Integrality ratio for Group Steiner Trees and Directed Steiner Trees

Transcription

1 Integraity ratio for Group Steiner Trees and Directed Steiner Trees Eran Haperin Guy Kortsarz Robert Krauthgamer Aravind Srinivasan Nan Wang Abstract The natura reaxation for the Group Steiner Tree probem, as we as for its generaization, the Directed Steiner Tree probem, is a fow-based inear programming reaxation. We prove new ower bounds on the integraity ratio of this reaxation. For the Group Steiner Tree probem, we show the integraity ratio is Ω(og k), where k denotes the number of groups; this hods even for input graphs that are Hierarchicay We-Separated Trees, introduced by Barta [Symp. Foundations of Computer Science, pp , 1996], in which case this ower bound is tight. This aso appies for the Directed Steiner Tree probem. In terms of the number n of vertices, our resuts for the Directed Steiner probem og impy an Ω( n (og og n) ) integraity ratio. For both probems, these are the first ower bounds on the integraity ratio that are superogarithmic in the input size. This exhibits, for the first time, a reaxation of a natura optimization probem whose integraity ratio is known to be superogarithmic but subpoynomia. Our resuts and techniques have been used by Haperin and Krauthgamer [Symp. on Theory of Computing, pp , 003] to show comparabe inapproximabiity resuts, assuming that NP has no quasi-poynomia Las-Vegas agorithms. We aso show agorithmicay that the integraity ratio for Group Steiner Tree is much better for certain famiies of instances, which heps pinpoint the types of instances (parametrized by optima soutions to their fow-based reaxations) that appear to be most difficut to approximate. Keywords and phrases. Group Steiner Tree, Directed Steiner Tree, fow-based reaxation, inear programming reaxation, integraity ratio, approximation agorithms. Proposed running head: Integraity ratio for Group Steiner Trees 1 Introduction The Group-Steiner-Tree probem is a network design probem that generaizes both Set-Cover and the Steiner-Tree probem. The Directed-Steiner-Tree probem is a further generaization of Group-Steiner-Tree. The natura reaxation for these two probems is a fow-based inear programming A preiminary version of this work appeared in the Proc. ACM-SIAM Symposium on Discrete Agorithms, pages 75 84, 003. Internationa Computer Science Institute, 1947 Center St., Suite 600, Berkeey, CA 94704, USA. Work was done whie this author was aso affiiated with the Computer Science Division of U.C. Berkeey, supported in part by NSF grants CCR and CCR and DARPA cooperative agreement F Emai: heran@cs.princeton.edu Department of Computer Sciences, Rutgers University, Camden, NJ 0810, USA. Emai: guyk@camden.rutgers.edu IBM Amaden Research Center, 650 Harry Road, San Jose, CA 9510, USA. Work was done whie this author was with the Internationa Computer Science Institute and with the Computer Science Division of U.C. Berkeey, supported in part by NSF grants CCR and CCR and DARPA cooperative agreement F Emai: robi@amaden.ibm.com Department of Computer Science and Institute for Advanced Computer Studies, University of Maryand, Coege Park, MD 074, USA. E-mai: srin@cs.umd.edu. Supported in part by NSF Awards CCR and CNS Morgan Staney, Inc., New York, NY, USA. Work done whie at the Department of Computer Science, University of Maryand, Coege Park, MD 074, USA. Supported in part by NSF Awards CCR and CNS

2 reaxation. We show a poyogarithmic (about og squared) ower bound on the integraity ratio of this reaxation. For both probems, these are the first such ower bounds that are superogarithmic in the input size, and our bounds are, in fact, neary tight in the important specia case of input graphs which are tree networks. Let n be the number of vertices and k the number of groups. Our probabiistic approach and our anaysis have been used in [HK03] to show that for every fixed ɛ > 0, the Group-Steiner-Tree and Directed-Steiner-Tree probems admit no efficient Ω(og ɛ k) and Ω(og ɛ n) approximations respectivey, uness NP has quasi-poynomia time Las-Vegas agorithms. We aso present improved approximation agorithms for certain famiies of instances of the Group-Steiner-Tree probem, shedding ight on the type of instances that appear to be most difficut