Approximation Algorithms for Directed Steiner Tree Problems *

Approximation Algor ithms for Directed Steiner Tree

Problems

y z x {

Moses Charikar Chandra Chekuri Ashish Goel Sudipto Guha

Dept of Computer Science

Stanford Univers ity

fmoses,chekuri,agoel,[email protected]

March 20, 1997

Ab stract

TheSteiner tree problem which i s known to b e NP-Completeisthe following. Givenawe ighted

undirecte d graph G =V; E, andaset X V of terminals,the ob jectiveisto nd a tree

of minimum cost which connects all theterminals. If the graph i s directed, in addition to

X ,we are given a ro ot r 2 V ,andthe ob jectiveisto nd a minimum cost arb ore scence

which connectstherootto eachoftheterminals. In the Generalize d Steiner tree problem, we

are givenaset X of pairs of vertice s, andthe goal i s to nda subgraph of minimum cost

suchthat each pair in X i s connected. In theundirecte d cas e, constant f actor algor ithms are

known for b oththevers ions [11, 14, 17, 1, 15], but e ss entially no approximation algor ithms

were known for the s e problems in the directe d cas e, other than the tr ivial O k -approximations.

We obtain the rst non-tr ivial approximation algor ithms for b oth problems in general directed

graphs. For the Directed Steiner tree problem, wede s ign a f amily of algor ithms thatachieve

an approximation ratio of O k intime O kn for any xe d >0, where k is thenumber of

terminals. For the Directe d Generalize d Steiner tree problem, we give an algor ithm thatachieves

1=3

2=3

an approximation ratio of O k log k , where k is thenumb er of pairs of vertice s that are to

b e connecte d. Relate d problems includingthe Group Steiner tree problem, theNodeWe ighted

Steiner tree problem andseveral others can b e re ducedinanapproximation pre s erving f ashion

tothe problems we solve, givingthe rst non-tr ivial approximations tothos e as well.

For the Directed Steiner tree problem, a re sult s imilar to ours has b een obtained indep endentl y in [6] byTo-yat

Cheung, Zuo Dai and MingLiofthe Department of Computer Science, City Univers ity of Hong Kong.

Supp orted byanARO MURI GrantDAAH04-96-1-0007 and NSF Award CCR-9357849, withmatchingfunds

f rom IBM, Schlumb erger Foundation, Shell Foundation, and Xerox Corp oration.

Supp orted byanARO MURI GrantDAAH04-96-1-0007 and NSF Award CCR-9357849, withmatchingfunds

f rom IBM, Schlumb erger Foundation, Shell Foundation, and Xerox Corp oration.

Supp orted byARO GrantDAAH04-95-1-0121 and NSF Grant CCR9304971

{

Supp orted byanARO MURI GrantDAAH04-96-1-0007 and NSF Award CCR-9357849, withmatchingfunds

f rom IBM, Schlumb erger Foundation, Shell Foundation, and Xerox Corp oration.

1 Intro duction

TheSteiner tree problem i s de ne d as follows: given a graph G =V; E with a cost function c on

the e dge s, andasubs et of vertice s X V calle d terminals, the goal i s to nd a minimum cost

tree that include s all thevertice s in X .The cost of the tree i s de ne d as thesumofthe costsofthe

edgesinthe tree. Notethatthe tree may includevertice s not in X as well.

TheSteiner tree problem i s known to b e NP-Hard even when the graph i s induce d by p ointsin

the plane[7], and i s MAXSNP-hard [4] for general graphs. The rst p olynomial timeapproximation

algor ithms for thi s problem were developed by Kou, Markowsky and Berman [11]andbyTakahashi

where k is the andMatsuyama [14]. The s e algor ithms achieve d an approximation f actor of 2

numberofterminals. Zelikovsky [17] obtained a ratio of andthi s has b een further improved by

Berman and Ramaiyer [3]and Karpinski and Zelikovsky [13]. The best ratio achievable currently i s

1:644. In a recent breakthrough re sult, Arora [2]has given a p olynomial timeapproximation scheme

for theSteiner tree problem on graphs induce d by p ointsinthe plane.

