Guarding Lines and 2-Link Polygons is APX-hard

Guarding Lines and

2

-Link Polygons is APX-Hard Björn Brodén

∗

Mikael Hammar

∗

Bengt J. Nilsson

†

Abstra t

Weprovethattheminimumline overingproblemand

the minimum guard overing problem restri ted to

2

-link polygonsareAPX-hard.

keywords: Computational Geometry, Polygon

De- omposition, Art Gallery Theorem, Minimum Guard

Covering,MinimumLineCovering.

1 Introdu tion

Pi ture yourself as the president of the national

mu-seum in your ountry holding invaluable treasures of

historyandart. Howmanysurveillan e ameraswould

you need installed at the museum to make you sleep

tightatnight? Beingas ientistyoumight ometothe

on lusionthatifeverypartofthegalleryisseenbythe

ameras,thiswouldfulll yourneeds. Theansweryou

seekisthesolutiontotheArtGalleryProblem[4,9,10℄:

howmany amerasdoweneedtoguardagivengallery

and how do we de ide where to pla e them? This

problemwasposedbyViktorKleein 1973inresponse

to a questionraisedby Va²ek Chvátal [6℄. The latter

showed in 1975 that

⌊n/3⌋

amerasare su ientand sometimesne essary to guardanygallery represented

asatwodimensionalsimplepolygon[3℄.

The algorithmi version of this problem is to nd

a minimum number of guards for a given art gallery.

Thisis alledtheminimumguard overingproblemand

wasshowntobeNP-hardbyAggarwal[1℄andLeeand

Lin[8℄. Thisisa tuallyapolygonde omposition

prob-lem. The visibility polygon from a point, that is the

areaagivenguard ansee,isstarshaped. Thismakes

theminimumguard overageproblemidenti altothat

of nding a minimum star over of a polygon. The

problem is known to be omputationally di ult [5℄

and it is strongly related to the minimum set over

problem.

In1995Joseph S.B. Mit hell raisedthe questionof

orresponden e between link diameter and

guardabil-ityofsimplepolygons.

1

-linkpolygonsare onvexand hen etriviallyguardableinpolynomialtime. The

orig-∗

DepartmentofComputerS ien e,LundUniversity,Box118,

S-22100Lund,Sweden. e-mail:{bjorn, mikael} s.lth.se

†

S hool of Te hnology and So iety, MalmöUniversity

Col-lege, Citadellsvägen 7, 205 06 Malmö, Sweden. e-mail:

inal NP-hardness proofs [1, 8℄ immediately give

NP-hardnessforpolygonswith linkdiameter

≥ 4

. Nilsson provedtheNP-hardnessfor

≥ 3

-linkpolygons[9℄. This leavesthe omplexityquestionfortheminimumguard

overingproblemfor

2

-linkpolygonsopen. Notethata large lassof

2

-linkpolygonsarestarshapedand there-foreguardableinpolynomialtime. Inthispaperweset

outtoanswerthe omplexity questionfor

2

-link poly-gons. Indoing soweanalyzearelatedproblem alled

the minimumline overingproblem whi h is

interest-ingin itsownright. Wegiveasu in tredu tionfrom

MAX

2

SAT proving that this problem is APX-hard. A similarresult was laimedby Kumaret. al.[7℄ but

our onstru tion givesan expli it lowerbound on the

approximationratio.

From theminimumline overingproblemthere is a

straightforwardgap preserving redu tion to the

mini-mumguard overingproblem,afa tthatMit helland

Kone£nýindependentlyobserved.

2 Problem Formulation

Before we prove our laim we give formal

deni-tions of the Minimum Guard Covering Problem, the

MinimumLineCoveringProblem,and

k

-linkpolygons.

Minimum GuardCovering Problem (MGCP)

Instan e: A polygon

P

.

Solution: A minimum set of points in

P

from whi h theentirepolygon,interiorandboundary anbeseen.

Twopoints

x, y ∈ P

seeea h other ifthestraightline segment

xy ⊂ P

.

TheMinimumLineCoveringProblem(MLCP)

Instan e: A set

L

ofnon-parallellinesin theplane. Solution: A minimumset

P

of pointssu h that there isatleastonepointin

P

onea hlinein

L

.