for the fow-based reaxation. (a). The Group Steiner Tree probem. The (undirected) Group-Steiner-Tree probem is the foowing. Given an undirected graph G = (V, E), a coection of subsets (caed groups) g 1, g,..., g k of V, and a weight w e 0 for each edge e E, the probem is to construct a minimum-weight tree in G that spans at east one vertex from each group. We can assume without oss of generaity that there is a distinguished vertex r V (caed the root) that must be incuded in the output tree. The case where g i = 1 for a i is just the cassica Steiner Tree probem; the case where G is a a star can be used to mode the Set-Cover probem (c.f., [GKR00]). A natura fow-based reaxation for this probem is the foowing. Find a capacity x e [0, 1] for each edge e E so that the capacities can support one unit of fow from r to g i, separatey for each g i (as opposed to supporting a unit fow simutaneousy for a g i ). Subject to this constraint, we want to minimize e w ex e. It is easy to check that the feasibe soutions which satisfy x e {0, 1} for a e, exacty correspond to feasibe soutions for the Group Steiner Tree probem; hence, the above fow-based reaxation is indeed a vaid inear programming (LP) reaxation for the probem. This is a natura reaxation for this probem (and for some of its generaizations), and is the main subject of investigation in this paper. We start with a usefu definition from [Bar96]. Definition 1.1. Let c > 1. A c-hierarchicay We-Separated Tree (c-hst) is a rooted weighted tree such that (i) a eaves are at the same distance from the root, (ii) the edges in the same eve are equa-weighted, and (iii) the weight of an edge is exacty 1/c times the weight of its parent edge. (Remark: Item (ii) is sighty stronger than the origina definition from [Bar96], but can be assumed without oss of generaity due to the anayses of [Bar96, Bar98, KRS01].) We simpy say HST when referring to a c-hst for an arbitrary constant c > 1. The first poyogarithmic approximation agorithm for the Group-Steiner-Tree probem was achieved in the eegant work of Garg, Konjevod and Ravi [GKR00]. A brief sketch of their O(og n og og n og N og k) approximation agorithm, where n = V and N = max i g i, is as foows. First, the powerfu resuts of [Bar98] are used to reduce the probem to the case where G is a tree T, with an O(og n og og n) factor oss in the approximation ratio. T can be furthermore assumed to be a c-hst for any desired constant c > 1. Next, sove the fow-based LP reaxation on T and round the fractiona soution into an integra soution for T by appying a nove randomized rounding approach that is deveoped in [GKR00]. It is estabished in [GKR00] that for any tree T, this randomized rounding eads to an O(og N og k) approximation. Thus, for the input graph G, we get an O(og n og og n og N og k) approximation. From a technica viewpoint, one of the main difficuties in [GKR00] is that a non-trivia anaysis of the randomized process is required. The anaysis uses Janson s inequaity in an interesting way. The work of [GKR00] has been extended and expanded in severa ways: Their agorithm was derandomized in [CCGG98, Sri01]; an aternative (combinatoria) agorithm is devised in [CEK06]; the oss incurred by the reduction to an HST is improved to O(og n) in [FRT04]. Since the first appearance of a poyogarithmic approximation for Group-Steiner-Tree (in the conference version of [GKR00] in 1998), there has been much interest in whether the approximation ratio can

3 be improved. One concrete notabe question in this regard has been the foowing: Can we achieve an approximation ratio better than O(og N og k) for trees? This is interesting for at east two reasons. First, since [GKR00] shows a reduction to the case of trees as seen above, an improved approximation for trees (or even for the case of c-hsts for some constant c > 1) woud directy ead to an improved approximation for genera graphs. Further, even the case where G is a star (which is a tree) captures the Set-Cover probem, for which o(og k) approximation is hard [LY94, Fei98, RS97], so there is an intriguing gap even on trees. Our main technica resut is that the integraity ratio of the fow-based reaxation for HSTs is Ω(og k). This bound is in fact tight an O(og k) bound on the integraity ratio hods for