The directed vers ion of theSteiner tree problem i s a natural extens ion of theundirected vers ion

andisde ne d as follows. Given a directe d graph G =V; A, a sp eci e d ro ot r 2 V ,and a s et of

terminals X V; jX j = kthe ob jectiveisto ndthe minimum cost arb ore scence ro oted at r and

spanning all thevertice s in X in other words r should havea pathtoevery vertex in X . The Directed

Steiner tree problem has s everal applications in networkde s ign, routinginmulticast networks and

other relate d areas. See [16] for a survey on the us e of Steiner tree problems in networks. From a

theoretical p oint of view it tur ns outto b e us eful in a numberofreductions of problems involving

connectivityandcover ing including, the Group Steiner tree problem in b oth directed andundirected

graphs, s everal intere sting problems in connecte d domination, namely Edge We ighte d Connected

Dominatingsets, Group or Set TSP,Nodewe ighted Steiner Connecte d domination, andothers.

For someofthereductions regarding connecte d domination s ee [9]. We also cons ider the Directed

Generalize d Steiner tree problem where instead of a ro ot and a s et of terminals we are given a s et of

k pairs of vertice s, andthe ob jectiveisto nd a minimum cost subgraph which connects each pair.

Constant f actor algor ithms are known for theundirecte d cas e [1,15].

A f airly easy re duction f rom the Set cover problem as has b een obs erved bymanyothers shows

thatitishard toapproximate Directed Steiner tree toafactor b etter than ln k where k is thenumber

of terminals. It i s also easy to obtain an approximation f actor of k ,by connectingevery terminal to

therootviaashortest path. The only known p olynomial timeapproximation algor ithm even for a

sp ecial cas e i s due to Zelikovsky [18], where he give s an approximation algor ithm of ratio O k for

any > 0 for directed acyclic graphs. No algor ithms were known for the cas e of general directed

graphs.

In thi s pap er, we presentafactor O k approximation algor ithm for any >0 for the Directed

Steiner tree problem. Thi s i s a new and s impler approach compare d to Zelikovsky's approach in [18].

log k

Our algor ithm can b e mo di e d to giveanO2 approximation in quas i-p olynomial time. Thi s

f act hintsatthe p oss ibility of obtaining p olylogar ithmic approximation guarantee s for the problem.

We also pre s ent an algor ithm for the Directe d Generalize d Steiner tree problem whichachieves an

3 3

approximation f actor of O k log k . Our re sults imply s imilar ratios for s everal other var iants

includingthe Group Steiner tree problem andtheNodeWe ighted Steiner tree problem in directed

graphs via somewell known s imple approximation pre s ervingreductions. Therestofthe pap er i s

organize d as follows. In Section 2 we s et up the notation and prove some bas ic lemmas. Sections 3

and4 de scr ib e the algor ithms for the Directed Steiner tree andthe generalize d vers ions re sp ectively.

We br ie y sketchthereductions of someoftheother var iants in Section 5. 1

2 Preliminar ie s

In thi s s ection we will prove some bas ic lemmas concer ningdecomp os ing tree s andhowtousethem

to obtain partial solutions for the problem athand. Wede ne a slightly more general vers ion of the

Directed Steiner tree problem b elow.

De nition 1 Given root r 2 V G, an integer k and a set X V of terminals with jX j k, the

problem D-Steinerk; r;X is to construct a treerootedat r,spanning any k terminals in X and

of minimum possible cost.

We giveaOk approximation totheD-Steinerk; r;X problem which implie s the sameratio

for the Directed Steiner tree problem. It i s also easy to s ee thatthi s implie s a s imilar ratio for

the generalization of the k -MST problem [5,8]to directe d graphs, where the ob jectiveisto ndan

arb ore scence on k vertice s of G of minimum cost.

Let cedenotethe cost of e dge e and let cT denotethe cost of a tree T ,orthesumofthe costs

of theedgesinT. Let k T denotethenumberofterminals in T ;inother words k T = T \X.

De nition 2 De ne dT , the dens ity of tree T to be the ratio of the cost of the tree to the number

of terminals in T ; in other words dT = cT=k T .

We will assume without loss of generalitythat all terminals are atthe leave s of anySteiner tree. We

can arrange thi s by connectingadummyterminal tothe real terminal with a zero cost e dge. The

following lemmashows thatwe can partition a tree intotwo almost equal parts.

