Introduction References

On The Watchman Route Problem and Its RelatedProblems

Dissertation Proposal

Ning Xu

The Graduate Center, The City University of New York

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Introduction References

Outlines

Introduction

The Watchman Route Problem (WRP)

The Touring Polygons Problem (TPP)

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Introduction References

Scenario: Night Watchman

Suppose a night watchman (a person or a robot) is required toguard a building.FThe night watchman has sensors (eyes or cameras).FIn the ideal case, the watchman can see infinite far away.

Figure: A night watchman in a building

Can we find a shortest closed path for the watchman toguard everywhere in the building?

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Introduction References

Visibility Region

VisibilityLet P be a polygon. Two points x , y ∈ P are visible fromeach other if the line segment xy lies within P; i.e., xy ⊂ P 1.

Visibility RegionThe visibility region of a point x ∈ P is the set of pointswithin P visible from x .

Figure: Visibility region: the yellow region (image taken from Wikipedia[1]).

1Here we assume the unlimited visibility4 / 27

Introduction References

Watchman Route Problem (WRP)

Watchman routeA watchman route T is a closed path within P so that everypoint in P is visible from some point on T ; i.e.,∀x ∈ P,∃y ∈ T : pq ∈ P.

The watchman route problem (WRP)The watchman route problem asks a shortest watchman routefor a known polygon P.

Figure: A shortest watchman route (image taken from [5])

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Introduction References

Related Problems: The Art Gallery Problem

The WRP was first introduced by Chin and Ntafos in 1986[6, 7], as a variation of the art gallery problem.

Art Gallery ProblemThe art gallery problem asks the minimum number ofstationary guards to watch over all paintings in a gallery of nwalls.

Figure: Four cameras cover thegallery (from Wikipedia [1])

Fdn3e cameras are sufficient and

sometimes neccessary (Chvatal 1975

[8]).

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Introduction References

Related Problems: The Euclidean TSP withNeighborhoods (TSPN)

Let R1,R2, . . . ,Rk be k geometric objects (maybe overlapped) of ntotal vertices in a plane. The TSPN asks a shortest closed pathvisiting all objects.

NP-hard (Papadimitriou 1977[19])

APX-hard (Gudmundsson and Levcopoulos 2000 [15])

O(log n) approximation (Mata and Mitchell 1995 [16], Elbassioni et al.

2006 [13], Gudmundsson and Levcopoulos 1999 [14])

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Introduction References

Related Problems: The Euclidean Traveling SalesmanProblem (ETSP)

Let p1, p2, . . . , pk be k points in a plane. The ETSP asks ashortest closed path visiting all points.

NP-hard (Papadimitriou 1977 [19])

PTAS (Arora 1997 [3], Mitchell 1999 [17], Rao and Smith 1998 [20])

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Introduction References

Outlines

Introduction

The Watchman Route Problem (WRP)

The Touring Polygons Problem (TPP)

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Introduction References

The WRP: Three Different Settings

On simple polygons (polygon without holes)

On polygonal domains (polygon with holes)

The WRP on arrangement of connected geometric objects.For example, arrangement of line segments, lines, and etc..

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Introduction References

The WRP on Simple Polygons

Two cases:

The fixed WRP, or the anchored WRPThe watchman route must pass a given point s, called theanchored point.The floating WRPNo such an anchor point is required, i.e., the general case.

s

(a) The fixed WRP (b) The floating WRP

Figure: The WRP on simple polygons: the red paths are theshortest watchman route.

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Introduction References

Exact Algorithms of WRP on Simple Polygons

Fixed WRP: O(n3 log n) time (Dror et al. 2003 [10])

Floating WRP: O(n4 log n) time (Dror et al. 2003 [10])

F An algorithm for the fixed WRP with running time O(T )admits an algorithm for the floating WRP with running timeO(nT ) (Tan 2001 [21]).

Problem to Investigate 1

Can the fixed WRP be solved in less than O(n3 log n) time?

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Introduction References

Approximation Algorithms of WRP on Simple Polygons

Fixed WRP:√

2 ratio, O(n) time (Tab 2004 [23])

Floating WRP: 2 ratio, O(n) time (Tan 2007 [25])

Problem to Investigate 2

Does the fixed WRP has an approximation algorithm betterthan

√2 ratio?

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Introduction References

The WRP on Polygonal Domains

NP-hard (Chin and Ntafos 1986 citeCN86 1988 [7], Dumitrescu and Toth

[12])

O(log2 n) approximation (Mitchell 2013 [18])

Impossible better than O(log n) ratio unless P 6= NP (Mitchell

2013[18])

Problem to Investigate 3

Can the WRP on polygonal domain be approximated betterthan O(log2 n) ratio?