HSTs, as we show in Section.7. Both bounds hod for c-hsts where c > 1 is any fixed constant. Reca that the upper bound of [GKR00] for trees in genera is O(og N og k); our methods show an Ω(og N og k/ og og N) ower bound on the integraity ratio, even for a cass of HSTs. The same ower bounds hod aso for trees where a weights are the same (i.e., unit-weight trees). Finay, we show randomized rounding agorithms for the fow-based reaxation that ead to improved approximation agorithms for certain specia famiies of HSTs; this sheds ight on the type of instances that are most difficut to approximate. A og-squared ower bound on the integraity ratio for trees had been conjectured circa 1998 by Uri Feige. The specific (randomy constructed) instance he suggested for this purpose (see Section for more detais) has proven to be quite difficut to anayze. Our ower bound is shown via a sighty different random instance, which eiminates one source of correation in the random choices, and makes the construction more amenabe to anaysis. Nevertheess, we are required to do intricate estimates that deve into ow-order terms, in particuar when crafting the precise induction hypothesis ying at the heart of the proof. (b). The Directed Steiner Tree probem. This is the directed version of the (undirected) Steiner- Tree probem. Given an edge-weighted directed graph G = (V, E) that specifies a root vertex r and k termina nodes v 1, v,..., v k, the goa is to construct a minimum-weight out-branching rooted at r, which spans a the terminas. This probem is easiy seen to generaize the undirected Group-Steiner-Tree probem, as we as to be equivaent to the directed Group-Steiner-Tree probem. Aside from intrinsic interest, this probem is aso of current interest, e.g., in the context of muticasting in the Internet (where inter-node distances are often not symmetric). The poynomia-time approximation ratio currenty known for this probem is k ɛ, for any constant ɛ > 0 [CCC + 99]; their agorithm extends to a poyogarithmic approximation ratio in quasi-poynomia running time. The fow-based reaxation here is simiar: insta for every edge e E a capacity x e [0, 1], so that a unit of fow can be shipped from r to v i, separatey for any given i. Intriguingy, it was recenty shown in [ZK0] that this reaxation has an integraity ratio of Ω( k), precuding a poyog(k) approximation agorithm based on this reaxation. However, the exampes og constructed in [ZK0] have k = Θ( n ); hence, the resut of [ZK0] does not excude an O(og n) (og og n) og integraity ratio. Our Group-Steiner-Tree ower-bound resut above proves aso an Ω( n ) ower (og og n) bound on the integraity gap for the Directed-Steiner-Tree probem. The ony ower bound known previousy for Directed-Steiner-Tree was Ω(og n), since this probem generaizes Set-Cover. As mentioned above, our resuts have paved the way to the improved hardness of approximation resuts of [HK03], which show that, for any fixed ɛ > 0, Group-Steiner-Tree cannot be approximated within ratio og ɛ k and Directed-Steiner-Tree cannot be approximated within ratio og ɛ n, uness NP has quasi-poynomia time Las-Vegas agorithms. The infuence of our work on these hardness resuts is threefod. First, our ower bounds on the integraity ratio has motivated working on a hardness of approximation resut. Second, the insights our anaysis provides regarding the (edge-weight) structure of instances that are difficut to approximate inspired specific detais of the hardness reduction. Third, our main technica emma (whose proof is rather non-trivia) is in fact made crucia use of in [HK03]. 3

4 Organization. Our ower bounds on the integraity ratio of Group-Steiner-Tree and Directed- Steiner-Tree are shown in Section. We then prove agorithmicay that the integraity ratio of the former probem is much better for certain famiies of instances in Section 3; this pinpoints the type of instances that appear difficut for this reaxation. Finay, concuding remarks are made in Section 4. Lower bounds on the integraity ratio In this section we prove a ower bound of Ω(og k) on the integraity ratio of the fow-based reaxation og of the Group-Steiner-Tree probem even on HSTs. In terms of n, the gap is Ω( n ). We start (og og n) (Section.1) by describing the inear programming reaxation and