0 0 0 0

Lemma1 We can partition any out-tree T into two trees T and T such that cT +cT cT

1 2 1 2

2 1

0 0 0

partition of T . and k T +kT kT and k T k T =3. Such a partition is cal leda

1 2

3 3

Pro of: For anynodeiin T , let T denotethesubtree ro oted at i. Among all no des i suchthat

kT kT, let x be thenode whichisfarthest away in terms of numb er of e dge s f rom theroot

r.Thenode x is well de ne d s ince k T =kT. Let x ;:::;x be thechildren of x and without

r 1 l

loss of generality assumethatkT are non-decreas ing. From thede nition of x, it follows that

1 2 2

k T for all x . Suppose there i s some i,1il,suchthat kTkT < kT. k T <

i x x

i i

3 3 3

0 0

Then T = T and T = T T clearly sati sfy the requirements. If there i s no suchnodewehave

x x

1 i 2 i

2 1

kT for all i. Cons ider the large st i suchthat kT < kT. We claim that kT <

x x

ji j i

3 3

P P

k T k T . Supp os e not. It must b e the cas e that i< l s ince k T = kT

x x x

j i j l

j j

kT. Wehave

X X

1 1 2

k T = kT +kT < k T + kT= kT

x x x

j j i+1

3 3 3

ji+1 ji

andthi s contradictsthechoice of i. Let T be the tree ro oted at x with T x ;:::;Tx asits

1 1 i

subtree s and let T be T T . It i s eas ily ver i e d that T and T sati sfy the conditions require d of

2 1 1 2

them.

We can us e thi s f act toshowthat, given a tree T ,there exi st smaller tree s whos e dens ityisat

most a xe d constanttime s more than thatofT.

Lemma2 Given a tree T , for any x k T , thereisatreeT such that k T x and cT <

x x x

3x dT . 2

Pro of: We construct a s equence of tree s T = T ;T ;:::;T as follows. T i s obtained from T

0 1 x i+1 i

1 2

0 0

in the followingway: Construct a partition of T to get the tree s T and T . Notethat

1 2

3 3

0 0

cT cT +cT

1 2

= dT

0 0

kT +kT kT

1 2

0 0 0 0

Without loss of generality, suppose dT dT . Then dT dT . Let T = T . Notethat

i i+1

1 2 1 1

dT dT . Also,

i+1 i

1 2

k T k T k T

i i+1 i

3 3

.Westopassoonas xkT<3xfor some i. Since the s equence fdT g i s non-increas ing,

i i

cT =k T =dTdT =dT

i i i 0

which implie s that

cT k T dT < 3x dT :

i i

T is the require d tree T .

i x

De nition 3 De ne a f k -partial approximation pro ce dure for D-Steinerk; r;X to beapro-

0 0

cedure which constructs a tree T rootedat r,spanning 1 k k terminals in X such that

OP T

dT f k , where C is the cost of an optimal solution to D-Steinerk; r;X.

OP T

If Ak; r;X i s a partial approximation pro ce dure for D-Steinerk; r;X, we can apply A re-

peate dly to obtain an approximation algor ithm B k; r;X for D-Steinerk; r;X as follows. B k; r;X

rst calls Ak; r;X. Suppose thi s retur ns a tree T spanning k terminals. If k = k , retur n T .

1 1 1 1

Els e, let X be the s et of terminals spanned by T . Call B k k ;r;X X and let T be the tree

1 1 1 1 2

retur ned bythi s pro ce dure. Retur n T [ T .

1 2

Lemma3 Given Ak; r;X,an fk-partial approximation for D-Steinerk; r;X where f x=x is

adecreasing function of x, the algorithm B k; r;Xis a g k -approximation algorithm for D-Steinerk; r;X

dx. where g k =

Pro of: We will provethe claim byinduction on k . The bas e cas e k = 1 follows as f 1

f x fx

dx bythedecreas ing prop ertyof . Supp os e it i s true for all value s le ss than k .We will

x x

provethe claim for k . Let T b e an optimal solution to D-Steinerk; r;X. Suppose the call to

OP T

Ak; r;X retur ns tree T ro oted at r spanning k terminals, i.e. k T = k .

1 1 1 1

cT cT

OP T 1

fk 1 dT =

k k

f x f k

cT k cT dx cT 2

1 1 OP T OP T

k x

where the last inequality follows f rom thedecreas ing prop ertyof .Ifk =k,the algor ithm retur ns

T .For thi s cas e, cT g k cT provingthatthe algor ithm gives a g k-approximation.

1 1 OP T

Suppose k

1 1 1 2

the recurs ive call to B k k ;r;XX . Since T spans k terminals in T , it spans at least k k

1 1 OP T 1

terminals in X X . Hence the minimum cost tree on k k terminals in X X has cost at most

1 1 1

cT . By theinductivehyp othesis, cT g k k cT , i.e.

