Exact and approximate algorithms for movement problems on (special classes of) graphs

TL;DR

This paper investigates motion planning problems on graphs for multiple objects, aiming to optimize various distance-based objectives, and provides tight approximation and inapproximability results for these problems.

Contribution

It introduces new algorithms and complexity bounds for movement problems on graphs, addressing objectives like minimizing total or maximum travel distance.

Findings

Several problems are shown to be approximable within specific bounds.

Some problems are proven to be hard to approximate beyond certain ratios.

The results include tight bounds for both algorithmic solutions and hardness.

Abstract

When a large collection of objects (e.g., robots, sensors, etc.) has to be deployed in a given environment, it is often required to plan a coordinated motion of the objects from their initial position to a final configuration enjoying some global property. In such a scenario, the problem of minimizing some function of the distance travelled, and therefore energy consumption, is of vital importance. In this paper we study several motion planning problems that arise when the objects must be moved on a graph, in order to reach certain goals which are of interest for several network applications. Among the others, these goals include broadcasting messages and forming connected or interference-free networks. We study these problems with the aim of minimizing a number of natural measures such as the average/overall distance travelled, the maximum distance travelled, or the number of objects that need to be moved. To this respect, we provide several approximability and inapproximability results, most of which are tight.