ptas via local search
DESCRIPTION
PTAS via Local Search. Rom Aschner and Or Caspi. Outline. Today you will see PTAS for: Geometric Minimum Hitting Set Terrain Guarding Maximum Independent Set of Pseudo Disks Minimum Dominating Set of Disk Graphs - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/1.jpg)
PTAS VIA LOCAL SEARCHRom Aschner and Or Caspi
![Page 2: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/2.jpg)
Outline Today you will see PTAS for:
Geometric Minimum Hitting Set Terrain Guarding Maximum Independent Set of Pseudo Disks Minimum Dominating Set of Disk Graphs
Using the technique discovered independently by Har-Peled & Chan (2009) and Mustafa & Ray (2009)
![Page 3: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/3.jpg)
Mustafa and Ray, 2009
Minimum hitting set problem
![Page 4: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/4.jpg)
Minimum hitting set problem Given a range space , The goal: compute the smallest that
has non empty intersection with each NP-Hard to approximate within a factor of
![Page 5: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/5.jpg)
Minimum hitting set problem For many geometric range spaces the
problem remains NP-hard. For example:
P – set of points, D – set of unit disks
![Page 6: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/6.jpg)
Minimum hitting set problem For many geometric range spaces the
problem remains NP-hard. For example:
P – set of points, D – set of disks
![Page 7: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/7.jpg)
Minimum hitting set problem For many geometric range spaces the
problem remains NP-hard. For example:
P – set of points, H – set of half-spaces in .
![Page 8: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/8.jpg)
Minimum hitting set problem
PTAS (Mustafa & Ray, 2009)
![Page 9: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/9.jpg)
The -level local search algorithm S While there is a subset of size that can
be replaced with subset of size s.t. is a hitting set. Then preform the swap Else, halt.
Example:
![Page 10: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/10.jpg)
The -level local search algorithm Running Time:
The number of iterations is bounded by . There are at most different local
improvements to verify. Checking whether a certain local
improvement is possible takes . The overall running time is
![Page 11: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/11.jpg)
- the set of points in the optimal solution - the set of points in the algorithm’s
output Assume:
We want to prove that
Approximation Analysis
![Page 12: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/12.jpg)
𝑅∩𝐵≠∅ We will see that Thus
![Page 13: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/13.jpg)
satisfies the locality condition if for any two disjoint subsets , it is possible to construct a planar bipartite graph s.t. for any , if and , then there exists two vertices and such that .
Locality condition
![Page 14: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/14.jpg)
The planar bipartite graph is the Delaunay Triangulation of without monochromatic edges.
Locality condition in disks
![Page 15: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/15.jpg)
For each disk , if and , then there exists two vertices and such that .
Locality condition in our graph
The locality condition holds!!
![Page 16: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/16.jpg)
Lemma: for any , is a hitting set of Proof:
If is only hit by the blue points in then one of them has red neighbor that hits
Otherwise, is hit by some point in
Locality condition
![Page 17: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/17.jpg)
- the set of points in the optimal solution - the set of points in the algorithm’s
output satisfies the locality condition We want to prove that The proof is based on the “Separation
lemma” (Fredrickson 1987)
Approximation Analysis
![Page 18: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/18.jpg)
Separation lemma (Fredrickson 1987): For any planar graph and a parameter , we can find a set of size at most and partition of into sets satisfying:
for
Separation lemma
𝑺
![Page 19: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/19.jpg)
Example : Then, we can find a set of size at most and a
partition of into sets satisfying:
for
Separation lemma
𝑂 (𝑛)
𝑂 (𝑛)
𝑂 (√𝑛)
![Page 20: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/20.jpg)
How can this separation lemma help us ?? We have a planar graph of This graph can be separated into disjoint subsets Next, we will see that by choosing the correct
value of the number of blue points inside each subset is not too big than the number of red points.
Using this observation, we will get that |
Approximation Analysis
![Page 21: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/21.jpg)
Let and Applying we have:
Assume, replace with – contradiction.
Approximation AnalysisFrom the locality condition lemma:If is a hitting set of X then is also a hitting set of X
![Page 22: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/22.jpg)
Therefore,
, and large enough constant
Approximation Analysis
![Page 23: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/23.jpg)
-approximation We proved the following theorem:
Let be a set of points and a set of disks. Then a -level local search algorithm returns a hitting set of size at most in Time.
