Steiner Tree Approximation via Iterative Randomized Rounding

JAROS LAWBYRKA

EPFL,Lausanne,SwitzerlandandUniversityofWro law,Poland

and

FABRIZIOGRANDONI

UniversityofRomeTorVergata, Roma,Italy

and

THOMASROTHVO

EPFL,Lausanne,Switzerland

and

LAURASANIT 

A

EPFL,Lausanne,Switzerland

TheSteinertreeproblemisoneofthe mostfundamentalNP -hardproblems: givenaweighted

undire tedgraphandasubsetofterminalnodes,ndaminimum- osttreespanningtheterminals.

Inasequen eofpapers,the approximationratioforthisproblemwasimprovedfrom2to1:55

[Robins,Zelikovsky-'05℄. All these algorithmsare purely ombinatorial. Along-standing open

problemiswhetherthereisanLPrelaxationofSteinertreewithintegralitygapsmallerthan 2

[Vazirani,Rajagopalan-'99℄.

Inthispaperwepresentan LP-basedapproximationalgorithmforSteinertree withan

im-provedapproximationfa tor.Ouralgorithmisbasedona,seeminglynovel,iterativerandomized

rounding te hnique. We onsideranLPrelaxationofthe problem,whi hisbasedonthenotion

ofdire ted omponents.Wesampleone omponentwithprobabilityproportionaltothevalueof

theasso iatedvariableinafra tionalsolution: thesampled omponentis ontra tedandtheLP

isupdated onsequently. Weiteratethispro essuntilallterminalsare onne ted. Ouralgorithm

deliversasolutionof ostatmostln(4)+"<1:39timesthe ostofanoptimalSteinertree.The

algorithm anbederandomizedusingthemethodoflimitedindependen e.

Asabyprodu t ofour analysis,weshowthat theintegralitygapof ourLPisat most1:55,

hen eansweringtothementionedopenquestion. Thismighthave onsequen esforanumberof

relatedproblems.

Categories and Subje t Des riptors: F.2.2 [Computations on dis rete stru tures℄:

Non-numeri alAlgorithmsandProblems

GeneralTerms: Algorithms,Theory

AdditionalKeyWordsandPhrases: Approximationalgorithms,linearprogrammingrelaxations,

networkdesign,randomizedalgorithms

1. INTRODUCTION

Givenanundire tedn-nodegraphG=(V;E),withedge osts(orweights) :E!

Q

+

, and asubset of nodes R V (terminals), theSteiner tree problem asks for

atree S spanning the terminals, of minimum ost (S) := P

e2S

(e). Note that

S might ontainsomeothernodes, besidestheterminals(Steiner nodes). Steiner

1

A preliminary version of this paper appeared in STOC'10 [Byrka et al. 2010℄.

Emails: J. Byrka jbyii.uni.wro .pl, F.Grandoni grandonidisp.uniroma2.i t, T. Rothvo

tree isoneof the lassi and, probably,most fundamental problemsin Computer

S ien e and Operations Resear h, with great theoreti al and pra ti alrelevan e.

Thisproblememergesinanumberof ontexts,su hasthedesignofVLSI,opti al

and wireless ommuni ation systems, as well as transportation and distribution

networks(see,e.g.,[Hwanget al.1992℄).

The Steiner tree problem appears already in the list of NP-hard problems in

thebook byGareyandJohnson[1979℄. Infa t,itis NP-hardtondsolutionsof

ost lessthan 96

95

timesthe optimal ost [Bernand Plassmann 1989; Chlebk and

Chlebkova2008℄. Hen e,thebestone anhopeforisanapproximationalgorithm

withasmallbut onstantapproximationguarantee.

Withoutlossofgenerality,we anrepla etheinputgraphbyitsmetri losure 2

.

AterminalspanningtreeisaSteinertreewithoutSteinernodes: su hatreealways

exists in the metri losure of the graph. It is well-known that aminimum- ost

terminalspanning tree isa 2-approximationfor theSteinertree problem [Gilbert

andPollak1968;Vazirani2001℄.

A sequen e of improved approximation algorithms appeared in the literature

[KarpinskiandZelikovsky1997; Promeland Steger2000; Zelikovsky1993℄,

ulmi-nating with the famous 1+ ln(3)

2

+" <1:55 approximation algorithm by Robins

and Zelikovsky [2005℄. (Here " > 0 is an arbitrarily small onstant). All these

improvementsarebasedonthenotionofk-restri tedSteiner tree,whi hisdened

asfollows. A omponent isatreewhoseleaves oin idewithasubsetofterminals.

A k-restri tedSteinertree S is a olle tionof omponents,with atmostk

termi-nals ea h (k- omponents), whose union indu es aSteiner tree. The ost of S is

thetotal ostofits omponents, ountingdupli atededges withtheirmultipli ity

(see [Bor hersand Du 1997℄ formore details). Thek-Steinerratio 

k

1is the

supremumoftheratiobetweenthe ostopt

k

oftheoptimalk-restri tedSteinertree

andthe ostoptoftheoptimal(unrestri ted)Steinertree. Thefollowingresultby

Bor hers and Du[1997℄showsthat, in order to have agood approximation,it is

suÆ ientto onsiderk-restri tedSteinertreesforalargeenough, onstantk.

Theorem 1. [Bor hers and Du 1997℄ Let r and s be the non-negative integers

satisfyingk=2 r +sands<2 r . Then  k = (r+1)2 r +s r2 r +s 1+ 1 blog 2 k :

The mentioned approximation algorithms exploit the notion of k- omponent

within alo al-sear h framework. Theystart with aminimum- ostterminal

span-ning tree (whi h is 2-approximate), and iterativelyimprove it. At ea h iteration,

theyaddto the urrentsolutionak- omponent, hosena ordingto somegreedy

strategy, and remove redundant edges. The pro ess is iterated until no further

improvementisa hievable. Dierentalgorithmsusedierentgreedy riteria.

Despitetheeortsofmanyresear hersinthelast10years,theaboveframework

didnotprovideanyfurtherimprovementafter[RobinsandZelikovsky2000;2005℄.

This motivated our sear h for alternativemethods. Onestandard approa h is to

2

Themetri losureofaweightedgraphisa ompleteweightedgraphonthesamenodeset,with

exploitaproperLPrelaxation(see,e.g.,[GoemansandMyung1993℄foralistofLP

relaxationsofSteinertree). Anaturalformulationfortheproblemistheundire ted

ut formulation (see [Goemans and Williamson 1995; Vazirani 2001℄). Here we

haveavariableforea hedgeofthegraphanda onstraintforea h utseparating

theset of terminals. Ea h onstraintfor es to pi kat leastone edge rossingthe

orresponding ut. ConsideringitsLPrelaxation,2-approximationalgorithms an

beobtainedeither usingprimal-dual s hemes[Goemans and Williamson1995℄ or

iterativerounding[Jain1998℄. However,thisrelaxationhasanintegralitygap of2

alreadyinthespanningtree ase,i.e.,whenR=V (seeexample22.10in[Vazirani

2001℄).

Another well-studied, but more promising, LP is the bidire ted ut relaxation

[Chakrabartyet al.2008;Edmonds1967;RajagopalanandVazirani1999℄. Letus

xanarbitraryterminalr (root). Repla e ea hedge fu;vgbytwodire tededges

(u;v)and(v;u)of ost (fu;vg). Foragiven utU V,deneÆ +

(U)=f(u;v)2

Eju2U; v2=Ugas thesetofedgesleavingU. Thementionedrelaxationis

min X e2E (e)z e (BCR) s:t: X e2Æ + (U) z e  1; 8U V nfrg:U\R6=;; z e  0; 8e2E:

We an onsider thevaluez

e

asthe apa itywhi h weare goingto installonthe

dire ted edgee. TheLP anthen beinterpreted as omputingtheminimum- ost

apa ities that support a ow of 1 from ea h terminalto the root. Ina seminal

work,Edmonds[1967℄showedthatBCR isintegralin thespanning tree ase.

Theorem 2. [Edmonds 1967℄ForR=V,the polyhedron of BCRisintegral.

Thebest-knownlowerboundontheintegralitygapofBCRwas8=7[Konemann

et al. 2007; Vazirani 2001℄. The best-known upper bound is 2, though BCR is

believedto haveasmallerintegralitygap thantheundire ted utrelaxation

[Ra-jagopalanandVazirani1999℄. Chakrabartyet al.[2008℄reportthatthestru ture

of the dual to BCR is highly asymmetri , whi h ompli ates a primal-dual

ap-proa h. Moreover,iterativerounding basedon pi king asingleedge annot yield

good approximations,aswaspointedoutin[RajagopalanandVazirani1999℄.

Finding a better-than-2 LP relaxation of the Steiner tree problem is a

long-standingopenproblem[Chakrabartyet al.2008;RajagopalanandVazirani1999℄.

WeremarkthatgoodLP-bounds,besidespotentiallyleadingtobetter

approxima-tionalgorithmsforSteinertree,mighthaveamu hwiderimpa t. This isbe ause

Steiner tree appears as a building blo k in several other problems, and the best

approximation algorithms for someof those problems are LP-based. Strong LPs

arealsoimportantin thedesignof(pra ti ally)eÆ ientand a urateheuristi s.

1.1 OurResultsandTe hniques

Themainresultofthispaperisasfollows.

Theorem 3. For any onstant " > 0, there is a polynomial-time (ln(4)+

(a)

r

(b)

Fig.1.(a)ASteinertreeS,wherere tanglesdenoteterminalsand ir lesrepresentSteinernodes.

(b)EdgesofS aredire tedtowards arootr. Thedire ted omponentsofS aredepi tedwith

dierent olors.

This anbeimprovedto73=60+"inthewell-studiedspe ial aseofquasi-bipartite

graphs (wherenon-terminalnodesarepairwise notadja ent).