OP T 2 1 OP T

k k

f x

dx cT 3 cT

OP T 2

0 3

Adding 2 and 3, we get

cT +cT gkcT

1 2 OP T

Thi s proves that for thi s cas e too, the algor ithm gives a g k-approximation.

3 Directed Steiner Tree

Before wede scr ib e our algor ithm formally,we will provide someintuition andanoutline of our

technique s. Thedens ity of a tree can b e interprete d as theaverage cost of connectinga terminal to

the ro ot. Lemma2shows that a partial approximation pro ce dure which nds a tree withdens ity

clos e tothatofoptimal, leads toagoodapproximation algor ithm. Our algor ithm to nd tree s of

good dens ity i s motivated bythe following.

A tr ivial algor ithm for the problem i s to computeshortest paths f rom eachoftheterminals to

therootand combinethem. It i s instructiveto cons ider an example for whichthi s algor ithm gives a

ratio of k . Cons ider a graph where there i s path of cost C f rom theroottoa vertex v ,and zero

OP T

cost e dge s f rom v to eachoftheterminals. In addition, there are paths of cost C f rom the

OP T

ro ot to eachoftheterminals. Thenaive algor ithm picks eachoftheshortest paths and do e s not us e

thepaththrough v ,thus incurr ing a cost k C .

OP T

Motivated bytheabove extreme example, we can think of ndingsubtree s of go o d dens ity which

havethe following structure. The tree cons i stsofapath f rom therootr toaninterme diatenode

v,andv i s connected to some s et of terminals us inga shortest pathto eachofthem. For obvious

reasons, we call such a structure a bunch.The advantage of choosingsuch a s imple structure i s that

we can compute in p olynomial time,abunch withthe best dens ity. Of cours e for thi s schemeto

work, wenee d to guarantee the exi stence of a bunch with good dens ity.Weusethedecomp os ition

lemma Lemma 2 for thi s purp os e. Cons ider a subtree of theoptimal, T with k T = x terminals

v v

anddens ityat most 3dT . Thi s subtree of theoptimal tree naturally de nesabunch. The

OP T

cost of thebunch i s comp os e d of two parts. The cost of connecting r to v is at most C .The

OP T

cost of connecting eachoftheterminals in T to v is at most cT . Therefore the cost of the

v v

bunchde ned by T is C + x cT =C +3x C =k ,andthedens ityofthebunchis

v OP T v OP T OP T

1=x +3x=k C . Since Lemma 2 guarantee s a subtree withthe require d prop ertie s for any x,we

OP T

k to minimize thi s expre ss ion. Thi s proves the exi stence of a bunch withdens ity can choose x =

p p

at most kC =k = kdT . Combine d withthe f act thatwe can ndthebunch withthe

OP T OP T

best dens ity in p olynomial time, we obtain a O k approximation.

An obvious way to improvethe re sultisto nd structure s which are more general than bunches

thatapproximatetheoptimal more clos ely. We obtain an improved Ok ratio us ingthi s ap-

proach, butthe s imple st way tode scr ib e it i s via recurs ion, whichwe pro cee d todonext. We

1=i

will de ne a s equence of algor ithms B k; r;XsuchthatB gives an Ok -approximation for

i i

D-Steinerk; r;X.

B k; r;X and A k; r;X op erate s as follows: Determinethek terminals in X clos e st to r .

1 1

Connect therootr to eachsuchterminal t bytheshortest path f rom r to t.The tree retur ne d i s s imply

theunion of all these shortest paths. Clearly,thisisak-approximation for D-Steinerk; r;X.

Havingde ned B k; r;X, we will construct a partial approximation A k; r;X for D-Steinerk; r;X

i1 i

1=i

and provethatAk; r;Xisan Ok -partial approximation. Applying Lemma3,we obtain

1=i

B k; r;X whichisan Ok -approximation algor ithm for D-Steinerk; r;X.

A k; r;Xworks as follows:

11=i

1. For all no des v , call B k ;v;X. Let T be the tree retur ned bythi s pro ce dure.

i1 v

2. Findthenoder for which dr;v+ cT i s minimize d.

v 4

3. Add thepath f rom r to r ,andthe tree T and retur n the re sultant tree.

1=i

Lemma4 Ak; r;X gives an O k partial approximation for D-Steinerk; r;X and hence

1=i

B k; r;X gives an O k approximation for D-Steinerk; r;X.

