algorithms for geometric covering and piercing problems robert fraser phd defence nov. 23, 2012
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Algorithms for Geometric Covering and Piercing Problems
Robert Fraser
PhD defence Nov. 23, 2012
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VignettesDiscrete Unit Disk Cover Problem
◦Various settings ◦Strip-based decomposition of the plane◦Hardness of problem even in a strip
Hausdorff Core Problem◦General simple polygons◦Polygons with a single reflex vertex
MST with Neighborhoods Problem◦Parameterized algorithms◦Hardness of problem
The Discrete Unit Disk Cover (DUDC) Problem
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unit disks with centrepoints , points . Select a minimum subset of which
covers .
Applications
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Line-separated DUDCand are separable by .Exact time algorithm.
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d1
d2
d4d3
d5
q1
q2
q4q3
q5
p1
p2 p4
p3 p5p6
p7
p9
p8
ℓ
Strip-Separated DUDC
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p1p2
p4
p3
p5
q1 q2q4
q3
q5 q6 q7q9
q8
ℓ2
ℓ1
All points contained in strip between lines , and disks centred outside strip.
Exact time algorithm.
[Ambühl et al. , 2006]
Assisted LSDUDC on one side of , above and below . All points in covered by Want of minimum cardinality.Best known is 2-approximation. Hardness open.
u1 u2 u4u3
u5
q1
q2
q4q3
q5
p1
p2 p4
p3 p5p6
p7
p9
p8
ℓ
l1 l2
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DUDC algorithm: work from top to bottom, then bottom to top, then cover what remains.
Uses assisted LSDUDC and WSDUDC algorithms.
Strip-Based Decomposition
Within-Strip DUDC
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unit disks with centre points , points . Strip , defined by and , of height which
contains and .Select a minimum subset of which
covers .
𝑠
h}ℓ2
ℓ1
Within-Strip DUDC If easy, implies a simple PTAS for DUDC based on shifting grids… MAX independent set of unit disk graph is easy in strips of fixed
height! Within-Strip Discrete Unit Square Cover is easy in strips of fixed
height!
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p1p2
p4
p3
p5
q1
q2q4
q3
q5
q6
q7
q9q8
ℓ2
ℓ1
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Hardness of WSDUDC (1/2) Reduce from Vertex Cover on
planar graphs of max degree 3.
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Hardness of WSDUDC (2/2)Each edge has even number of
added points:
Convert to instance of WSDUDP:
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-Approximate WSDUDC Algorithm
Find the rectangle of height circumscribed by each disk, then take the intersection with the strip:
Rectangles define intervals, the remainder are gaps.
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Covering GapsAll disk centres are
outside of the gap. Instance of SSDUDC, so
use time algorithm to cover gap optimally.
Optimality is lost when combining solutions from multiple gaps.
p1p2
p4
p3
p5
q1 q2q4
q3
q5 q6 q7q9
q8
ℓ2
ℓ1
ℓ2
ℓ1
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3-Approximate AlgorithmConsider all single disks, as well
as pairs and triples.Run a dynamic program from left
to right on the strip.Runs in time.Each gap & interval is covered
optimally!
Summary of DUDC Results
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LSDUDC algorithm, time.2-approximate assisted LSDUDC
algorithm.Strip-based decomposition providing
DUDC algorithms as follows:◦ 18-approximate, time◦ 16-approximate, time◦ 15-approximate, time
WSDUDC is NP-hard.
Trade-off in DUDC Algorithms
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The Hausdorff Core ProblemGiven a simple polygon P, a Hausdorff
Core of P is a convex polygon Q contained in P that minimizes the Hausdorff distance between P and Q.
We denote the Hausdorff core as .
Does there exist a convex polygon Q contained in P such that ?
Note: the 1-centre of a polygon P is the point c which minimizes .
Shrinking Disks
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Why is this challenging?
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Hausdorff Core Approximation Algorithm for Simple PolygonsApproximation scheme based upon a
discretization of the search space.
Intuition:1. Divide the disks into discrete
segments.2. Grow the disks slightly such
that at least one segment of each expanded disk will be contained in the optimal solution.
3. Check for a solution among all intervals on all disks.
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A Simpler ProblemWhat if the polygon that we want
to approximate has only one reflex vertex?
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One Vertex SolutionFind angle where max distance is
minimized.
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Summary of Hausdorff Core ResultsFPTAS for Hausdorff core on
simple polygons:◦O((n3+n2ε-6)∙log(ε-1)) time.
O(n3) time exact algorithm for polygons containing a single reflex vertex.
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The (Minimum Weight) MST with Neighborhoods Problem
...
..
..
.
.
..
..
. ..MSTN
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The Maximum Weight MST with Neighborhoods Problem
..
..
..
.
max-MSTN
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Parameterized Algorithms
= separability of the instance◦min distance between any two
disks
𝑟𝑚
𝑘=0.25
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Parameterized max-MSTN Algorithm
.
. .. .
...
– factor approximation by choosing disk centres
..
. ..
... .
. ..
....
Topt Tc Tc’
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Parameterized max-MSTN Algorithm
.
. .. .
...
– factor approximation by choosing disk centres
..
. ..
... .
. ..
....
Topt Tc Tc’
Consider this edge
𝑟 𝑖
𝑟 𝑗
weight = weight
𝑑+𝑟 𝑖+𝑟 𝑗
𝑑+2𝑟 𝑖+2𝑟 𝑗
≥𝑘𝑟𝑚+𝑟 𝑖+𝑟 𝑗
𝑘𝑟𝑚+2𝑟 𝑖+2𝑟 𝑗
≥𝑘𝑟𝑚+𝑟𝑚+𝑟𝑚𝑘𝑟𝑚+2𝑟𝑚+2𝑟𝑚
¿𝑘+2𝑘+4
¿1−2
𝑘+4
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Hardness of MSTN
Reduce from planar 3-SAT
clause
variable
clause
clause
clause
variable
variable
variable
variable
(with spinal path)
Create instance of MSTN (resp. max-MSTN) so that:- Weight of the optimal
solution may be precomputed for any instance;
- Weight of solution corresponding to a non-satisfiable instance is greater than (resp. less than) the optimal solution by a significant amount.
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Summary of MST ResultsMSTN
◦NP-hard◦ – factor approximation
max-MSTN◦NP-hard◦ – factor approximation
2-GMST◦NP-hard in 2 dimensions◦Exact solution if topology is known
.
.
.. .
.
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Nice Results and Open Problems Best known fast DUDC approximation algorithm. Hardness of WSDUDC. OPEN Is assisted LSDUDC NP-hard? OPEN Is there a nice PTAS for WSDUDC?
First known solutions for the Hausdorff Core problem. OPEN Are there classes of polygons for which finding the
Hausdorff Core is easy?
Best known results for MSTN. First results for max-MSTN. Hardness of 2-GMST. OPEN Is there a PTAS for max-MSTN (even on unit disks)? OPEN Is there a combinatorial approximation algorithm
for 2-GMST (particularly with an approximation factor of <4)?
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Thanks!