Our algorithm is based on the following dire ted- omponent ut relaxation for

theSteinertreeproblem(asimilarrelaxationis onsideredin[PolzinandV

ahdati-Daneshmand2003℄). ConsideranysubsetofterminalsR 0 R ,andanyr 0 2R 0 . Let

C betheminimum- ostSteinertreeonterminalsR 0

,withedges dire tedtowards

r 0

(dire ted omponent). For a given dire ted omponent C, we let (C) be its

ost, and sink(C) be its unique sink terminal. We all the remaining terminals

sour es(C):=V(C)\Rnfsink(C)g. Thesetof omponentsobtainedthiswayis

denotedbyC

n

. Wesaythatadire ted omponentC2C

n

rosses asetU Rif

C hasat leastonesour einU andthesinkoutside. ByÆ +

Cn

(U)wedenotetheset

ofdire ted omponents rossingU. Furthermore,we hooseanarbitraryterminal

rasaroot. OurLPrelaxationisthen:

min X C2C n (C)x C (DCR ) s:t: X C2Æ + Cn (U) x C  1; 8URnfrg; U 6=;; x C  0; 8C2C n :

DCRistriviallyarelaxationoftheSteinertreeproblem. Infa t,one andire tthe

edgesoftheoptimalSteinertreeS 

towardsterminalr,andsplittheedgesetofS 

atinteriorterminals. Thisyieldsasetofdire ted omponentsCC

n

(seeFigure

1). ObservethatanyC2CmustbeanoptimalSteinertreeonterminalsR\V(C).

Consequently,settingx

C

=1foranyC2C,and theremainingvariablestozero,

providesafeasiblesolutiontoDCRof ost P

C2C

(C)= (S 

)=opt.

Unfortunatelythe ardinalityofC

n

isexponential.However,wewillseethat,for

any onstant">0,one an omputea(1+")-approximate fra tionalsolutionto

DCR inpolynomialtime. Thisis a hievedbyrestri tingC

n

to thedire ted

om-ponentsC

k

that ontainat mosta(big) onstantnumberk ofterminals(dire ted

k- omponents).

We ombineourLPwitha(tothebestofourknowledge)noveliterative

random-izedrounding te hnique. WesolvetheLP(approximately),sampleone omponent

C with probabilityproportionalto its valuex

C

in the near-optimalfra tional

so-lutionx, ontra tC intoitssink nodesink(C),andreoptimizetheLP.Weiterate

by the sampled omponents. A fairly simple analysis provides a 3=2+" bound

on the approximation ratio. With a rened analysis, we improve this bound to

ln(4)+". Ouralgorithm anbederandomized.

Weremarkthatouralgorithm ombinesfeaturesofrandomizedrounding (where

typi allyvariablesareroundedrandomly,but simultaneously)anditerative

round-ing(wherevariablesareroundediteratively,butdeterministi ally). Webelievethat

ouriterativerandomizedroundingte hniquewill alsondotherappli ations, and

ishen eforthofindependentinterest.

Thekeyinsightinouranalysisistoquantifytheexpe tedredu tionofthe ostof

theoptimalSteinertreein ea hiteration. Toshowthis,weexploit anovelBridge

Lemma, relating the ost of terminal spanning trees with the ost of fra tional

solutionstoDCR.TheproofofthelemmaisbasedonTheorem2[Edmonds1967℄.

In our opinion, our analysis is simpler (or at least more intuitive) than the one

in[Robins andZelikovsky2005℄.

Asaneasy onsequen eofouranalysis,weobtainthattheintegralitygapofDCR

is at most1+ln(2) < 1:694,hen eansweringto thementioned open problem in

[Chakrabartyet al.2008;RajagopalanandVazirani1999℄. CombiningourBridge

LemmawiththealgorithmandanalysisbyRobinsandZelikovsky[2005℄,weobtain

thefollowingimprovedresult.

Theorem 4. Forany ">0, thereis analgorithm for the Steiner treeproblem

whi h omputesasolutionof ostatmost1+ ln(3)

2

+"timesthe ostofthe optimal

fra tional solution to DCR. The running time of the algorithm is polynomial for

onstant ".

Theabovetheoremimmediatelyimpliesa1+ln(3)=2<1:55upperbound onthe

integrality gap of DCR, by letting " tend to zero (the running time is irrelevant

withthat respe t). Asmentionedbefore, integralitygap resultsof thistypeoften

providenewinsightsintovariantsandgeneralizationsoftheoriginalproblem. We

expe tthatthiswillbethe asewiththeabovetheoremaswell,sin eSteinertree

appearsasabuildingblo kin manyotherproblems.

WealsoshowthattheintegralitygapofDCRandBCRareatleast8=7>1:142

and36=31>1:161,respe tively.

1.2 RelatedWork

A sign of importan e of the Steiner tree problem is that it appears either as a

subproblem or as a spe ial ase of many other problems in network design. A

( ertainlyin omplete) list ontainsSteiner forest [Agrawalet al.1995; Goemans

and Williamson1995℄,prize- olle ting Steiner tree [Ar heret al. 2009; Goemans

and Williamson 1995℄, virtual private network [Eisenbrand and Grandoni 2005;

Eisenbrandet al.2007;GrandoniandRothvo2010; Guptaet al.2001;Rothvo

and Sanita 2009℄,single-sink rent-or-buy [Eisenbrand et al. 2008; Grandoni and

Rothvo2010;Guptaet al.2007;JothiandRaghava hari2009℄, onne tedfa ility

lo ation [Eisenbrandet al.2008;SwamyandKumar2004℄,andsingle-sink

buy-at-bulk [GrandoniandItaliano2006;Guptaet al.2007;GrandoniandRothvo2010;

Talwar2002℄.

Boththepreviously itedprimal-dualanditerativeroundingapproximation

rounding te hniqueintrodu ed by Jain[1998℄providesa 2-approximationfor the

Steiner network problem, and theprimal-dual framework developedby Goemans

and Williamson [1995℄ gives the same approximation fa tor for a large lass of

onstrainedforest problems.

RegardingtheintegralitygapofLPrelaxationsoftheSteinertreeproblem,upper

boundsbetterthan2areknownonlyforspe ialgraph lasses. Forexample,BCR

hasanintegralitygapsmallerthan2onquasi-bipartitegraphs,wherenon-terminal

nodesindu eanindependentset. Forsu hgraphsRajagopalanandVazirani[1999℄

(seealso [Rizzi2003℄)gaveanupperbound of3=2on thegap. Thiswasre ently

improvedto 4=3byChakrabarty,Devanurand Vazirani [2008℄. Still,forthis lass

of graphs the lowerbound of 8=7 holds [Konemann et al. 2007; Vazirani 2001℄.

Konemann,Prit hardand Tan[2007℄showedthat for adierentLPformulation,

whi hisstrongerthanBCR,theintegralitygapisupper-boundedby 2b+1

b+1

,whereb

isthemaximumnumberofSteinernodesinfull omponents. AllthementionedLPs

anbesolved in polynomialtime, while wesolveDCR onlyapproximately: from

ate hni al point ofview, weindeed solveexa tly arelaxationof thek-restri ted

Steinertreeproblem.

Finally,weremarkthat underadditional onstraints,Steinertreeadmits better

approximations. Inparti ular,aPTAS anbeobtainedbythete hniqueofArora

[1998℄ifthenodesarepointsin axed-dimensionEu lidean spa e,and usingthe

algorithmofBorradaile,Kenyon-MathieuandKlein[2007℄forplanargraphs.

1.3 Organization

Therestofthispaperisorganizedasfollows. InSe tion2wegivesomedenitions

and basi results. In Se tion3 weshowhowto approximate DCR andproveour

BridgeLemma. InSe tion4wepresentasimpleexpe ted(1:5+")-approximation

for the problem. This result is improved to ln(4)+" in Se tion 5. The spe ial

ase of quasi-bipartite graphs is onsidered in Se tion 5.1. We derandomize our

algorithminSe tion6. Finally,inSe tion7wedis usstheintegralitygapofDCR,

and ompareDCRwithBCR.

2. PRELIMINARIES

We useOpt to denote the optimalintegral solution,and opt = (Opt). The ost

of an optimal solution to DCR (for the input instan e) is termed opt

f

. We will

onsider algorithms onsisting of a sequen e of iterations, ea h one onsidering

dierent subproblems. Wewill use anapexto denote the onsidered iterationt.

Forexample,opt t

f

denotesthe ostofanoptimalfra tionalsolutionatthebeginning

ofiterationt.

For a given (dire ted orundire ted) omponent C, R (C) := R\V(C) is the

set ofitsterminals. Re allthat DCRhasanexponentialnumberof variablesand

onstraints. Forthisreason,ouralgorithmswill onsiderapproximatesolutionsto

DCR with apolynomial-size support. Therefore, it is notationally onvenient to

representasolutiontoDCRasapair(x;C),whereCC

n isasubsetofdire ted omponentsandx=fx C g C2C

denotesthevaluesthatareasso iatedtoea hsu h

omponent. (Othervariablesareassumedtohavevaluezero).

Let T bea minimum- ostterminal spanning tree. It isa well-knownfa t that

proof,thisboundalsoholdsw.r.t. ourLPrelaxation.

Lemma 5. Onehas (T)2
opt

f .

Proof. Let(x;C)beanoptimalfra tionalsolutiontoDCR.Forea h omponent

C2C,obtainanundire tedTSPtouronR (C)of ostatmost2 (C),removeone

edgeofthetour,anddire ttheremainingedgestowardssink(C). Install apa ity

x

C

umulativelyonthedire tededgesof theresultingarbores en e. This indu es

afra tionalsolutionto DCRof ostat most2
opt

f

,with thepropertythat only

omponentswith2terminalsandwithoutSteinernodesareused. Thisalsoprovides

afeasiblefra tionalsolutionto BCRofthesame ost. Sin eBCR withoutSteiner

nodesisintegralbyTheorem2,the laimfollows.

Let R 0

be a subsetof k terminals. Consideragiven Steinertree S, with edge

weights , ontainingtheterminalsR 0

. Theweightfun tion asso iatedtoS,ifnot

spe ied,willbe learfrom the ontext. Letus ollapsetheterminalsR 0

into one

node, and allG 0

theresulting(possibly,multi-)graph. LetS 0 S beaminimum spanning tree of G 0 . Observe that S 0

will ontain all the edges of S but k 1

edges,sin e ollapsingR 0

de reasesthenumberofnodesinS byk 1. We allthe

latteredgesthebridges of S w.r.t. R 0

,anddenotethem byBr

S (R 0 ) 3 . Intuitively,

if we imagine to add zero ost dummy edges betweenthe terminals R 0 , Br S (R 0 )

is amaximum- ostsubset of edgesthat we ould removefrom S andstill havea

onne tedspanningsubgraph. Inotherterms,

Br S (R 0 )=argmax n (B)jBS; SnB[ R 0 2
onne tsV(S) o : Letusabbreviatebr S (R 0 ):= (Br S (R 0

)). Fora(dire tedorundire ted) omponent

C 0 , we use Br S (C 0 ) and br S (C 0 ) as short uts for Br S (R (C 0 )) and br S (R (C 0 )), respe tively.