Pro of: byinduction on i.

The claim i s clearly true for i =1.We will provethe claim for i. Cons ider A k; r;X. Let T

i OP T

b e an optimal solution to D-Steinerk; r;Xand let C be its cost. k T k.Then,

OP T OP T

OP T

11=i

Setting T = T and x = k in Lemma2,we knowthatthere exi sts a tree T suchthat

OP T 1

11=i

kT k

OP T

cT 3x dT 3

1 OP T

1=i

0 0 11=i 0

Let T b e ro oted at r . Notethatdr;r cT . Theoptimal solution to D-Steinerk ;r ;X

1 OP T

0 1=i1

OP T

By theinductivehyp othesis, B k; r ;X gives an Ok has cost at most cT 3

i1 1

1=i

11=i 0

approximation to D-Steinerk ;r ;X, hence the cost of the tree T retur ned bythe call to

11=i 0

B k ;r ;Xis

11=i 1=i1

cT Ok cT

1=i

O k cT

O C

OP T

dr;r + cT OC

r OP T

Hence the minimum cost tree T retur ned by A k; r;Xhas cost O C . Also, the tree spans at

i OP T

11=i

least k terminals. Hence, thedens ityofthe tree

OP T

dT O

11=i

0 1=i

dT O k dT

OP T

1=i

Thi s proves that A k; r;X gives an Ok partial approximation to D-Steinerk; r;X.

B k; r;X uses A k; r;Xto construct an approximation for D-Steinerk; r;X. Us ing Lemma3,

i i

we knowthatB k; r;X gives a g kapproximation to D-Steinerk; r;Xsuchthat

gk= dx

1=i 1=i

where f x=Ox . Hence g k = Ok . Thi s completes theinductivestep.

An analys i s of the constant hidden in the O k notation reveals thatthe constant grows f airly

1=i

rapidly with i. Suppose B k; r;X gives a c k approximation. Suppose, in A k; r;X, we

i1 i1 i

0 0 11=i

invoke B k ;v;X with k = x k .Then, bytheanalys i s in theabove pro of, the cost of the

tree retur ned is at most

OP T

11=i 11=i

xk C + c xk

OP T i1

i=i1

=1+3c x C

i1 OP T 5

Hence thedens ityofthe tree i s at most

i=i1

C 1 + 3c x C 1

OP T i1 OP T

1=i 1=i1

k +3c x

11=i

x k

i1=i

Choosing x = , soasto minimize thi s dens ity,we get that A k; r;X gives an f k

partial approximation where

i1=i

1=i

k f k = i

Substitutingthi s in g k , we get

i1=i

2 1=i

g k = i k

1=i

Thi s means that B k; r;X gives a c k approximation where

i i

i1=i

c = i

i 1

Solving for c ,we get

1=i

x=i

x i1=2

c = i x 3

x=1

i=2

or c 3i .

11=i i

Lemma5 The running time of algorithm A k; r;X is O k n and that of B k; r;X is

i i

O k n .

Pro of: Weshall provethe claim byinduction on i. Assume it i s true for i 1. Weshall

11=i

provethe claim for i. A k; r;X calls B k ;v;X for all no des v .Thenumb er of calls to

i i1

11=i 11=i i1

B k ;v;Xisat most n, each witha runningtimeofOk n , givingarunningtime

11=i i

of O k n for A k; r;X.

11=i i

B k; r;Xinvokes A k; r;X rep eate dly, eachinvocation takes time O k n and further,

i i

11=i i

thevalue of k drops by k .Thus, we can charge a runningtimeofOn p er unit dropinthe

value of k . Thi s gives a total runningtimeofOkn for B k; r;X.

i 1=i

Theorem 1 For every i> 0, there is an algorithm which gives an approximation ratio of 3i k

for the Directed Steiner treeproblem and runs in time O kn .

log k

Corollary 1 There is a quasi-polynomial time algorithm which gives an approximation of O 2

for the Directed Steiner treeproblem.

