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!