Intheanalysis,itisoften onvenienttoturnagivenSteinertreeS intoarooted,

possiblynon- omplete, binarytreeasfollows(see,e.g.,[Karpinskiand Zelikovsky

1997℄). Byadding dummynodes and dummy edges of ost zero, we anassume

that theleavesofS oin ide withitsterminals, and that internal(Steiner) nodes

havedegreeexa tlythree. Thenwesplitanyedgebyaddingadummynodev and

a dummy edge of ost zero, and we root the tree at v. Note that the resulting

rooted binarytree hasheightat mostjR j 1. Given thisredu tion, itis easy to

provethefollowingstandardresult.

Lemma 6. ForanySteiner treeS on terminalsR ,br

S (R )

1

2 (S):

Proof. TurnS into arootedbinary treeasdes ribedabove. Forea hSteiner

nodeofS,markthemostexpensiveedgeoutoftheedgesgoingtoits2 hildren. Let

BS bethesetofmarkededges. Observethat (B) 1

2

(S). Furthermore,after

ontra tingR ,one anremoveB whilekeepingS onne ted. From thedenition

ofbridgesitfollowsthat br

S

(R ) (B) 1

2 (S).

Throughoutthis paper,wesometimes identifyasubgraphG 0

withitsset ofedges

E(G 0

).

3

3. ADIRECTED-COMPONENTCUTRELAXATION

Inthisse tionweshowhowtosolveDCRapproximately(Se tion3.1),and prove

ourBridgeLemma(Se tion3.2).

3.1 ApproximatingDCR

We next show how to ompute a (1+")-approximate solution to DCR, for any

given onstant ">0,in polynomialtime. This is a hieved in two steps. Firstof

all,weintrodu earelaxationk-DCRofthek-restri tedSteinertreeproblem. This

relaxation an be solvedexa tlyin polynomialtimefor any onstantvalue ofthe

parameterk(Lemma8). Thenweshowthattheoptimalsolutionstok-DCRand

DCRare loseforlarge-enoughk(Lemma7).

LetC

k C

n

denotethe set of dire ted omponentswith at mostk terminals,

and let Æ + C k (U) :=Æ + C n (U)\C k

. Bythe sameargumentsasfor the unrestri ted

ase,thefollowingisarelaxationofthek-restri tedSteinertreeproblem:

min X C2C k (C)x C (k-DCR ) s:t: X C2Æ + C k (U) x C  1; 8U Rnfrg; U 6=;; x C  0; 8C2C k : Let opt f;k

be the value of the optimal fra tional solution to k-DCR. Trivially,

opt

f;k opt

f

sin eanyfeasiblesolutiontok-DCRisalsofeasibleforDCR.We an

exploittheresultbyBor hersandDu[1997℄to showthatopt

f;k

isindeed loseto

opt

f

forlargek.

Lemma 7. Onehas opt

f;k  k
opt f .

Proof. Let(x;C)beanoptimalfra tionalsolutionforDCR.Weshowhowto

onstru tasolution(x 0

;C 0

)tok-DCRwiththe laimedproperty. Forany

ompo-nent C 2 C, we an apply Theorem 1 to obtain alist of undire ted omponents

C

1 ;:::;C

`

su hthat: (a) S

`

i=1 C

i

onne tstheterminalsinC,(b)anyC

i

ontains

at mostkterminals,and ( ) P ` i=1 (C i ) k

(C). Next,wedire t theedgesof

allC

i

's onsistentlytowardssink(C)andin reasethevalueofx 0 Ci byx C forea h C i

. Theresultingsolution(x 0

;C 0

)satisesthe laim.

Itremainstosolvek-DCRfork=O(1). Foranyxedk,inpolynomialtimeone

an onsider any subsetR 0

R of at mostk terminals, and omputeanoptimal

Steinertree Z onR 04 . By onsidering ea h r 0 2R 0

, anddire ting the edgesof Z

towardsr 0

,oneobtainsallthedire ted omponentsonterminalsR 0 . Consequently, jC k j=O(kn k

)andthek- omponents anbelistedinpolynomialtime.

Lemma 8. Theoptimal solution tok-DCR an be omputedinpolynomialtime

for any onstantk.

4

Were allthat,givenkterminals,thedynami -programmingalgorithmbyDreyfusandWagner

[1972℄ omputesanoptimalSteinertreeamongtheminO(3 k n+2 k n 2 +n 3 )worst- asetime. A

Proof. Wedene adire ted auxiliarygraphG 0 =(V 0 ;E 0 ), onnodeset V 0 = R[fv C j C 2 C k

g. For every omponent C, insert edges (u;v

C

) for any u 2

sour es(C),andoneedgee

C =(v

C

;sink(C)). Weobservethatk-DCRisequivalent

toanon-simultaneousmulti ommodity owproblem,whereanyterminalinRsends

oneunit of owtotherootandedgese

C

have ost (C).

Morepre iselyk-DCRisequivalenttothefollowing ompa tLP:

min X C2Ck (C)x C s:t: X e2Æ + (v) f s (e) X e2Æ (v) f s (e) = 8 > < > : 1 ifv=s; 1 ifv=r; 0 ifv2V nfr;sg; 8s2Rnfrg; f s (e C )  x C ; 8s2Rnfrg;C2C k ; f s (e);x C  0; 8s2Rnfrg;e2E 0 ;C2C k : Heref s

(e)denotesthe owthatterminalssendsa rossedgeeandthe apa ityon

edge e C is x C =max s2Rnfrg f s (e C

). An optimalsolutionof thelatterLP anbe

omputedinpolynomialtime,seee.g. [Kha hiyan1979;Grots helet al.1981℄ 5

.

Puttingeverythingtogether,weobtainthedesiredapproximatesolutiontoDCR.

Lemma 9. For any xed " >0, a (1+")-approximate solution (x;C) to DCR

an be omputedin polynomial time.

Proof. It issuÆ ient tosolvek-DCR fork :=2 d1="e

with thealgorithm from

Lemma8. Observethat

k

1+"(seeagainTheorem1). The laimfollowsfrom

Lemma7.

3.2 TheBridgeLemma

WenextproveourBridgeLemma,whi histheheartofouranalysis. Thislemma

relatesthe ostofanyterminalspanningtreetothe ostofanyfra tionalsolution

toDCRviathenotionofbridges,anditsproofisbasedonEdmonds'Theorem2.

Akeyingredientintheproofofourlemmaisthe onstru tionofaproperweighted

terminalspanning treeY. ConsideraSteiner treeS onterminalsR . Wedenea

bridgeweightfun tion w:RR!Q

+

asfollows: Foranyterminalpairu;v2R ,

the quantity w(u;v) is themaximum ost of any edge in the uniqueu-v path in

S. Re allthatBr

S (R

0

)isthesetofbridgesofS withrespe ttoterminalsR 0 ,and br S (R 0

)denotesits ost.

Lemma 10. LetS beany Steiner treeon terminalsR , andw:RR!Q

+ be

the asso iatedbridge weightfun tion. Forany subsetR 0 R of terminals,thereis atreeY R 0 R 0 su hthat (a) Y spans R 0 . (b) w(Y)=br S (R 0 ).

( ) Foranyfu;vg2Y,the u-vpathinS ontainsexa tlyoneedgefromBr

S (R

0

).

5

NotethatthisLP anevenbesolvedinstrongly-polynomialtimeusingtheFrank-Tardos

3 4 b1 7 1 b3 6 1 1 b 2 2 b 4 8 1 e 1 7 e 2 2 e 4 8 e 3 6

Fig.2. Steinertree S isdrawn inbla k. Terminalsof R 0

aregrayshaded. Bold bla kedges

indi ateBr S (R 0 )=fb 1 ;:::;b 4

g. The orrespondingedgese

1 ;:::;e

4

ofY aredrawningrayand

labeledwithw(e

i

). Notethat w(e

i )= (b

i

). Observealso thatb

3

isthe uniquebridgeonthe

y le ontainedinS[fe 3 g. Proof. Let Br S (R 0 ) = fb 1 ;b 2 ;:::;b k 1

g be the set of bridges. Observe that

S nBr S (R 0 ) is a forest of trees F 1 ;:::;F k , where ea h F i

ontains exa tly one

terminalr i 2R 0 . Ea h bridgeb i

onne tsexa tlytwotreesF

i 0 and F i 00 . Forea h b i , weadd edge e i =fr i 0 ;r i 00

gto Y. Observethat Y ontainsk nodesand k 1

edges. Assumeby ontradi tionthatY ontainsa y le,saye

1 ;e 2 ;:::;e g . Repla e ea he i =fr i 0 ;r i 00 gwith F i 0 [F i 00 [fb i

g: theresultinggraphis a y li subgraph

ofS,a ontradi tion. Hen e Y isaspanningtreeonR 0 . ThepathP i betweenr i 0 andr i 00 ontainsb i

andnoother bridge. Hen e b

i isa maximum- ostedgeonP i ,andw(e i )= (b i

)(seeFigure2). The laimfollows.

Lemma 11. [Bridge Lemma℄ Let T be aterminalspanning treeand(x;C) bea

feasiblesolutiontoDCR. Then

(T) X C2C x C
br T (C):

Proof. Forevery omponentC2C,we onstru taspanningtreeY

C

onR (C)

with weightw(Y

C )= br

T

(C) a ordingto Lemma 10. Then we dire t the edges

ofY

C

towardssink(C). Letus install umulatively apa ityx

C

onthe(dire ted)

edgesof Y

C

, forea hC 2C. This wayweobtain adire ted apa ityreservation

y : RR ! Q + , with y(u;v) := P Y C 3(u;v) x C

. The dire ted tree Y

C

supports

at leastthesame owas omponentCwith respe ttoR (C). Itthenfollowsthat

y supports one unit of ow from ea h terminal to the root. In other terms, y is

a feasible fra tional solutionto BCR. ByEdmond's Theorem 2, BCR is integral

when no Steiner node is used. As a onsequen e there is an (integral) terminal

spanningtreeF thatisnotmore ostlythanthefra tionalsolutiony,i.e.,w(F)

P

e2RR

w(e)y(e).