DenitionApolygon

P

haslinkdiameter

k

if

k

isthe minimum integer value su h that for all points

x, y ∈

P

there exists a path from

x

to

y

that does not ross the boundary of

P

and that onsists of

k

straight line segments. Wesaythat apolygon is

k

linkifithaslink diameter

k

.

The Minimum Guard Covering Problem is trivial for

Covering Problem onstru tsa polygon instan e

hav-ing link diameter

4

[1, 8℄ and this was strengthened to

3

-linkpolygonsby Nilsson [9℄. In the nextse tion weprovethattheMinimumLine CoveringProblemis

APX-hard. This,inturn,leadsustotheAPX-hardness

resultfortheMinimumGuardCoveringProblemin

2

-link polygoninstan es, strengthening theprevious

re-sults.

3 The Redu tion

The redu tion is made from a spe ial ase of MAX

SAT where ea h lause ontainstwoliterals and ea h

literalo ursatmosttwi e.

MAX

2

SAT(

2

L)

Instan e: A set

U

of variables and a olle tion

C

of disjun tive lauses withexa tly

2

literals. Ea h literal anappear at mosttwi e in

C

. A literalis avariable oranegatedvariablein

U

.

Solution: A truth assignment for

U

that satises as many lauses aspossiblein

C

.

This problem is APX-hard,a dire t onsequen eof

thefollowinglemma:

Lemma3.0.1(Berman& Karpinski [2℄) For

ev-ery

ǫ > 0

,itishardtoapproximate

3

-OCC-MAX

2

SAT withinfa tor

2012/2011 − ǫ

.

Here

3

-OCC-MAX

2

SAT is the MAX

2

SAT problem where ea h variable an o urat most three times, a

spe ial aseofMAX

2

SAT(

2

L).

The redu tion is divided into two parts. The rst

partredu es anarbitraryMAX

2

SAT(

2

L)instan e to theminimumline overingproblem(MLCP).The

se -ond part redu es the MLCP to the minimum guard

overing problem for

2

-link polygon instan es. Note that the instan es overed by the MLCP all onsists

of non-parallellines. This is important in the se ond

partof the redu tion, sin e we an onstru t a

2

-link polygonfrom su hanarrangement.

3.1 The Redu tion to MLCP

Let

(U, C)

beaMAX

2

SAT(

2

L)instan e. Westartthe redu tionby reating, forea h variable

x

i

∈ U

, a set

L

i

of eight lines with interse tions asin Fig. 1. Ea h set

L

i

forms a onsisten y gadget and they are built so that no two lines are parallel. Furthermore, there

are no interse tions between three lines ex ept those

expli itlydes ribed. Notethat thelinesin Fig.1that

appeartobeparallelreallyarenot. Inea h

L

i

thelines

a

and

c

represent theliteral

x

i

, and thelines

b, d

the literal

¬x

i

. We allthese linesliterallines.

A B C D a d

Figure1: The onsisten ygadget.

Forea h lause

(l

i

, l

j

) ∈ C

we reate aline, denoted lauselinethatinterse tsallotherlinesinthe

onstru -tion.

Furthermore,forea hliteralinea h lausewe reate

anadditionalline. This linepasses throughthe

inter-se tion point between the lause line and the literal

line of the onsisten y gadget representing the

orre-sponding variable; seeFig.2. Note that thereare two

possible literal lines to hoose from and that a literal

in the MAX

2

SAT(

2

L) instan e ano ur in at most two lauses. Thus, there isalwaysafreeliteralline to

hoosefromatanytimeinthe onstru tion. Notethat

the onstru tion ontainsinterse tionsoftwoandthree

linesonly,andthatnolinesareparallel. The

onstru -tion ontainsatotalof

8|U |

linesin onsisten ygadgets and

3|C|

additionallines forthe lauses, foratotalof

3|C| + 8|U |

lines. Let

L

be the set of lines thus on-stru ted.

Aninterestingset

S

for

L

isamaximalsetofpoints su hthateverypoint

x ∈ S

istheinterse tionofthree lines in

L

and every line in

L

ontains at most one pointfrom

S

. Notethatonevery lauseline thereare two interse tion points that ould be in luded in an

interesting set and in every onsisten y gadget there

are four interse tion points. No other points an be

in luded.