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Introduction References

The WRP on Arrangement of Connected Line Segments

Figure: The WRP on arrangement of line segments: the red paths arethe shortest watchman route.

A special case of the WRP on polygonal domain, where thepolygon collapses into an arrangement of line segments.

NP-hard (Xu 2012 [27])

O(log3 n) approximation (Dumitrescu, Mitchell and Zylinski [11])

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Introduction References

The WRP on Arrangement of Connected Line Segments

Figure: The WRP on simple polygons: the red path shows the shortestwatchman route.

Problem to Investigate 4

What is the lower bound of the approximation ratio for theWRP on arrangement of line segments?

Problem to Investigate 5

Does the WRP on arrangement of axis-parallel line segmentsadmit a constant factor approximation algorithm, or even aPTAS?

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Introduction References

The WRP on Arrangement of Lines

Can be solved in O(n8) time (Dumitrescu, Mitchell and Zylinski [11])

NP-hard in 3D space (Dumitrescu, Mitchell and Zylinski [11])

Figure: A watchman route (in bold) on arrangement of lines (imagetaken from [11]).

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Introduction References

Outlines

Introduction

The Watchman Route Problem (WRP)

The Touring Polygons Problem (TPP)

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Introduction References

The Touring Polygons Problem (TPP)

Let P1, . . . ,Pk be k polygons of n totoal vertices in the plane,possibly intersecting. Let s and t be two points in the plane. TheTPP asks a shortest path from s to t visiting P1, . . . ,Pk in order,possibly subject to fence constrainsFFences Simple polygons F0, . . . ,Fk so that Pi ∪ Pi+1 ⊆ Fi .

stP

P

P

F

F

F

F1

2

3

0

1

2

3

F If there is no fence, the TPP is unconstrained.

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Introduction References

Results on the Unconstrained TPP

Convex polygons: O(kn log(n/k)) time (Dror et al. 2003 [10])

Non-convex polygons: NP-hard (Ahadi, Mozafari, Zarei [2])

Non-convex polygons but the path cannot enter polygons:O(k2n log n) time (Ahadi, Mozafari, Zarei [2])

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Introduction References

Results on Constrained TPP

NP-hard (Dror et al. 2003 [10])

the part of Pi ’s boundary from which shortest routes maybounce is convex: O(kn2 log n) time (Dror et al. 2003 [10])

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Introduction References

The Zookeeper’s Problem

Given a simple polygon P and a set S of convex disjoint convexpolygons inside P called cages, each sharing one edge of P,compute a shortest closed path within P that visits every cage butdoes not enter the interior of any cage.FThe fixed zookeeper’s problem The route are required to passthrough a given point s ∈ ∂P

s

(a) The fixed zookeeper’s prob-lem

(b) The general zookeeper’sproblem

Figure: The zookeeper’s problem

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Introduction References

Exact Algorithms of the Zookeeper’s Problem

The fixed case: O(n log n) time (Bespamyatnikh 2003 [4])

The general case: O(n2) time (Tan 2001 [22])

Problem to Investigate 6

Can the fixed zookeeper’s problem be solved in O(n) time?

Problem to Investigate 7

Does the zookeeper’s problem admits a sub-quadratic timealgorithm, for example, O(n log2 n) time as conjectured byTan [22]?

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Introduction References

Approximation Algorithms of the Zookeeper’s Problem

The fixed case:√

2 ratio (Tan 2004 [23])

The general case: 2 ratio (Tan 2006 [24])

Problem to Investigate 8

Can we find an algorithm for the fixed zookeeper’s problembetter than

√2 ratio?

Problem to Investigate 9

Can we find an approximation algorithm for the zookeeper’sproblem better than 2 ratio?

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Introduction References

The Safari Problem

Given a simple polygon P and a set S of convex disjoint convexpolygons inside P called cages, each sharing one edge of P,compute a shortest closed path within P that visits every cage.FIn the safari problem, the route can enter cagesFThe fixed safari problem The route are required to passthrough a given point s ∈ ∂P.

s

(a) The fixed safari problem (b) The general safari problem

Figure: The safari problem

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Introduction References

Results on the safari Problem

The fixed case: O(kn log n) time (Dror et al. 2003 [10])

The general case: O(kn2 log n) time (Dror et al. 2003 [10])

F An algorithm for the fixed safari problem with running timeO(T ) admits an algorithm for the general safari problem withrunning time O(nT ) (Tan and Hirata 2003 [26]).

Problem to Investigate 10

Can we improve the current best result, O(kn log n), for thefixed safari problem?

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Introduction References

The Aquarium-keeper’s Problem

Given a simple polygon P, find a shortest close path that visitsevery edge of P.

Figure: The aquarium-keeper’s problem

F Optimal algorithm: O(n) time (Czyzowicz et al. 1991 [9])

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Introduction References

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