Re allthatw(u;v),foru;v2R ,isthemaximum ostofanyedgeoftheunique

y leinT[fu;vg. Itfollowsfromthe lassi y leruleforminimumspanningtree

(1) Fort=1;2;::: (1a) Computea(1+ " 2 )-approximatesolution(x t ;C t

)toDCR(w.r.t.the urrentinstan e).

(1b) Sampleone omponentC t ,whereC t =Cwithprobabilityx t C =

P

intoitssink.

(1 ) Ifasingleterminalremains,return

S

C i

.

Fig.3. A(ln(4)+")-approximationalgorithmforSteinertree.

Altogether X C2C x C br T (C)= X C2C x C w(Y C )= X e2RR w(e)y(e)w(F) (T):

4. ITERATIVERANDOMIZEDROUNDING

Inthisse tionwepresentourapproximationalgorithmforSteinertree. Tohighlight

thenovelideasoftheapproximationte hniquemorethantheapproximationfa tor

itself, we present asimplied analysis providing a weaker3=2+" approximation

fa tor(whi hisalreadyanimprovementonthepreviousbest1:55approximation).

Themore omplexanalysisleadingtoln(4)+"ispostponedto Se tion5.

Theapproximationalgorithmfor Steinertree is des ribedin Figure 3. InStep

(1a) we use the algorithm from Lemma 9. Re all that the ardinality of C t

is

upperboundedbyavalueMwhi h,foranyxed">0,isboundedbyapolynomial

in n. Contra ting a omponent C t

means ollapsing allitsterminals intoits sink

sink(C t

),whi h inheritsalltheedgesin identto C t

(in aseofparalleledges, we

only keep the heapest one). We letOpt t

denote the optimal Steinertree at the

beginningofiterationt,andletopt t

beits ost. Byopt t

f

wedenotethe ostofthe

optimalfra tionalsolutionatthebeginningofiterationt.

Observethat P C2C t x t C

M,and thisquantitymightvary overtheiterations

t. Inorderto simplifytheanalysis,weaddadummy omponent 

C formedbythe

rootonly(hen eof ostzero)toensurethatin fa t

X C2C t x t C =M; 8t1:

Note thatadding su h dummy omponent orrespondsto insertingidleiterations

into the algorithm. But the expe ted ost of the produ ed solution remains the

same.

Theexpe ted ostoftheprodu edsolutionis:

X t1 E[ (C t )℄ X t1 X C2C t E h x t C M (C) i  1+ " 2 M X t1 E[opt t f ℄ 1+ " 2 M X t1 E[opt t ℄ (1)

Thus, in order to obtainagood approximationguarantee, itsuÆ es to providea

good boundonE[opt t

℄.

4.1 Arstbound

Asimple onsequen eoftheBridgeLemmaisthatthe ostoftheminimumterminal

spanning tree de reases by a fa tor (1 1

M

) per iteration in expe tation. This

impliesanupperboundonopt t

f

viaLemma5(whilelaterboundswillholdforopt t

only). Theboundonopt t

f

guaranteeofouralgorithm(duetothefa tthatopt t

f opt

t

)andontheintegrality

gapofourLP.

Lemma 12. Onehas E[opt t f ℄ 1 1 M
t 1
2opt f . Proof. LetT t

betheminimum- ostterminalspanningtreeatthebeginningof

iterationt. ByLemma 5, (T 1

)2opt

f

. Foranyiterationt>1,theredu tionin

the ostofT t w.r.t. T t 1 isat leastbr T t 1(C t ). Therefore: E[ (T t )℄  (T t 1 ) E[br T t 1(C t 1 )℄ = (T t 1 ) 1 M X C2C t 1 x t 1 C
br T t 1(C) BridgeLem11   1 1 M 
(T t 1 ): Byindu tion E[opt t f ℄E[ (T t )℄  1 1 M  t 1
2opt f :

ObservethattheboundfromLemma12improvesoverthetrivialboundopt t

f 

opt

f

fort>M
ln(2). NeverthelessitsuÆ estoprovethefollowingresult.

Theorem 13. For any xed" > 0, there is a randomized polynomial-time

al-gorithm whi h omputes asolution to the Steiner treeproblem of expe ted ostat

most(1+ln(2)+")
opt

f .

Proof. Assume without lossof generality that M
ln(2) is integral.

Combin-ing(1)withLemma12, theexpe tedapproximationfa toris

E " P t1 (C t ) opt f #  1+"=2 M X t1 E " opt t f opt f #  1+"=2 M X t1 min ( 1;2  1 1 M  t 1 )  1+"=2 M 0  M
ln(2)+ X tM
ln(2)+1 2  1 1 M  t 1 1 A   1+ " 2  ln(2)+2  1 1 M  M
ln(2) !   1+ " 2  ln(2)+2e ln(2)  1+ln(2)+":

Above we used the equation P tt0 x t = x t 0 1 x

for 0 < x < 1 and the inequality

(1 1=x) x

1=eforx1.

ObservethatTheorem13impliesthattheintegralitygapofDCRisatmost1+ln(2).

4.2 Ase ondbound

In order to further improve the approximation guarantee we show that, in ea h

iteration,the ostopt t

oftheoptimal(integral)Steinertreeofthe urrentinstan e

de reases by a fa tor (1 1

2M

) in expe tation. We remark that it is not known

whether thisboundholds alsoforopt t

f

. Alsoin this asetheproofrelies ru ially

ontheBridgeLemma.

Lemma 14. LetS be anySteiner treeand(x;C)beafeasiblesolution toDCR.

Sample a omponent C2C su hthatC=C 0 withprobability x C 0 =M. Then there isasubgraphS 0 S su hthat S 0 [C spansR and E[ (S 0 )℄  1 1 2M 
(S):

Proof. It suÆ es to prove that E[br

S (C)℄ 

1

2M

(S). Turn S into a rooted

binarytreewiththeusualpro edure. Then, foranySteinernodein S, hoosethe

heapest edge going to oneofits hildren. Theset H S ofsu h sele tededges

has ost (H) 1

2

(S). FurthermoreanySteinernodeis onne tedtooneterminal

usingedges ofH. Considertheterminal spanningtreeT thatemergesfrom S by

ontra tingH. BytheBridgeLemma11,

E[br T (C)℄= 1 M X C 0 2C x C 0
br T (C 0 ) 1 M (T):

ObservealsothatBr

T

(C)isafeasibleset ofbridgesforS withrespe tto C,and

thusbr S (C)br T (C). Altogether: E[br S (C)℄E[br T (C)℄ 1 M (T)= 1 M ( (S) (H)) 1 2M (S):

IteratingLemma14yieldsthefollowing orollary.

Corollary 15. Foreveryt1,

E[opt t ℄  1 1 2M  t 1
opt:

Wenowhavealltheingredientstoshowa(3=2+")-approximationfa tor.

Theorem 16. For any " >0, there is a polynomial-time randomized

approxi-mation algorithm forSteiner treewithexpe tedapproximation ratio3=2+".

(1)withLemma12andCorollary15,theexpe tedapproximationfa toris: E " P t1 (C t ) opt #  1+"=2 M X t1 E  opt t opt   1+"=2 M X t1 min ( 2  1 1 M  t 1 ;  1 1 2M  t 1 )  1+"=2 M 0  M
ln(4) X t=1  1 1 2M  t 1 + X tM
ln(4)+1 2  1 1 M  t 1 1 A =  1+ " 2 
2 2
 1 1 2M  M
ln(4) +2  1 1 M  M
ln(4) !   1+ " 2 
 2 2
e ln(4)=2 +2e ln(4)  3=2+":

Aboveweexploitedtheequation P t0 t=1 x t 1 = 1 x t 0 1 x

for0<x <1. Wealsoused

thefa t that (1 1 y ) ln(4)y (1 1 2y ) ln(4)y

is anin reasing fun tion ofy >1, and

thatlim y!1 (1 1 y ) y = 1 e .

5. AREFINEDANALYSIS

Inthisse tionwepresentarened(ln (4)+")approximationboundforourSteiner

treealgorithm.

We rst give a high-level des ription of our analysis. Let S 

:= Opt be the

optimal Steiner tree for the original instan e (in parti ular, (S 

) = opt). Ea h

time we sample a omponent C t

, we will delete a proper subset of edges from

S 

. Consider the sequen e S  = S 1  S 2  ::: of subgraphs of S  whi h are

obtainedthisway. Wewill guaranteethat atanyiterationt,theedgeset S t

plus

thepreviouslysampled omponentsyieldsasubgraphthat onne tsallterminals.

Furthermore, we will provethat a xed edge e2 S 

is deleted after an expe ted

numberofatmostln(4)
Miterations. Thisimmediatelyimpliestheapproximation

fa torofln(4)+".

Inordertotra kwhi hedges anbesafelydeletedfromS 

,wewill onstru tan

arti ialterminal spanningtreeW (the witnesstree)and assignarandomsubset

W(e) of edgesof W to ea h edge e2S 

(the witnesses of e). At ea h iteration,

when omponent C t

is sampled, we mark a proper random subset Br

W (C

t

) of

edgesof W. Assoon asall theedgesof W(e)are marked, edgee is deleted from

S 

. Summarizing,we onsiderthefollowingrandompro ess:

Fort=1;2;:::, sampleone omponentC t

from (x t

;C t

)and mark the

edges in Br

W (C

t

). Delete an edge e from S 

as soon asall edges in

W(e)aremarked.

The subgraph S t

is formed by the edges of S 

whi h are notyet deleted at the

beginningofiterationt.