constructing a famiy of -HST instances, accompanied by an overview of the anaysis; then, the main technica parts (Sections. and.3) anayze the fractiona and the integra soutions of this inear program. We aso show how simpe modifications to this construction extend the integraity ratio to unit-weight trees (Section.4) and to c-hsts for an arbitrary constant c > 1 (Section.5). We further point out how this immediatey eads to a ower bound og of Ω( n ) on the integraity ratio for the Directed-Steiner-Tree probem (Section.6). Finay, (og og n) the ower bound of Ω(og k) is shown to be tight (in Section.7)..1 The reaxation and the instance The cut-based reaxation (that is equivaent to the fow-based reaxation) for Group-Steiner-Tree is as foows. (Here, δ(s) is the set of edges with exacty one endpoint in S V.) Minimize w e x e e E x e 1, S V s.t. r S and S g j = for some group g j e δ(s) 0 x e 1, e E (1) Let T n be a -HST tree with n nodes defined by the foowing random process. Let the coection of groups be G = {g 1, g,..., g k }. The groups are defined by a random process. The vaue of k, as we as those of two other parameters H and d, wi be defined shorty (in terms of n). The height (i.e., depth) of T n is H, and every non-eaf vertex has d chidren. The root of T n is denoted r. The eve of a vertex is its depth; r is at eve 0, and there are H + 1 eves. An edge is said to be at eve i iff it connects a vertex at eve i 1 to a vertex at eve i. Each edge at eve i has weight 1/ i ; thus, for instance, edges incident at r have weight 1/. Each group g j is a subset of the eaves, described as foows. We sha associate a subset A() G of the groups with each eaf, and define each group g j to be the set of eaves for which g j A(). Thus, a soution that reaches a eaf by a path from r, covers a groups in A(). To define A() for each eaf, we now recursivey and randomy define a set A(v) for every node v in the tree (incuding non-eaf nodes), as foows. Proceed independenty for each group g j as foows. We start by etting g j A(r) with probabiity 1. In genera, if g j A(u) for some non-eaf node u, then for each chid v of u, we independenty put g j in A(v) with probabiity 1/. Thus, this random process goes top-down in the tree, independenty for each group. Note that the number of vertices in T n is n d H, where H is the height of the tree. Ceary, the expected size of every group is d H / H. Parameters and notation. We set d = c 0 og n for some universa constant c 0 > 0; this wi be used in some Chernoff bound arguments in Section.. It then foows that H = og n og d = Θ(og n/ og og n). We further set k = H ; thus, og k = Θ(og n/ og og n). Throughout, with high probabiity means with 4

5 probabiity that is at east, say, 1 1/n. A probabiities refer to the randomness in constructing the instance T n. Feige s instance. The construction suggested by Feige circa 1998 is the foowing: Take a compete tree of arity 4 (i.e., every non-eaf vertex has 4 chidren) and height og k; now generate k groups, each containing k eaves, by an independent randomized branching process that starts from the root and randomy picks two out of four chidren unti the eaves are reached. This random instance differs from T n in its degree, which is constant rather than ogarithmic, and in that the choices made (when generating a singe group) at each chid of a vertex are correated (rather than independent). Overview of the anaysis. We first show (Section.) that with high probabiity the instance T n has a feasibe fractiona soution of cost O(H). In this soution, a edges e in the same eve of the tree are assigned the same vaue x e, and this vaue is chosen so that the tota cost of every eve in the tree is the same, namey, O(1). The feasibiity of this soution is shown by empoying a sequence of Chernoff bound arguments, and this is the reason why the degree d of the tree must be (at east) ogarithmic. This is in contrast to Feige s instance, where the feasibiity (of a simiar fractiona soution) is guaranteed by construction, i.e., with probabiity 1, and thus the tree can have a constant degree. We then show (Section.3) that with high probabiity the cost of any integra soution for T n is ower bounded by Ω(H og k). At a high eve, we imitate the argument known for Set-Cover; we show that for any