Minimizing Movement: Fixed-Parameter Tractability

TL;DR

This paper investigates the computational complexity of movement minimization problems involving multiple agents, establishing a fixed-parameter tractability framework based on treewidth, and identifying which problems are efficiently solvable.

Contribution

It introduces a general approach to determine the tractability of movement problems using treewidth, providing a clear boundary between tractable and intractable cases.

Findings

Many movement problems are fixed-parameter tractable based on treewidth.

The complexity boundary depends on the treewidth of minimal configurations.

Several concrete movement problems are shown to be efficiently solvable.

Abstract

We study an extensive class of movement minimization problems which arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). The only previous theoretical results about this class of problems are about approximation, and mainly negative: many movement problems of interest have polynomial inapproximability. Given that the number of mobile agents is typically much smaller than the complexity of the environment, we turn to fixed-parameter tractability. We characterize the boundary between tractable and intractable movement problems in a very general set up: it turns out the complexity of the problem fundamentally depends on the treewidth of the minimal configurations. Thus the complexity of a particular problem can be determined by answering a purely combinatorial question. Using our general tools, we determine the complexity of several concrete problems and fortunately show that many movement problems of interest can be solved efficiently.