The hoi eof Br

W (C

t

)guarantees that, deterministi ally, the unmarkededges

W 0

3 1 1 2 1 3 1 2 2 e0 1 2 1 1 1 f0 f 1 (a) (b)

Fig.4.(a)OptimalSteinertreeS 

inbla k,whereboldedgesindi atethe hosenedges ~

B,andthe

asso iatedterminalspanningtreeWingray.EdgeseinS 

arelabeledwithjW(e)j.Forexample

W(e0)=ff0;f1g. (b)MarkededgesinWatagiveniterationtaredrawndotted;thenon-deleted

edgesinS 

(i.e.,edgesofS t

)aredrawninbla k. Non-markededgesofW andnon-deletededges

ofS 

supportthesame onne tivityonR.

ensures that, deterministi ally, if W 0

plus the sampled omponents onne t all

the terminals,then the sampled omponentsplus theundeleted edges S t =fe2 S  jW(e)\W 0

6=;g dothe same. Hen e theS t

's havethe laimed onne tivity

properties. Theanalysisthenredu estoshowthatalltheedgesinW(e)aremarked

withinasmallenoughnumberofiterations(inexpe tation).

Wenext dene W, W(
), andBr

W

(
). TurnS 

into arooted binarytree with

theusual pro edure. Re all that theheight ofthe binarytree isat mostjR j 1.

For ea h Steiner node, hoose uniformly at random one of the two edges to its

hildren. Let ~

B denote the hosenedges. Clearly Pr[e2 ~ B℄= 1 2 foranye2 S  . LetP uv S 

betheuniqueu-vpathinS 

. Thewitnesstreeis

W := n fu;vg2 R 2
jjP uv \ ~ Bj=1 o :

SimilarlytoargumentsinLemma10,W isaterminalspanningtree. Forea hedge

e2S  ,dene W(e):=ffu;vg2W je2P uv g:

SeeFigure4(a)foranillustration. Aswewillsee,W(e)issmallin expe tation. It

remainsto deneBr

W

(
). Foragiven omponentC 2C,letthe setof andidate

bridgesB W (C)be B W (C):=fBW jjBj=jR (C)j 1;(WnB)[C onne tsV(W)g: Intuitively, B W

(C) is the family of bridge sets of W with respe t to C that one

obtainsforvarying ostfun tions. ThesetBr

W (C t )is hosenrandomlyinB W (C),

a ordingtoaproperprobabilitydistribution w

C :B

W

(C)![0;1℄,whi h will be

des ribedin the following. Observethat Br

W

(C)2B

W

(C). Theintuitivereason

forusingarandomelementofB

W

(C)ratherthanBr

W

(C)isthatwewishtomark

theedgesofW inamoreuniformway. This,in ombinationwiththesmallsizeof

W(e),guaranteesthat edgesaredeleted qui klyenough.

Nextlemmashowsthattheundeletededgesplusthesampled omponents onne t

Lemma 17. The graphS t [ t 1 t 0 =1 C t spansR . Proof. Let W 0

 W be the set of edges whi h are not yet marked at the

beginning of iteration t (see also Figure 4(b)). By the denition of B

W (C) and beingBr W (C)2B W (C),W 0 [ S t 1 t 0 =1 C t 0

spansR . Consideranyedgefu;vg2W 0

.

Thenfu;vg2W(e)foralle2P

uv

. Hen enoedgeonP

uv

isdeleted. Thereforeu

andv arealso onne tedin S t

. The laimfollows.

Notethat 1jW(e)jjR j 1. Observealsothat jW(e)j=1ife2 ~

B. Indeed,

theexpe ted ardinalityofW(e)issmallalsofortheremainingedges.

Lemma 18. Foranyedgee2S 

atlevelk

e

jR j 1(edgesin identtotheroot

areatlevel one), onehas

Pr[jW(e)j=q℄= 8 > < > : 1=2 q if 1q<k e ; 2=2 q if q=k e ; 0 otherwise :

Proof. Considerthepathv

0 ;v

1 ;:::;v

ke

frometowardstheroot. Inparti ular,

e=fv 0 ;v 1 g. If (v q 1 ;v q

) is therstedge from ~

B onthis path, thenjW(e)j=q.

Thisisbe ause,forea hnodev

j

,j1,thereisonedistin tpathP

uv

withfu;vg2

W that ontainse(seealsoFigure4(a)). Thiseventhappenswithprobability1=2 q

.

If there is no edge from ~ B on the path v 0 ;v 1 ;:::;v ke , jW(e)j = k e by a similar

argument. Thelattereventhappenswithprobability1=2 k

e

. The laimfollows.

Nextlemmaprovestheexisten eofrandomvariablesBr

W

(
)su hthatea hedge

ofW ismarkedatea hiterationwithprobabilityatleast 1=M. Itsproofisbased

ona ombinationofFarkas'LemmawithourBridgeLemma.

Lemma 19. There is a hoi e of the random variables Br

W

(
) su h that ea h

edge e2W ismarkedwithprobability atleast1=M atea h iteration.

Proof. Consideranygiveniteration. Let(x;C)bethe orrespondingsolutionto

DCR,and C 

bethesampled omponentin thatiteration. Inparti ular, C  =C with probability x C =M = x C = P C 0 2C x C

0. In this iteration we mark the edges

Br W (C  ),wherePr[Br W (C  )=B℄=w C (B) foranyB2B W (C  ). Wewillshow

thatthere isa hoi eofthew

C 's,C2C,su hthat X (C;B):B2BW(C);e2B x C
w C (B)1; 8e2W:

Thisimpliesthe laimsin e

Pr[e2Br W (C  )℄= X (C;B):B2B W (C);e2B x C M
w C (B) 1 M :

Suppose by ontradi tion that su h probability distributions w

C

Thenthefollowinglinearsystemhasnosolution 6 : X B2B W (C) w C (B)  1; 8C2C; X (C;B):B2BW(C);e2B x C
w C (B)  1; 8e2W; w C (B)  0; 8C2C;8B2B W (C): Farkas'Lemma 7

yieldsthat thereisave tor(y; )0with

(a) y C  P e2B e x C ; 8C2C;8B2B W (C); (b) P C2C y C < P e2W e :

Let usinterpret asan edge ost fun tion. Inparti ular, (W) := P e2W e and br W

(C)isthe ostofthebridgesofW withrespe tto omponentCandthis ost

fun tion. Onehas

y C (a)  x C
maxf (B)jB2B W (C)g=x C
br W (C): Then X C2C x C
br W (C) X C2C y C (b) < X e2W e = (W);

whi h ontradi tstheBridgeLemma 11.

We next show that, for small jW(e)j, all the edges of W(e) are marked (and

hen ee is deleted) within asmall number ofiterations. A handwavingargument

worksas follows. LetjW(e)j=q. SimilarlytotheCouponsColle torproblem (see

e.g.[Mitzenma herandUpfal2005℄),ittakesinexpe tation M

q

iterationsuntilthe

rstedgeismarked,then M

q 1

iterationstohitthese ondoneandsoforth. Finally

alledgesaremarkedafteranexpe tednumberofM
( 1 q + 1 q 1 +:::+1)=H q
M iterations. (HereH q := P q i=1 1 i

denotestheq-thharmoni number). However,this

argumentdoesnotre e tthefa t thatasetBr

W (C

t

)might ontainseveraledges

fromW(e). Amore arefulargumentin orporatesthis ompli ation.

For ~

W  W, let X( ~

W) denote the rst iteration when all the edge in ~

W are

marked. Observethat S t =fe2S  jX(W(e))tg. Lemma 20. Let ~

W W. Then the expe tednumber ofiterationsuntilalledges

in ~

W aremarkedsatises

E[X( ~ W)℄H j ~ Wj
M: Proof. Letq =j ~ Wj. By m q

we denote thebest possible upperbound onthe

expe ted number of iterations until all edges of ~

W are marked (over all feasible

probabilitydistributions). Wewillprovethat m

q H

q

M byindu tiononq.

6

We anrepla ethe\=" onstraintwith\"withoutae tingfeasibilitysin eall oeÆ ientsof

w C (B)arenon-negative. 7 9x0:Axb _ _9z0:z T A0;z T b<0.

For q = 1, the only edge in ~

W is marked with probability at least 1

M

at ea h

iteration,hen em

1

M. Next,letq>1and onsider therstiteration. Suppose

that 

i

is theprobabilitythat at least imany edgesare marked in thisiteration.

Sin e the expe ted number of marked edges must be at least q
1

M

in the rst

iteration, this distribution has to satisfy P q i=1  i  q M . Note that  0 = 1 and  q+1

=0. Fornotational onvenien e, letm

0 :=0.

If we ondition on the event that i 2 f0;:::;qg edges are marked in the rst

iteration,weneedinexpe tationatmostm

q i

moreiterationsuntiltheremaining

q iedgesaremarked. Hen e weobtainthefollowingbound:

m q  1+ q X i=0 Pr 

exa tlyiedgesmarked

attherstiteration 
m q i indu tive hypothesis  1+M
q X i=1 ( i  i+1 )H q i +(1  1 )m q = 1+M
q X i=1  i
(H q i H q i+1 ) | {z }  1=q + 1 H q M+(1  1 )m q  1 1 q M
q X i=1  i |{z } q=M + 1 H q M+(1  1 )m q   1 H q M+(1  1 )m q : From 1 >0weobtainm q H q

M. The laimfollows.

Nowwehavealltheingredientstoprovetheexpe ted(ln (4)+")approximation

fa tor.

Theorem 21. For any onstant " >0, there is a polynomial-time randomized

approximation algorithm for the Steiner tree problem with expe tedapproximation

ratioln(4)+".

Proof. Foranedge e2S 

, wedene D(e)=maxftje2S t

inwhi heisdeleted. Onehas

E[D(e)℄ = ke

X

q=1

Pr[jW(e)j=q℄
E[D(e)jjW(e)j=q℄

Lem20  k e X q=1 Pr[jW(e)j=q℄
H q
M Lem18 = ke 1 X q=1  1 2  q
H q
M+ 2 2 ke
H ke
M  X q1  1 2  q
H q
M = M
X q1 1 q X i0  1 2  q+i = M
X q1 1 q  1 2  q 1 =ln(4)
M:

Theexpe ted ostoftheapproximatesolutionsatises

E h X t1 (C t ) i  X t1 1+"=2 M E  opt t f   1+"=2 M X t1 E  (S t )  = 1+"=2 M X e2S 

E[D(e)℄
(e)(ln(4)+")
opt:

The laimfollows.

5.1 A( 73

60

+")-ApproximationforQuasi-BipartiteGraphs

Inthisse tionwe onsiderthespe ial aseofquasi-bipartitegraphs.Re allthatwe

allagraphG=(V;E)quasi-bipartiteifnopairofnon-terminalnodesu;v2VnR

is onne tedbyanedge. Weshowthat ouralgorithmhasanapproximationratio

ofat most 73

60

+"<1:217(for"smallenough). This improvesoverthepreviously

bestknownfa torof1.28in [RobinsandZelikovsky2005℄. NotethatGroplet al.