Lemma3.1.1 The size of a solution to the MLCP

onstru tionis

 3|C| + 8|U | − |S|

2



,

where

S

isthemaximuminterestingsetinthesolution.

Proof: We an view the solution to the MLCP as

follows. Every point in the maximum interesting set

x

i

x

j

x

i

¬x

i

¬x

i

¬x

j

x

j

x

j

¬x

j

(¬x

i

, x

j

)

x

i

Figure2: Wherethelinesfrom theliteralsin a lause rossthelineforthat lauseweaddanewline.

an overat mosttwolines ea h. Thetotalnumberof

linesinthe onstru tionis

3|C| + 8|U |

sothesolution's sizeis

 3|C| + 8|U | − 3|S|

2



+ |S| =

 3|C| + 8|U | − |S|

2



.



Weusetheinterestingsettodes ribeatruth

assign-ment to the orresponding MAX

2

SAT(

2

L) instan e and to ount the number of lauses satised by the

assignment. Theinterestingset restri tedto a

onsis-ten y gadgethasasize nolargerthan two;see Fig.1.

Thesemaximalsetsare

{A, C}

and

{B, D}

. Whenthey o urinaninterestingsettheyrepresentthevaluefalse

(

{A, C}

) or true (

{B, D}

) of the orresponding vari-able. Wesaythatthe onsisten ygadgetinthis aseis

seteither tofalseortrue(bytheinterestingset). Itis

importantto noti ethat three-wayinterse tionpoints

indierent onsisten ygadgetsareindependentofea h

other in the sense that all onsisten ygadgets anbe

set arbitrarilyto either trueorfalsewithout violating

the denition of interesting sets. However, they will

put restri tions on therest of the points in the

inter-esting set. These points alllie on lauselines and we

usethemto ountthenumberof lausessatisedbythe

truth assignment. Let us assumethat the interesting

set ontainsades riptionofatruthassignment,thatis,

every onsisten y gadgetis set to somevalue. Bythe

denition of interesting set there an be at most one

point in

S

on ea h lause line. This point represents a literal that satises the lause. It an be in luded

in the interestingset ifand only if the orresponding

onsisten ygadgetissettoavaluethat makesthe

lit-eral satisfy the lause; see Fig. 3. This implies that

the truth assignment des ribed by the interesting set

satises

c

lausesof theinstan eif andonly iftheset ontains

c

pointsfrom the lauselines, i.e. the sizeof theinterestingsetis

2|U |+c

,stillundertheassumption thatevery onsisten ygadget issettosomevalue.

Figure3: Theinterse tionpointonthe lauseline an

bein ludedintheinterestingsetifandonlyifthe

on-sisten ygadgetisset (totruein thisexample).

there is always a maximum interesting set onsistent

withatruthassignmentofthevariables. Ifthisistrue

thenwehaveadire t orresponden ebetweenthesize

of the over and the number of satised lauses.

As-sume that there is no maximum interesting set that

orrespondstoatruthassignment. Considerthe

maxi-muminterestingset,

S

,that ontainsthelargest num-berofpointsfromthe onsisten ygadgets. Thereisat

least one onsisten y gadget

G

that is not set, i.e., it ontainsless than twopointsin

S

. Set thegadget

G

by xing oneor two points appropriately. Let

p ∈ G

be oneof these points. The reasonfor

p

not being in

S

is that the literal line that ontains

p

also ontains a point

q ∈ S

. If we remove

q

from

S

then we an in lude

p

in

S

,therebygettinganewinterestingset

S

′

with

|S| = |S

′

_|

butwithonemorepointfromthe

onsis-ten ygadgets, ontradi tingour hoi eof

S

andhen e ourassumption that there is no maximum interesting

set orrespondingtoatruthassignment.

s ription of the lines is polynomial in the size of the

3

-OCC-MAX

2

SAT instan e. To see this we give a des ription of how the lines are pla ed in the plane.

