Loading…
A Colored Path Problem and Its Applications
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 th...
Saved in:
Published in: | ACM transactions on algorithms 2020-09, Vol.16 (4), p.1-48 |
---|---|
Main Authors: | , |
Format: | Article |
Language: | English |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Summary: | 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. |
---|---|
ISSN: | 1549-6325 1549-6333 |
DOI: | 10.1145/3396573 |