[2002℄showtheboundof 73

60

forthemorerestri ted aseofuniformquasi-bipartite

graphs,where alledges in ident to anon-terminal node havethe same ost. For

this lass the integrality gap of the hypergraphi LP relaxation by Chakrabarty

et al.[2010a℄ analsobeboundedby 73

60 .

Again let S 

be an optimal Steiner tree, whi h now is quasi-bipartite. Let

Z

1 ;:::;Z

`

be its (undire ted) omponents. In parti ular, ea h Z

i

is a star with

asingle Steiner node as enter and terminalsas leaves. We an improvethe

ap-proximationguaranteeby hoosing thewitnesstreeW in amoree onomi alway,

exploitingthestru tureofS 

. Forea hi=1;:::;`,weaddtothe hosenedges ~

B

alltheedgesof Z

i

letP

uv

betheuniqueu-vpathinS  . Welet W :=ffu;vg2 R 2
jjP uv \ ~ Bj=1g:

ObservethatW willinfa tbeaterminalspanningtree. Theanalysisisnowmu h

simpler.

Theorem 22. For any onstant " >0, there is a polynomial-time randomized

approximation algorithmfor theSteinertreeproblemonquasi-bipartite graphswith

expe tedapproximation ratio 73

60 +".

Proof. We still onsider the algorithm in Figure 3. For an edge e 2 S 

, we

deneD(e)=maxftje2S t

gastheiterationinwhi he isdeleted. Letk bethe

number of terminals in the star Z

i

that ontainse. With probability 1

k

onehas

jW(e)j=k 1,andotherwisejW(e)j=1. Hen e,byLemma20,

E[D(e)℄ 1 k
H k 1
M+  1 1 k 
H 1
M=  1 k
H k 1 + k 1 k 
M 73 60
M:

Inthelastinequalityweusedthefa tthat 1 k
H k 1 + k 1 k ismaximizedfork=5.

The laimfollowsalongthesameline asinTheorem 21.

6. DERANDOMIZATION

In this se tion, we show how to derandomize the resultfrom Se tion 5using the

methodoflimited independen e (see,e.g.,[AlonandSpen er2008℄). Thisway,we

proveTheorem3.

Westart(Se tion6.1)bypresentinganalternative,phase-basedalgorithm,whi h

updates the LP only a onstant number of times (the phases). Then we show

(Se tion6.2)howto sample omponentsin ea hphasewithalogarithmi number

ofrandombits.

6.1 APhase-BasedRandomizedAlgorithm

Consider the algorithm from Figure 5. The basi idea behind the algorithm is

groupingiterations into phases. In ea h phase, wekeepthe LPun hanged. The

detailsonhowto sample omponentsin ea h phasearegivenlater.

(1) Forphases=1;2;:::;1=" 2

(1a) Computea (1+")-approximate well-rounded solution (x s ;C s ) to DCR (w.r.t. the urrentinstan e). (1b) Sample s omponentsC s;1 ;:::;C s; s fromC s a ordingtox s

,and ontra tthem.

(2) Computeaminimum- ostterminalspanningtreeT intheremaininginstan e.

(3) OutputT[

S

S

Fig.5. Phase-basedsamplingalgorithm

Wemayassumethatthe omputedDCR solution(x s ;C s )iswell-rounded,i.e.,  jC s

j=mforaprime numberm,

 x s C = 1 N forallC2C s

This an be a hieved as follows: One omputes a (1+ "

2

)-approximate solution

(x;C). Sayh=jCj. Then weround upall entries in xto thenearestmultiple of

1

N

forN :=8h="andtermtheobtainedsolutionx 0

. Usingthegenerousestimate

(C)2opt

f

(followingfromLemma5) weobtain,for"1,

X C2C x 0 C
(C)  1+ " 2 
 opt f + X C2C " 8h (C)  (1+")opt f :

Next, repla e a omponent C by x 0

C

N many opies. Let m 0

be thenumber of

obtained omponents( ountedwithmultipli ities). Thenwe an omputeaprime

number m 2 [m 0

;2m 0

℄ (see e.g. [Niven et al. 1991℄) and add m m 0

dummy

omponents C ontaining only the root, ea h one with x 0 C := 1 N . This yields a

feasiblewell-roundedsolutionasdesired. Wefurthermoreassume 8 thatmN=" 2 and1="isinteger. For ~ W W, let  X( ~

W)denote therstphasewhenalledgesin ~

W are marked.

Analogously, 

D(e)isthephasewhenalltheedgesin W(e)aremarked. For

nota-tional onvenien e, weinterpretStep(2) asanalphasewhenalltheedges ofW

aremarked(sothat 

X( ~

W)and 

D(e)arewelldened). Thenextlemmaisasimple

adaptationof Lemma20.

Lemma 23. Let ~

W W. Supposeea hedgeismarkedatea h phasewith

prob-ability atleastp2℄0;1℄. Then the expe ted number of phases until all edges in ~

W

aremarkedsatises

E[  X( ~ W)℄H j ~ Wj
1 p :

Proof. Bya oupling argument, we anassumethat the numberof phasesis

unbounded. The laim follows along the same line as the proof of Lemma 20,

repla ingthenotionofiterationwiththenotionofphaseandtheprobability1=M

withp.

Wenext bound theapproximationfa torof the algorithmfor ageneri sampling

pro edure(satisfyingsomeproperties).

Lemma 24. SupposethatStep (1b)satisesthe following twoproperties:

(a) Ea h omponentC issampledwithprobability atmost
x s

C

(b) Ea h edge einthe witnesstreeismarkedwith probability atleast.

Thentheapproximationfa torofthealgorithminFigure5isatmostln(4)
( (1+")  + 2" 2  ).

Proof. AsintheproofofTheorem21,onehas

E[  D(e)℄ = k e X q=1 Pr[jW(e)j=q℄
E[  D(e)jjW(e)j=q℄ Lem18+23  X q1  1 2  q
H q
1  =ln(4)
1  : (2) 8 If1 T x= m N

=O(1)andjCj=O(1)forC2C,thenthenumberofterminalswouldbebounded

Let opt s

bethe ostof an optimalSteiner tree at thebeginning ofphase s. The

expe ted ostofthesampled omponentssatises

E h 1=" 2 X s=1 s X i=1 (C s;i ) i  1=" 2 X s=1 X C2C s E[
x s C
(C)℄  (1+")
1=" 2 X s=1 E[opt s ℄  (1+") X e2S  E[  D(e)℄
(e) (2) ln(4)
(1+")  opt: LetS 0 :=fe2S  j  D(e)>1=" 2

gbeafeasibleSteinertree at theend ofthe last

phase. ByMarkov'sinequalityand(2),

Pr[  D(e)>1=" 2 ℄ln(4) " 2  : ThereforeE[ (S 0 )℄ln(4) " 2 

opt. Theminimum- ostterminalspanningtreeisat

mosttwi ethatexpensive,hen eE[ (T)℄2ln(4) "

2

opt. The laimfollows.

Lemma24suggestsanalternativewaytoimplementthealgorithmfromSe tion5.

ConsiderthefollowingnaturalimplementationofStep(1b):

(IndependentPhaseSampling)

Sample s ="
M omponentsC s;1 ;:::;C s; s

independently(with

rep-etitions),whereC s;i =C2C s withprobabilityx s C =M.

TheIndependent PhaseSamplingsamplesa omponentC with aprobability

ofatmost s
1 M
x s C ="
x s C

. Ontheotherhand,theprobabilitythatedgee2W

is marked is essentiallylower bounded by ". Inspe ting Lemma 24, we see that

",whi h givesthefollowing orollary.

Corollary 25. The algorithm fromFigure5whi h implementsStep(1b)with

theIndependent PhaseSamplingis(ln (4)+O("))-approximate inexpe tation.

This provides a1:39-approximationalgorithm thatneeds to solvejust a onstant

(rather than polynomial) number of LPs. In parti ular, its running time might

be ompetitivewiththe better-than-2 approximationalgorithmsin theliterature.

Butadrawba koftheIndependent PhaseSamplingimplementationisthatit

needstoomany(namelypolynomiallymany)randombits: hen eit isnoteasyto

derandomize. Forthisreasonweintrodu eamore omplexsamplingpro edure in

thenextsubse tion.

6.2 ADependentSamplingPro edure

Wenextdes ribeanalternativeimplementationofStep(1b),whi hstillguarantees

 ", andrequires onlyO(logn)randombits. Wefo us onaspe i phase

sand anedgee2W. Let(x;C):=(x s

;C s

). Werenumberthe omponentssu h

thatC=(C 0 ;:::;C m 1 ),andweletx j :=x C j = 1 N .

(i) Choose A 2f0;:::;m 1g and B 2 f1;:::;m 1guniformly and independentlyat random. (ii) Sele tC j withj 2J:=fA+i
B modmji=1;:::;b " N m g.

Observethat Step(ii)requires onlyO(logm)randombits. Sin em=n O(1)

, this

numberofbitsis O(logn).

Wewill showthat: (1)any omponentC

j

issampled with probabilityno more

thanx

j

,:="and(2)edgeeismarkedwithprobabilityatleast:="(1 2").

Therst laimiseasytoshow.

Lemma 26. Implementing Step (1b) with the Dependent Phase Sampling,

ea h omponent C

j

issampledwithprobability atmost"
x

j .

Proof. Forany omponentC

j ,Pr[j 2J℄= 1 m
b " N m  " N ="
x j .

Showing laim(2)ismoreinvolving.

Lemma 27. Implementing Step (1b) with the Dependent Phase Sampling,

ea hedgee2W ismarkedwith probability atleast"(1 2").

Proof. Letw

Cj

betheprobabilitydistributionfor omponentC

j asinLemma19. Re allthatPr[Br W (C j )=B℄=w Cj (B)and Æ:= m 1 X j=0 x j X B2B W (C j ):e2B w Cj (B)1: Let y j := P B2BW(Cj):e2B w Cj

(B) denote the probabilitythat eis marked, given

that C j is sampled. Sin ex j = 1 N , we have P m 1 j=0 y j

=ÆN. There liesno harm

in assumingthat Æ=1,sin etheprobabilitythat eis markedisin reasingin the

y

j 's.

