PTAS VIA LOCAL SEARCHRom 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

Using the technique discovered independently by Har-Peled & Chan (2009) and Mustafa & Ray (2009)

Mustafa and Ray, 2009

Minimum hitting set problem

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

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

Minimum hitting set problem For many geometric range spaces the

problem remains NP-hard. For example:

P – set of points, D – set of disks

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 .

Minimum hitting set problem

PTAS (Mustafa & Ray, 2009)

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:

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

- 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

𝑅∩𝐵≠∅ We will see that Thus

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

The planar bipartite graph is the Delaunay Triangulation of without monochromatic edges.

Locality condition in disks

For each disk , if and , then there exists two vertices and such that .

Locality condition in our graph

The locality condition holds!!

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

- 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

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

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)

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

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

Therefore,

, and large enough constant

Approximation Analysis

-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 …

Gibson, Kanade, Krohn and Varadarajan, 2009

Terrain Guarding

Terrain A polygonal chain in the plane that is x-

monotone.

Terrain Guarding

G: Possible Guard locations X: Target points that need to be guarded

Terrain Guarding

Objective: find smallest subset of guards such that every point is seen by at least one guard

Terrain Guarding

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 …

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.

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

- 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

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

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

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

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

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

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 )

Chan & Har-Peled, 2009

Independent Set

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

Intersection graph – there is an edge between two pseudo-disks if they intersect

Max Independent Set of Pseudo Disks

Max independent set– no pair of objects intersect

Max Independent Set of Pseudo Disks

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

Gibson & Pirwani, 2010

Dominating Set

Min. Dominating Set of Disk Graphs

Intersection graph – there is an edge between to points if their disks intersect

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

Probably yes….

More ?