Firstthe onsisten ygadgetsarepla edevenlyspa ed

asin Figure2. Next werotate therstgadget

45/|U |

degrees, the se ond one twi e as mu h, the third one

three timesasmu h and soforth. Sin etheangle

be-tweentwolinesinthe onsisten ygadgetisnolessthan

45

degreesthismeansthatalllinesinterse t. Wepla e the

|C|

lauselinesinsu hawaythat notwolinesare parallel. This an bedone easily. Now, we pla e the

extralinesattheinterse tionpointsbetweenthe lause

linesandtherelevantliterallines. This learlygivesa

polynomial des ription of the lines. We have proved

thefollowingtheorem.

Theorem3.1.2 Forevery

δ > 0

,itishardto approx-imateMLCP within fa tor

28169/28168 + δ

.

Proof: Given an instan e to MAX

2

SAT(

2

L), let

c

denotethemaximumnumberof lausessatisable. We

an bound the number of variables to

|U | ≤ 2|C|

, sin e ea h lause ontains

2

literals. From Lem-mas3.0.1, 3.1.1, and thefollowingdis ussion we infer

thattheinapproximabilityratiois

l

_{3|C|+6|U|−2011c/2012+ǫ}

2

m

l

_{3|C|+6|U|−c}

2

m

≥

28169/28168 + δ,

sin e theworst ase o urs when

c = |C|

. The

δ

de-pends on

ǫ

.



3.2 The Redu tion to MGCP

In the se ond part of the redu tion we are given an

arrangementofnon-parallellines. A re tangle

R

su- iently large to ontainall interse tions between lines

in

L

is reated. Consideraline segmentin

R ∩ L

. At oneendpointweputaspikeinthere tangle;seeFig4.

The ones visiblefrom anytripletof spikesshould

in-terse t if and only if the orresponding line segments

ross. If this riterion is met then the interse tions

betweenline segmentswill orrespondto areasin this

polygon. Thus,fromaguardsetinthepolygonweget

apoint set in the MLCP. That is, ifwe an solve the

minimumguard overingproblemthenweimmediately

getasolutiontotheMLCP.

Theorem3.2.1 The minimum guard overing

prob-lemisAPX-hard.

4 Con lusion

Wehaveprovedthattheminimumguard overing

prob-lem restri tedto

2

-linkpolygonsis APX-hard. In ad-dition wegive anexpli itlowerbound onthe

approx-Figure4:Thesetoflinesareembeddedinsidea

re tan-gle.Wheretheresultinglinesegmentsmeetthe

bound-arywe reateanarrowspike.

A knowledgments We would like to thank Piotr

Bermanforprovidingus withvaluableinformationon

the

3

-OCC-MAX

2

SATproblem.

Referen es

[1℄ A. Aggarwal. The Art Gallery Theorem: Its

Varia-tions,Appli ationsandAlgorithmi Aspe ts. PhD

the-sis,JohnsHopkinsUniversity,1984.

[2℄ P.Bermanand M. Karpinski. Onsome tighter

inap-proximability results. In Pro . of the 26th

Interna-tional Colloquium on Automata, Languages and

Pro-gramming,pages200209,1999.

[3℄ V.Chvátal. A ombinatorialtheoreminplane

geome-try.J.Combin.Theory Ser.B,18:3941, 1975.

[4℄ M. de Berg, M. van Kreveld, M. Overmars, and

O.S hwarzkopf. Computational Geometry. Springer

Verlag,2000.

[5℄ S.Eidenbenz. Inapproximability results for guarding

polygons without holes. In Pro . 9th Annual

Inter-nationalSymposiumonAlgorithms andComputation,

pages427436,1998.

[6℄ R.Honsberger. Mathemati la GemsII. Mathemati al

Asso iationofAmeri a,1976(104110).

[7℄ V.S.A. Kumar, S.Arya,and H.Ramesh. Hardness

ofset overwithinterse tion

1

. InPro .27th Interna-tionalColloquium,ICALP,pages624635,2000.

[8℄ D.T.LeeandA.K.Lin. Computational omplexityof

artgalleryproblems. IEEETransa tions on

Informa-tionTheory,IT-32:276282,1986.

[9℄ B.J. Nilsson. Guarding Art Galleries  Methods for

MobileGuards. PhDthesis,LundUniversity,1995.

[10℄ J.O'Rourke. Art Gallery Theorems and Algorithms.