LetE

j

betheeventthatC

j issampledand e2Br W (C j ). ItissuÆ ienttoshow that Pr[ S m 1 j=0 E j

℄"(1 2"). The ru ialinsightis to obtainalower bound on

Pr[E

j

℄ andanupper bound onPr[E

j \E j 0 ℄forj6=j 0

. Firstofall,wehave

Pr[E j ℄=y j
Pr[j 2J℄=y j
j "m N k
1 m "(1 ") y j N ; (3) using that "m N  1 "

by assumption. Se ondly, letj;j 0

2f0;:::;m 1gbedistin t

omponentindi es. Thenj;j 0

2J ifandonlyifthesystem

j  m A+Bi (4) j 0  m A+Bi 0 hasasolutioni;i 0 . Butsin eZ m

isaeld,foranydistin tpairi;i 0

2f1;:::;b "m

N g,

there ispre iselyonepair(A;B)2f0;:::;m 1gf1;:::;m 1gsatisfying(4).

Hen e Pr[E j \E j 0 ℄y j
y j 0
j "m N k
j "m N k 1 
1 m
(m 1) y j
y j 0
" 2 N 2 : (5)

Bythe in lusion-ex lusionprin iple(see, e.g., CorollaryA.2 in [Arora and Barak 2009℄), Pr h m 1 [ j=0 E j i  m 1 X j=0 Pr[E j ℄ m 1 X j=0 X j 0 6=j Pr[E j \E j 0 ℄ (3)+(5)  m 1 X j=0 "(1 ") y j N " 2 N 2 m 1 X j=0 y j | {z } =N
X j 0 6=j y j 0 | {z } N  "(1 ") " 2 ="(1 2");

whi hprovesthe laim.

Adeterministi (ln (4)+")-approximationalgorithmeasilyfollows.

Proofof Theorem 3. Consider the algorithm from Figure 5 whi h

imple-ments Step (1b) with the Dependent Phase Sampling. This algorithm an

bederandomized by onsidering all the possible out omesof random variables A

and B in ea h phase, whi hare at most m 2="

2

. The laimon theapproximation

followsfromLemmas24, 26,and27.

We ansimilarlyderandomizetheresultforquasi-bipartitegraphs.

Theorem 28. Forany onstant">0,thereisadeterministi polynomial-time

algorithmfor theSteinertreeproblemonquasi-bipartitegraphswithapproximation

ratio 73

60 +".

Proof. ConsiderthesamealgorithmasinTheorem3. Lemmas23, 26,and27

stillhold. UnderthesameassumptionsasinLemma24,andbythedierent hoi e

ofthewitnesstreeW in this ase,wenowhave

E[  D(e)℄ 1 k
H k 1
1  +  1 1 k 
H 1
1  =  1 k
H k 1 + k 1 k 
1   73 60
1  :

Then the expe ted ost of the sampled omponents satises E[ P s;i (C s;i )℄  73 60 (1+") 

opt. Similarly, the expe ted ost of the nal spanning tree satises

E[ (T)℄  2
73 60 " 2 

opt. Altogether, the approximation fa tor from Lemma 24

now redu es to 73 60
( (1+")  + 2" 2 

). The laim follows along the same line as in

Theorem3.

7. INTEGRALITYGAP

In this se tion weupper bound (Se tion 7.1) and lower bound (Se tion 7.2) the

integralitygapofDCR.Furthermore,we ompareDCRwithBCR(Se tion7.3).

7.1 AnUpperBound

Note that, despitethe fa t that ouranalysis is basedon an LPrelaxationof the

problem,itdoesnotimplyaln(4)(norevena1:5)upperboundontheintegrality

gap of the studied LP. It only provides a 1+ln(2) upper bound, as shown in

theiterationsofthealgorithm,anditssolutionisonlyboundedwithrespe ttothe

initialoptimalintegralsolution. Inthisse tionweprovethatourLPhasintegrality

gap at most 1+ln(3)=2 < 1:55. Before pro eeding with our (fairly te hni al)

argument,letus remarkthat,afterthe onferen eversionofthispaperappeared,

ashorterandperhapsmoreelegantproof(stillbasedontheBridgeLemma)ofthe

same laimwasre entlygivenin [Chakrabartyet al.2010b℄.

Inorderto provethe1:55upperbound ontheintegralitygapof DCR, laimed

in Theorem 4, we onsider the algorithm R&Z by Robins and Zelikovsky [2005℄.

We show that this algorithm produ es solutionsof ost bounded with respe t to

the optimal fra tionalsolutions to k-DCR (and hen eof DCR). This is a hieved

by ombiningthe original analysis in [Robins and Zelikovsky2005℄ with ournew

Bridge Lemma 11. This approa hwas, to someextent, inspired byan argument

in [Charikar and Guha 2005℄ in the ontext of fa ility lo ation. We leave it as

an interesting open problem to prove a ln (4) (or even 1:5) upper bound on the

integrality gap of DCR (if possible). This might involve the development of a

fra tionalversionofLemma 14.

AlgorithmR&Zworksasfollows. It onstru tsasequen eT 0 ;T 1 ;:::;T  of

ter-minalspanning trees, where T 0

is a minimum- ostterminal spanning tree in the

originalgraph. Atiterationt wearegivenatreeT t

and a ostfun tion t

onthe

edgesofthetree(initially 0

 ). Thealgorithm onsidersany andidate

ompo-nentC with atleast 2andat mostk terminals(k- omponent). LetT t

[C℄ denote

theminimumspanningtreeofthegraphT t

[C,wheretheedgese2Chaveweight

0and theedges f 2T t

weight t

(f). The subsetofedges in T t but notin T t [C℄ aredenoted byBr T t(C). Infa t,Br T

t(C)is thesetofbridgesofT t

withrespe t

to R (C) andthe aboveweightfun tion. Foragiven omponentC, we denote as

Loss(C)the minimum- ostsubforestof C withthe propertythat there is apath

between ea h Steiner node in C and someterminal in R (C). In theterminology

from Se tion 3,Loss(C)is the omplementof theset ofbridgesof thesubtreeC

after ontra tingR (C). Weletloss(C)= (Loss(C)).

Itis onvenienttodenethefollowingquantities:

gain t (C)=br T t(C) (C) and sgain t (C)=gain t (C)+loss(C):

Thealgorithmsele tsthe omponentC t+1

whi h maximizesgain t

(C)=loss(C). If

thisquantityisnon-positive,thealgorithmhalts. Otherwise,it onsidersthegraph

T t

[C t+1

,and ontra tsLoss(C t+1

). Thetree T t+1

is aminimum- ostterminal

spanning tree in the resulting graph. In asethat parallel edges are reatedthis

way,thealgorithmonlykeepsthe heapestofsu hedges. Thiswayweobtainthe

ostfun tion t+1

ontheedgesofT t+1

.

Lemma 29. [RobinsandZelikovsky2005℄Fort=1;2;:::;, t (T t )= t 1 (T t 1 ) sgain t 1 (C t ): LetApx k

betheapproximatesolution omputedbythealgorithm, andapx

k =

(Apx

k ).

Lemma 30. [Robins andZelikovsky2005℄ Forany `,

apx k  ` X t=1 loss(C t )+ ` (T ` ):

Re allthatopt

f;k

isthe ostoftheoptimalfra tionalsolutiontok-DCR.Let(x;C)

beanoptimalfra tionalsolutiontok-DCR.Deneloss

f;k := P C2C x C loss(C). Corollary 31. loss f;k  1 2 opt f;k .

Proof. FromLemma 6,foranyC2C,loss(C)= (C) br

C (R (C)) 1 2 (C). Asa onsequen e,loss f;k  1 2 P C2C x C
(C)= 1 2 opt f;k . Corollary 32.  (T  )opt f;k .

Proof. Usingthefa tthat gain 

(C)=br

T 

(C) (C)0forany omponent

C,  (T  ) BridgeLem11  X C2C x C br T  (C) X C2C x C (C)=opt f;k :

ByCorollary32,andsin e t

(T t

)isanon-in reasingfun tionoft,theremustbea

valueof`su hthat:

` 1 (T ` 1 )>opt f;k  ` (T ` ): (6)

Inthefollowingwewillbound P ` t=1 loss(C t )+ ` (T `

). ByLemma30,thiswillgive

abound onapx k . Let gain t f := t (T t ) opt f;k and sgain t f :=gain t f +loss f;k : Lemma 33. Fort=1;2;:::;, sgain t 1 (C t ) loss(C t )  sgain t 1 f loss f;k :

Proof. Werstnotethat

gain t 1 f loss f;k = t 1 (T t 1 ) P C2C x C (C) P C2C x C loss(C) BridgeLem11  P C2C x C (br T t 1(C) (C)) P C2C x C loss(C) = P C2C x C gain t 1 (C) P C2C x C loss(C) max C2C  gain t 1 (C) loss(C)   gain t 1 (C t ) loss(C t ) ;

whereinthelastinequalityweusedthefa tthatC t

maximizesgain t 1

(C)=loss(C)

overallthek-restri ted omponentsC. Itfollowsthat

sgain t 1 (C t ) loss(C t ) =1+ gain t 1 (C t ) loss(C t ) 1+ gain t 1 f loss f;k = sgain t 1 f loss f;k :

Weneedsomemorenotation. Letsgain ` 1 (C ` )=sgain 1 +sgain 2 su hthat sgain 1 = ` 1 (T ` 1 ) opt f;k (6) > 0: (7)

Wealsoletloss(C ` )=loss 1 +loss 2 su hthat sgain ` 1 (C ` ) loss(C ` ) = sgain 1 loss 1 = sgain 2 loss 2 : (8) Eventually,wedene sgain `1 f :=sgain ` 1 f sgain 1 (7) = ` 1 (T ` 1 ) opt f;k +loss f;k ( ` 1 (T ` 1 ) opt f;k )=loss f;k : (9) Lemma 34. P ` 1 t=1 loss(C t )+loss 1 loss f;k ln  sgain 0 f sgain `1 f  : Proof. Foreveryt=1;2;:::;` 1, sgain t f =sgain t 1 f sgain t 1 (C t ) Lem33  sgain t 1 f  1 loss(C t ) loss f;k  : Furthermore sgain ` 1 f loss f;k Lem33  sgain ` 1 (C ` ) loss(C ` ) (8) = sgain 1 loss 1 ; fromwhi h sgain `1 f =sgain ` 1 f sgain 1 sgain ` 1 f  1 loss 1 loss f;k  : Then sgain `1 f sgain 0 f   1 loss 1 loss f;k  ` 1 Y t=1  1 loss(C t ) loss f;k  :

