Maximum thick paths in static and dynamic environments

We consider the problem of finding a maximum number of disjoint paths for unit disks moving amidst static or dynamic obstacles. For the static case we give efficient exact algorithms, based on adapting the "continuous uppermost path" paradigm. As a by-product, we establish a continuous anaMenger's Theorem. (In this extended abstract we only state these results.) Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. We observe that (unless P=NP), for any , > 0, one cannot decide in polynomial time whether there exist K paths for disks of radius R, where K is the maximum number of paths for radius-R disks. The problem is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying "dual" approximations, compromis...

• Type: Conference
• Year: 2008
• Where: COMPGEOM
• Authors: Esther M. Arkin, Joseph S. B. Mitchell, Valentin Polishchuk
• Keywords: COMPGEOM 2008, Discrete Geometry, Maximum Number, Time Intervals, Unit Disks