Affiliation:
1. Royal Holloway, University of London, Egham, United Kingdom
2. School of Computing, DePaul University, Chicago, IL, USA
Abstract
Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than
k
different obstacles? Equivalently, can we remove
k
obstacles so that there is an obstacle-free path between the two designated points? This is a fundamental NP-hard problem that has undergone a tremendous amount of research work. The problem can be formulated and generalized into the following graph problem: Given a planar graph
G
whose vertices are colored by color sets, two designated vertices
s
,
t
∈
V
(
G
), and
k
∈ N, is there an
s
-
t
path in
G
that uses at most
k
colors? If each obstacle is connected, then the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.
We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, including a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms. We also show that our hardness results translate to the geometric instances of the problem.
We then focus on graphs satisfying the color-connectivity property. We design an FPT algorithm for this problem parameterized by both
k
and the treewidth of the graph and extend this result further to obtain an FPT algorithm for the parameterization by both
k
and the length of the path. The latter result implies and explains previous FPT results for various obstacle shapes.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
2 articles.
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