Let WβR2 be a planar polygonal environment
(i.e., a polygon potentially with holes) with a total of n vertices, and let
A,B be two robots, each modeled as an axis-aligned unit square, that can
translate inside W. Given source and target placements
sAβ,tAβ,sBβ,tBββW of A and B, respectively, the goal is to
compute a \emph{collision-free motion plan} Οβ, i.e., a motion
plan that continuously moves A from sAβ to tAβ and B from sBβ to
tBβ so that A and B remain inside W and do not collide with
each other during the motion. Furthermore, if such a plan exists, then we wish
to return a plan that minimizes the sum of the lengths of the paths traversed
by the robots, β£Οββ£. Given W,sAβ,tAβ,sBβ,tBβ and a parameter Ξ΅>0, we present an
n2Ξ΅βO(1)logn-time (1+Ξ΅)-approximation algorithm
for this problem. We are not aware of any polynomial time algorithm for this
problem, nor do we know whether the problem is NP-Hard. Our result is the first
polynomial-time (1+Ξ΅)-approximation algorithm for an optimal motion
planning problem involving two robots moving in a polygonal environment.Comment: The conference version of the paper is accepted to SODA 202