Zelikovsky conjecture s in [18]that Directed Steiner tree problem cannot haveasubp olynomial

approximation guarantee unle ss P = NP . Contrary to hi s conjecture, we b elievethat Corollary 1

give s evidence that p olylogar ithmic approximation guarantee s are p oss ible. One reason for our b elief

log n

is the f act thatthere i s no problem known whichhas a hardnessof2 for some xed >0. 6 y x1 1 x y 2ab2

xppy

Figure 1: Thebunch < a;b;Y >

4 Directe d Generalize d Steiner Tree s

In thi s s ection we extend some of our technique s f rom Section3to obtain an approximation algor ithm

for the Directe d Generalize d Steiner tree problem. Formally the problem i s the following. Given a

directe d graph GV; Eanda setX =fg of k no de pairs, ndthe minimum cost subgraph

i i

H of G suchthat for eachnode pair 2 X,there exi sts a directed path f rom u to v in H .

i i i i

The cost of thesubgraph H is thesumofthe cost of all theedgesinH. Thi s problem has b een well

studie d in undirecte d graphs and constant f actor approximations are pre s ente d in [1,15].

As b efore, wework with a slightly more general problem we call DG-Steiner k; X whichis

the problem of nding a minimum cost subgraph H of G thatsati s e s at least k no de pairs f rom the

set X . Clearly, Directe d Generalize d Steiner tree i s a sp ecial cas e of DG-Steiner k; X . In thi s

1 2

log s ection, we presentan Ok k approximation totheDG-Steiner k; X problem.

Let C k; X denotethe cost of theoptimal solution to DG-Steiner k; X . De ne k H to

OP T

be the minimumofkandthenumber of node pairs in X that get sati s e d by H .Further, let cH

denotethe cost of H ,anddH= cH=k H bethedens ityof H.We will omit the parameters k; X

where their value s are clear f rom the context. A f k partial approximation to DG-Steiner k; X

can b e de nedinamanner s imilar tothat for D-Steinerk; X .

We assume without loss of generalitythatbetween every pair of vertice s ,there exi stsan

e dge of cost equal totheshortest pathdistance f rom u to v in G.Themain idea of the algor ithm i s

s imilar tothat for thesteiner tree problem. We can obtain a tr ivial k approximation by connecting

each pair withashortest pathbetween them. As b efore, if optimal's solution i s much

i i

better than thi s, it must b e the cas e thatthere exi sts a longpath whichisshare d bymany pairs.

Our ob jectiveisto ndsuchapathandasetofterminals whichshare it, thus obtaininga subgraph

of go o d dens ity.However, unlikeintheroote d cas e, the structure of thebunches we lo ok for are

di erentand provingthe exi stence of go o d bunches involve s more work. We formally de newhat

wemean bya bunch b elow.

De nition 4 Let Y = f;:::;< u ;v >g be a subset of X containing p k node

1 1 p p

pairs. Also let a and b be vertices of G.Abunch B =< a;b;Y > isagraph with the vertex set

fa; b; x ;x ;:::x ;y ;y ;:::y g. B has an edge of cost identical to that of the edge

1 2 p 1 2 p

in the graph G. B also has edges and , with costs identical to those of

i i i

and in G.

Figure 1 illustrates the structure of a bunch. Notice thatthex sandy s are unique even though

i i

the u sandv smay not b e. So B may not b e a subgraph of G.However, givenabunch B we

i i

can ndasubgraph of G thathas cost no more than cB andsati s e s eachofthekBnode pairs

sati s e d by B .

In Figure 2 we give an algor ithm C k; X that ndsabunch of minimumdens ity.Wethen prove

log k . Thi s allows the exi stence of a bunch B k; X thathas dens ity no more than O dH k

OP T 7

1 d1; B

2 for all pairs 2 VGVG

3 for each pair 2X, s[u; v ]c+c

4 sort X in increasing order of s. Let refer to the j th pair in

j j

this sorted list.

5 for p going from 1 to k

6 C c+s[u ;v ]+s[u ;v ]+ :::+ s[u ;v ]

1 1 2 2 p p

7if C=p d then dC=p; B ;:::;< u v >g

1 1 ; p

8output B

Figure 2: Algor ithm C k; X

3 3

us to claim that C provides an Ok log k partial-approximation to DG-Steiner k; x. Wethen

apply Lemma3to obtain an approximation to DG-Steiner k; x.

4.1 FindingtheBestBunch

To ndthe best bunch, we ndthelowe st cost bunch for all p oss ible value s of a, b,andp.We

choose the minimumdens itybunchoutofthese at most n k lowe st cost bunche s. Figure 2 formally

de scr ib e s an algor ithm C k; X which nds the best bunch.

