Revisiting the Complexity Analysis of Conflict-Based Search: New Computational Techniques and Improved Bounds

TL;DR

This paper provides a refined complexity analysis of Conflict-Based Search (CBS) for Multi-Agent Path Finding, introducing new bounds and techniques that significantly improve understanding of its worst-case computational performance.

Contribution

The paper introduces novel analytical methods to tighten the worst-case complexity bounds of CBS, including decision diagram bounds and recurrence relations with generating functions.

Findings

New upper bounds on CBS complexity for various cases

Improved bounds on common benchmarks by a factor of at least 2^{10^7}

Enhanced understanding of parameters influencing CBS runtime

Abstract

The problem of Multi-Agent Path Finding (MAPF) calls for finding a set of conflict-free paths for a fleet of agents operating in a given environment. Arguably, the state-of-the-art approach to computing optimal solutions is Conflict-Based Search (CBS). In this work we revisit the complexity analysis of CBS to provide tighter bounds on the algorithm's run-time in the worst-case. Our analysis paves the way to better pinpoint the parameters that govern (in the worst case) the algorithm's computational complexity. Our analysis is based on two complementary approaches: In the first approach we bound the run-time using the size of a Multi-valued Decision Diagram (MDD) -- a layered graph which compactly contains all possible single-agent paths between two given vertices for a specific path length. In the second approach we express the running time by a novel recurrence relation which bounds the algorithm's complexity. We use generating functions-based analysis in order to tightly bound the recurrence. Using these technique we provide several new upper-bounds on CBS's complexity. The results allow us to improve the existing bound on the running time of CBS for many cases. For example, on a set of common benchmarks we improve the upper-bound by a factor of at least $2^{10^{7}}$.