Minimizing Movement: Fixed-Parameter Tractability
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by
Erik D. Demaine, MohammadTaghi Hajiaghayi, Dániel Marx
2012
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.
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1205.6960v2
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