The pro of of the following lemma i s quite s imple andhence we omit it.

Lemma6 The algorithm C k; X nds a minimum density bunch.

4.2 Low Dens ityBunche s Exi st

2 1

For convenience, wede ne hk ;k > 1tobethe quantity6k log k . Also, let h1 = 1.

Theorem 2 There exists a bunch B k; X such that dB C =k hk .

OP T

Pro of: Suppose there exi sts a no de pair 2X suchthatthedistance b etween u and v is

no more than C =k hk . Then g > is the require d bunch, andwe are done.

OP T

Notice thatif k=1 then suchanode pair always exi sts. Therefore we will implicitly assume k 2

for therestofthe pro of.

If there exi stsnosuchnode pair, then in theoptimal solution H k; X eachnode pair must b e

s eparated byadistance of at least C =k hk . Also, no no de pair can b e s eparated by more

OP T

than C .For i 1, let X denotethe s et of no de pairs suchthatthedistance f rom u

OP T i

h i

C C

i1 i

OP T OP T

to v in H k; X lie s in the range 2 hk ; 2 hk . Let I denotethemaximum

MAX

k k

value of i for which X i s non empty. Since k 2,

& '

I log log k

MAX

6 log k

.Therefore there exi stsani suchthatjX j . Let H b e a minimum cost subgraph of H

log k

0 0

sati sfying all no de pairs in X .For eachnode pair 2X , let P u; v betheshortest path

i i

i 8

between u and v in H .Ifthere are more than oneshortest paths b etween u and v ,choose oneof

them arbitrar ily.

i 1

Claim 1 There exists an edge e 2 E H such that at least 2 of the paths pass through e.

log k

i 1

OP T

Thesumoftheshortest paths b etween all thenode pairs in X i s greater than 2 hk

. Butthe cost of H i s no more than C .Thus there has to b e an e dge whichisshare d by

OP T

log k

i 1

at least 2 paths, provingtheabove claim.

log k

Wenow concentrateonthe e dge e = guarantee d bytheabove claim, andthesetX of

no de pairs suchthatP a; b pass e s through e. Eachofthese paths can b e split intothree

parts, P a; u, the e dge and P v; b. Let H be theunion of paths of the form P a; u ie.

the rst comp onents and H be theunion of paths of the form P v; b ie. thethird comp onents.

Clearly cH C and cH C . Let T be a shorte st-incoming-path tree ro oted at u

1 OP T 2 OP T 1

in the graph H and T be a shorte st-outgoing-path tree ro oted at v inthe graph H .

1 2 2

Let dist a; ubethe cost of thepath f rom a to u in T . dist v; bisde ne d s imilarly. Let

T 1 T

1 2

D = MAX dist a; u+ dist b; v . By de nition of thesets X, D i s no more than

2X T T i

1 2

C =k hk 2 . Wenow dividethe tree s T and T . Let c repre s entthe quantity

OP T 1 2

log k

intoat most c s egments, eachofdepth C =c.Nodeab elongs to s egment i of T if dist a; u 2

OP T 1 T

h i h i

C C C C

OP T OP T OP T OP T

i 1 . Similarly,nodebb elongs to s egment j of T if dist v; b 2 j 1 . ;i ;j

2 T

c c c c

De nition 5 We de ne k 1 i c; 1 j ctobe the number of node pairs 2 X

ij e

such that a belongs to segment i of T and b belongs to segment j of T .Also, let n i 2be the

1 2

number of nodes a 2 T which satisfy the fol lowing properties:

1. a belongs to segment i 1 of T .

2. There exists no vertex a in segment i 1 of T such that a lies on the path from a to u.

3. There exists a vertex a in segment i of T such that a lies on the path from a to u.

1 2

De ne n to be 1. The quantities n are similarly de ned on the tree T .Let n be the product

2 ij

1 2

of n and n .

i j

Informally, n repre s entsthenumb er of branche s of the tree T that cross theentire i 1-th

s egmentof T .

i 1 2

Lemma7 There exist 1 i c and 1 j c such that k =n 2 =c .

ij ij

log k

Pro of of Lemma7: To provethi s claim, wemakethe following obs ervations:

hk 0

i 1

k 2

log k

1i;j c

n c

1i;j c

The rst obs ervation follows f rom the f actsthat eachnode pair that b elongs to X contr ibutes

to k for some pair i; j . For the s econd obs ervation, weusethe f act that is bounded by c, n

i 9

s ince thenumb er of branches that cross an entire s egmentofdepth C =c can b e no more than c

OP T

remember thatthe cost of T can b e no more than C . n i s s imilarly b ounded. Now,

1 OP T

X X

1 2

n = n n

i j

1i;j c 1i;j c

X X

1 2

n n

i j

1ic 1j c

P P

i 1 2

2 =c ,we are guarantee d thatthere exi sts a pair Since k = n

ij ij

1i;j c 1i;j c

log k

i; j ; 1 i; j c whichsati s e s the require d condition. Thi s completes the pro of of Lemma7.

