflict is detected, the subset of agents that are in conflict $I_C(l)$ is back propagated to the ancestor labels. (f) After the back propagation, these ancestor labels and re-opened and re-expanded while considering all possible actions of all agents in conflict (f1,f2,f3). During the re-expansion, for each of the successors, DMS* first uses a fast-to-compute yet roughly estimated cost-to-go as the heuristic of the generated labels to avoid solving a mHPP. For both labels (f2) and (f3), this rough heuristic is $(9,5)-(1,1)=(8,4)$, where $(9,5)$ is the heuristic of (f1), the parent of (f2) and (f3), and $(1,1)$ means each agent can move at most one step towards their goals. When either (f2) or (f3) is popped from OPEN and before being expanded, DMS* re-computes a new heuristic and checks if the popped label should be expanded or re-added to OPEN for future expansion. To obtain this new heuristic, DMS* first attempts to let the agents follow the previously computed target sequences of the parent label and check if this worsens the makespan. For the successor shown in (f2), the resulting makespan is 9 (f4), which is no worse than the previous makespan 9, and DMS* will not call mHPP solver to save computational effort. For the successor shown in (f3), the resulting makespan is 10 (f5), which is worse than the previous makespan 9, and DMS* have to call mHPP solver to find a new joint sequence from that successor (f3) to the goals. Finally, with the joint sequence, new policy for the successors can be built and the search continues as in (e).
• Figure 3: The success rates and runtime of DMS* and MS* (baseline) with varying number of agents and targets in a random 32x32 map. DMS* has higher success rates and less runtime on average than MS*.
• Figure 4: The success rates and runtime of both DMS* and MS* (baseline) with varying number of targets in maps of different sizes. DMS* achieves higher success rates than MS* while requiring less runtime on average.

Theorems & Definitions (9)

• Definition 1: MCPF-max Problem
• Definition 2: Binary Dominance
• Definition 3: Label Dominance
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4