Takingthelogarithmof bothsidesandre allingthat xln(1+x),

ln sgain 0 f sgain `1 f !  1 loss f;k ` 1 X t=1 loss(C t )+loss 1 ! :

Wenowhavealltheingredientstobound theapproximationfa torof the

algo-rithmwith respe t to opt

f;k . Let mst= (T 0 )= 0 (T 0

). The followingtheorem

and orollaryare straightforwardadaptationsofanalogous resultsin [Robins and

Zelikovsky2005℄. Theorem 35. apx k opt f;k +loss f;k ln  mst opt f;k +loss f;k lossf;k  :

Proof. Sin esgain t 1

(C t

)loss(C t

),itfollowsfrom (8)that

sgain 2

loss 2

Puttingeverythingtogetherweobtain apx k Lem30  ` X t=1 loss(C t )+ ` (T ` ) Lem29 = ` 1 X t=1 loss(C t )+loss(C ` )+ ` 1 (T ` 1 ) sgain ` 1 (C ` ) = ` 1 X t=1 loss(C t )+loss 1 +loss 2 + ` 1 (T ` 1 ) sgain 1 sgain 2 (10)  ` 1 X t=1 loss(C t )+loss 1 + ` 1 (T ` 1 ) sgain 1 (7) = ` 1 X t=1 loss(C t )+loss 1 +opt f;k Lem34  opt f;k +loss f;k ln sgain 0 f sgain `1 f ! (9) = opt f;k +loss f;k ln  mst opt f;k +loss f;k loss f;k  :

Lemma 36. For any onstant k  2, there exists a polynomial-time algorithm

forSteinertreewhi h omputesasolutionof ostatmost1+ln(3)=2timesthe ost

ofthe optimal fra tional solution tok-DCR.

Proof. AstraightforwardadaptationofLemma5impliesthat

mst2opt

f;k :

CombiningtheinequalityabovewithTheorem35,weobtain

apx k opt f;k +loss f;k ln  1+ 2opt f;k opt f;k loss f;k  :

Theright-handsideoftheinequalityaboveisanin reasingfun tionofloss

f;k . By Corollary31,loss f;k  1 2 opt f;k ,whi himplies apx k opt f;k + 1 2 opt f;k ln  1+2 2opt f;k opt f;k opt f;k  =opt f;k  1+ ln(3) 2  : Theorem4follows.

Proofof Theorem 4. From Lemma 7,opt

f;k 

k
opt

f

. The laim follows

fromLemma 36andTheorem1by hoosingalarge-enoughk.

7.2 ALowerBound

Thebest-knownlowerbound ontheintegralitygapofBCR (priorto ourwork)is

8=7[Konemannet al.2007℄.We anusethesamefamilyofinstan estoprovethe

001 2 010 2 011 2 100 2 101 2 110 2 111 2 0012 0102 0112 1002 1012 1102 1112

Fig.6. Skutella'sgraph. Nodesarelabeledwiththeirindi esinbinaryrepresentation.

Theorem 37. The integrality gapof DCRisatleast8=7>1:142.

Proof. Wewill use Skutella'sgraph [Konemann et al. 2007℄. ConsideraSet

Coverinstan e with elements U = f1;:::;7g and sets S

1 ;:::;S 7 . Let b(i) be a ve torfrom Z 3 2

that is thebinary representationof i, forexampleb(3)=(0;1;1).

We dene the sets by S

j

:= fi2 U j b(i)
b(j) 

2

1g. Note that this is exa tly

thedenitionoftheinstan ewhi hyieldsa(logn)lowerboundontheintegrality

gap of Set Cover forn = 7[Vazirani 2001℄. The riti al propertyis that for our

parti ularinstan eoneneeds3setsto overallelements,but hoosingea hset to

anextentof1=4givesafra tionalSetCoversolutionof ost7=4.

Nextwedene agraph where ea h elementforms a terminaland ea h set isa

non-terminalnode onne tedtotherootandtothe ontainedelementsbyunit ost

edges(seeFigure6).

If wedire t all the edges upwards, the graph anbe de omposed into 7

edge-disjoint omponents, ea h one ontainingonenon-terminal node and the 5edges

in ident into it. Onone hand installing1=4on ea h of these omponentsgivesa

fra tional solutionof ost 35=4, while onthe other hand at least 3Steiner nodes

mustbein ludedforanintegersolution. Consequentlyopt=10andweobtainthe

promisedgapof 10 35=4 = 8 7 . 7.3 ComparisonwithBCR

WestartbyobservingthatDCR isarelaxationstri tlystrongerthanBCR.

Lemma 38. Letopt

DCR

andopt

BCR

betheoptimal fra tional solutionstoDCR

and BCR, respe tively, for a given input instan e. Then opt

DCR  opt

BCR and

thereareexamples wherestri tinequality holds.

Proof. Anyfeasiblesolutionto DCR anbeturned into afeasiblesolutionto

BCR of the same ost. In fa t, it is suÆ ient to split ea h omponent into the

orrespondingset ofedges. Thisprovestherstpartofthe laim. Anexampleof

stri tinequalityisgivenin Figure7.

Observethatthe1:55upperboundontheintegralitygap ofDCRdoesnotimply

the samebound on theintegrality gap of BCR.It remains asa hallengingopen

v1 v 2 v3 r t 11 t 12 t 13 t 21 t 22 t 23 t 31 t 32 t 33 s 11 s 12 s 13 s 21 s 22 s 23 s 31 s 32 s 33

Fig.7. Alledgeshave ost1. TheuniqueoptimalsolutiontoBCR,of ost15,installs apa ity

1=4onthe entraledgesand apa ity1=2ontheremainingedges(alwaysdire tedupwards).The

heapest solutiontoDCRhas ost7=4
9=15:75. Theoverall apa ityreservedonea hedgeis

thesameasintheBCR ase,ex ludingthetopedges,wherethe apa ityis3=4. The(integral)

optimalSteinertreehas ost17.

Thebest-knownlowerboundontheintegralitygapofBCRis8=7>1:142[K

one-mannet al.2007;Vazirani2001℄. Inparti ular,thefamilyofinstan eswhi h

pro-videsthisboundisthesameasin Se tion7.2. Wenextpresentanimprovedlower

boundof36=31ontheintegralitygapofBCR.

Theorem 39. The integrality gapof BCRisatleast36=31>1:161.

Proof. Thebasi ideaisgeneralizingthe onstru tionusedin Se tion7.2. Let

p2 N be aparameter. We reate agraphwith p+2levels and unit ost edges.

For i2f1;:::;pgonehas7 i

non-terminalnodesontheithlevel,ea hrepresented

byave torfrom U i

,where U =f1;:::;7g. Furthermore wehavearootterminal

onlevel0and7 p

terminalsonthe(p+1)thlevel,representedbyve torsfromU p

.

We onne ttherootto allnodesin therstlevel. Fori=1;:::;p, onsidernodes

u=(u 1 ;:::;u i )2 U i onleveli andv =(v 1 ;:::;v i+1 )2U i+1 onleveli+1. We

onne tuandv byanedgeif(u

1 ;:::;u i 1 )=(v 1 ;:::;v i 1 )andb(u i )
b(v i ) 2 1.

We onne tthenon-terminalnodeu2U p

onlevelpwithterminalv2U p

onlevel

p+1inasimilarmanner,namelyifandonlyif(u

1 ;:::;u p 1 )=(v 1 ;:::;v p 1 )and b(u p )
b(v p ) 2

1. Observethat,forp=1,weobtainexa tlySkutella'sgraph. The

graphobtainedforp=2isdepi tedin Figure8.

Let us onsider any integer optimal solution, of ost opt, and dire t the edges

towards r 

. Ea h time we havean edge going from aleveli downwards to level

i+1 we an repla eit by anedge to level i 1 without dis onne ting the tree.

Observethat, fori=0;:::;p 1,weneedatleast3
7 i

edgesbetweenleveliand

i+1 and that 7 p

r  1 2 3 4 5 6 7 11121314151617 21222324252627 31323334353637 41424344454647 51525354555657 61626364656667 71727374757677 11121314151617 21222324252627 31323334353637 41424344454647 51525354555657 61626364656667 71727374757677 0 1 2 3

Fig. 8. Instan e for p = 2. Nodes are labeled with the orresponding ve tor in abbreviated

notation;alledgeshaveunit osts. Theoptimalfra tionalsolution onsistofinstalling apa ity

1=16onea hedgefromlevel2tolevel1and apa ity1=4otherwise(alwaysdire ted\upwards"),

thusopt f =7 2 +7 2

=4+7=4=63.Ontheotherhandforanintegersolutiononeneeds3+3
7+7 2

=

73edges. Thegapforthisinstan eis onsequently 73

63

1:158.

edgesisalsosuÆ ient,thus

opt=3
(7 0 +7 1 +:::+7 p 1 )+7 p = 3 2
7 p 1 2 :

Considernowthe optimal fra tional solutionto BCR forthe sameinstan e. Let

opt p

f

denoteits ost. Thissolutioninstalls apa ity1=4ontheedgesin identtothe

rootand tothe terminals,and apa ity1=16ontheremainingedges (alldire ted

upwards). Hen e opt p f = 4 4 7 p + 4 16
(7 1 +7 2 +:::+7 p )= 31 24
7 p 7 24 :

The laimfollowssin e

lim p!1 opt opt p f = 36 31 : ACKNOWLEDGMENTS

Theauthors wish to thankC.Chekuri, F.Eisenbrand, M. X.Goemans, J.K

one-mann,D.Prit hard,F.B.Shepherd,andR.Zenklusenforhelpfuldis ussions. The

se ondauthorisgratefultoF.EisenbrandforsupportinghisvisitatEPFL(during

whi h partof this paper was developed). We thank the anonymous reviewersof

the onferen eversionof this paperfor many suggestionsonhowto improvethe