0 0 0

Claim 2 There exist nodes a and b in T and T ,respectively, and a set X X such that

1 2 e

0 i 1 2 0

jX j 2 =c and for each node pair 2X the fol lowing properties hold:

log k

0 0

1. a lies on the path from a to u in T and b lies on the path from v to b in T .

1 2

0 0

2. dist a; a 2C =c and dist b ;b 2C =c

T OP T T OP T

1 2

Thi s claim follows f rom Lemma7.

0 0 0 0 0 0

Let thebunch B be de ned as . Let k = k B bethenumber of node pairs

i 1 2 0 0

2 =c . Also, let cB bethe cost of thi s bunch. Theshortest sati s e d bythi s bunchk

log k

0 0

path f rom any rst comp onentof X to a is at most 2C =c,by construction. Theshortest path

OP T

0 0 0 0

f rom a to b is at most D .Theshortest path f rom b toany s econd comp onentof X i s also at most

0 0 0

2C =c.Therefore, cB D +4k C =c.Wenow giveaboundonthedens ityofB . Recall

OP T OP T

1 2 1

3 3 3

that D C =k hk 2 , c = ,andhk=6k log k .

OP T

log k

0 0 0

dB = cB =k

D=k +4C =c

OP T

C =k hk 2

OP T

+4C =c

OP T

0 hk

i 1 2

2 =c

log k

OP T

= 4k=c +2c log k

OP T

hk =

Thi s completes the pro of of Theorem 2. The following corollary follows f rom Theorem 2 and

Lemma6.

Corollary 2 The algorithm C k; X produces a hk partial approximation to DG-Steiner k; x.

Theorem 3 C k; X can be used to obtain a polynomial time algorithm which gives an approximation

2 1

ratio of O k log k for the Directed Generalized Steiner treeproblem.

Pro of: Follows f rom Corollary 2 and Lemma3. 10

5 Group Steiner Tree andOther Relate d Problems

In thi s s ection weshowhowthe re sults in Section 3 can b e us e d to obtain O k -approximations for

several relate d problems.

5.1 The Group Steiner Tree Problem

The Group Steiner Tree Problem i s the following: Given a graph GV; E, a ro ot r ,and a collection

of groups X = fg ;g ; :::; g g;g V , ndthe minimum cost tree T that connectsat least onevertex

1 2 k i

in eachofthe groups g tothe ro ot. There i s also a directed vers ion of thi s problem. Also, as in the

Directed Steiner Tree Problem, there can b e a var iantwhere jX j >k butwenee d to connect only

k of the groups tothe ro ot.

It i s easy to s ee thattheUndirecte d Group Steiner Tree Problem i s at least as hard as the

Minimum Connecte d Dominating Set Problem in undirecte d graphs andistherefore unlikely tobe

approximated belowafactor of log n. Also, theUndirecte d as well as Directe d Group Steiner Tree

Problem re duce s tothe Directed Steiner Tree Problem as follows: For each group g intro duce a

new dummyvertex x . Draw directe d zero cost e dge s f rom x to eachofthevertice s in g .We

i i i

nowhave a directe d graph even if westarted out f rom an undirected one. Construct an instance

of the Directed Steiner Tree Problem on thi s new graph withthe given ro ot r and with x :::x

i k

as thevertice s thatnee d to b e connecte d. It i s easy to s ee thatany solution tothe Group Steiner

Tree Problem corre sp onds to an equal-cost solution of the Directed Steiner Problem, and vice versa.

Thi s gives an Ok approximation for the Directe d as well as theUndirected vers ions of the Group

Steiner Tree Problem.

5.2 No deWe ighted var iants

For directe d problems there i s a s imple approximation pre s ervingreductions b etween thenode

we ighted andthe e dge we ighted cases. Therefore all our re sults carry over tothenodewe ighte d cas e

as well.

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