True also for different regions, such as: Same height rectangles Translates of convex shapes …
![Page 24: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/24.jpg)
Gibson, Kanade, Krohn and Varadarajan, 2009
Terrain Guarding
![Page 25: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/25.jpg)
Terrain A polygonal chain in the plane that is x-
monotone.
Terrain Guarding
![Page 26: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/26.jpg)
G: Possible Guard locations X: Target points that need to be guarded
Terrain Guarding
![Page 27: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/27.jpg)
Objective: find smallest subset of guards such that every point is seen by at least one guard
Terrain Guarding
![Page 28: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/28.jpg)
Terrain Guarding Previous results:
NP-Hard 4-approximation
Applications: Placing guards/cameras along borders Constructing line-of-sight networks for radio
broadcasting Placing street lights along roads Placing fire trucks on the Carmel mountain …
![Page 29: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/29.jpg)
The -level local search algorithm
While there is a subset of size that can be replaced with subset of size s.t. guards . Then preform the swap Else, halt.
![Page 30: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/30.jpg)
Terrain Guarding via local search - the set of guards in the optimal solution - the set of guards in the algorithm’s output We can assume that We want to prove that the locality condition
holds. This means we need to find a planar bipartite
graph in which for each , there is an edge between guards and that both see .
As before, using the separation lemma on this graph will show that
![Page 31: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/31.jpg)
- the leftmost guard that sees among points in
For every guard, we shoot a ray upwards. Let be the first segment in A1 that it hits.
Construction 1
x1
x2
x3
v1
v2
v3
v4 v5
![Page 32: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/32.jpg)
Construction 1 Why are there no crossing edges? Thanks to the ‘order claim’:
Let be four points on the terrain in increasing order according to -coordinate. If sees and sees
Then sees .A
B
C
D
![Page 33: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/33.jpg)
Construction 2 - The flip Create and in the same way, using the rightmost guards What if the new edges cross the previous ones?
x1
x2
x3
v1
v2v3
v4 v5
![Page 34: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/34.jpg)
Construction 3 Finally, for every point x, add an edge in if they are of
opposite colors. Embed this edge along and to remain planar. The final graph is , planar bipartite.
x1
x2
x3
v1
v2
v3
v4 v5
![Page 35: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/35.jpg)
Locality condition Still need to show that the locality
condition holds: For every point there are guards and that both see and they are connected in G.
If and are of opposite colors, we are done because they are connected in .𝝀 (𝒙)
𝝆 (𝒙)
x
![Page 36: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/36.jpg)
Locality condition Otherwise, assume w.l.g. there are only guards to
the left of , and that is red. Since both and guard , there is also a blue
guard that sees , call the leftmost one . Because is between and , is above . If it also the first such segment, then
x𝜆 (𝑥)
b
![Page 37: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/37.jpg)
Locality condition Otherwise, let be the first segment in above . From the order claim on sees . From the choice of as the leftmost blue guard
that sees , is red!
x
y
𝜆 (𝑥)
b
𝜆 ( 𝑦 )
![Page 38: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/38.jpg)
Chan & Har-Peled, 2009
Independent Set
![Page 39: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/39.jpg)
A set of objects is a collection of pseudo-disks, if the boundary of every pair of them intersects at most twice.
Max Independent Set of Pseudo Disks
![Page 40: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/40.jpg)
Intersection graph – there is an edge between two pseudo-disks if they intersect
Max Independent Set of Pseudo Disks
![Page 41: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/41.jpg)
Max independent set– no pair of objects intersect
Max Independent Set of Pseudo Disks
![Page 42: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/42.jpg)
The bipartite graph is simply the Intersection graph of
Is it planar ? Yes Embed the edges
along the intersections Locality condition ? Yes
If any two red-blue pseudo disks intersect then
Max Independent Set of Pseudo Disks
![Page 43: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/43.jpg)
Gibson & Pirwani, 2010
Dominating Set
![Page 44: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/44.jpg)
Min. Dominating Set of Disk Graphs
Intersection graph – there is an edge between to points if their disks intersect
![Page 45: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/45.jpg)
Min. Dominating Set of Disk Graphs
Minimum Dominating Set - the smallest subset s.t. each vertex is either in or is adjacent to vertex in
![Page 46: PTAS via Local Search](https://reader038.vdocument.in/reader038/viewer/2022103102/56816945550346895de0cf87/html5/thumbnails/46.jpg)
Probably yes….
More ?