singe ow-cost integra soution (i.e., subtree of T n ), with high probabiity over the randomness in constructing the groups this soution is infeasibe (i.e., does not cover a the groups), and then we take a union bound over a possibe integra soutions. In fact, any singe vertex of T n together with its chidren is essentiay a standard Set-Cover instance with integraity ratio Ω(og k). The main technica work is to estimate the probabiity that an arbitrary (but fixed in advance) integra soution is feasibe. Unike in the Set-Cover scenario, where this is a straightforward cacuation, in the Group-Steiner-Tree instance T n the soution s structure comes into pay. For instance, the integra soution might not be a reguar subtree of T n, and its cost need not be spit eveny among the different eves (of T n ). We prove some upper bound on the probabiity that this soution for T n is feasibe. Our anaysis shows that to maximize this upper bound on the probabiity, it is essentiay best to have an even spit of the cost used under a vertex v (i.e., the edges of the soution that beong to the subtree rooted at v). Whie we do not caim that this is the worst-case for the exact probabiity of feasibiity, our upper bound gives a good enough estimate. However, the anaysis does not specify how many chidren of v shoud have a non-zero cost under them; in fact, this vaue is very sensitive to ower-order tradeoffs between different eves in the tree. The main difficuty in the anaysis is to disti the effect of H, the height of the tree, on the feasibiity probabiity. Our proof is by induction on the height of the tree, and uncovers a very deicate tradeoff between the height of a subtree and its cost. This tradeoff eventuay transates to the cost of the integra soution for T n having, on top of the og k term which comes from Set-Cover (i.e., a singe eve), aso a inear dependence on the height H. Due to seemingy technica imitations this proof works ony for H 1 og k, but we show in Section.7 that this is unavoidabe. Interestingy, there is an anaogy with the approximation agorithm of [GKR00], whose rounding procedure pays, at some intermediate stage, an O(H og k) factor, and then shows that, in effect, H can be upper bounded by O(og N).. The fractiona soution Reca that d = c 0 og n. We start with a coupe of propositions which show that if the constant c 0 is sufficienty arge, then certain quantities reated to our randomy chosen groups stay cose to their mean. Henceforth, the phrase with high probabiity wi mean with a probabiity of 1 o(1). 5

6 Proposition.1. Let c 0 be a sufficienty arge constant. Then, with high probabiity, a groups have size at east (d/) H /3. Proof. Fix j. We now show that if c 0 is arge enough, then Pr [ g j < (d/) H /3 ] 1/n. We may then appy the union bound over a j to concude the proof. Let δ = 1/4. Let X 1 be the number of vertices u at eve 1 (i.e., chidren of r) such that g j A(u). Then X 1 has Binomia distribution X 1 B(d, 1/), so by a Chernoff bound on the ower-tai (see e.g. [MR95]), [ Pr X 1 (1 δ) d ] e 1 δ d. Let X be the number of vertices u at eve such that g j A(u). Then X has binomia distribution X B(X 1 d, 1/). Suppose that X 1 > (1 δ)ie[x 1 ] = (1 δ) d. Then, it is immediate that X (X 1 > (1 δ) d ) stochasticay dominates a random variabe X B((1 δ) d d, 1/), i.e., Pr[X t] Pr[X t] for a t. By appying the Chernoff bound on X we get Pr [X (1 δ ] )(1 δ)(d ) e 1 ( δ ) (1 δ)( d ) Continue simiary for i = 3,..., H, by defining X i to be the number of vertices u at eve i such that g j A(u), and by assuming that X i 1 > (1 δ )... (1 δ i )(1 δ)( d )i. We get by the Chernoff bound that [ Pr X i (1 δ i 1 )... (1 δ ] )(1 δ)(d )i e 1 ( δ δ i 1 ) (1 i )... (1 δ )(1 δ)( d )i. () For any 0 < δ 1 we have 1 δ 1 1+δ e δ. Thus, (1 δ )... (1 δ δ i 1 )(1 δ) e i... δ δ e 4δ > 1 3. It foows that the tai-bound obtained in the righthand side of () is at most e Ω((d/8)i). Appying the union bound on these H events we get that with high probabiity none of them happens (if the constant c 0 is sufficienty arge), and in particuar, X H (1 δ )... (1 δ H 1 )(1 δ)( d )H 1 3 ( d )H. This concudes the proof of Proposition.1. The foowing proposition has a simiar proof; the main difference is that we wi now empoy Chernoff bounds on the upper-tai. Proposition.. Suppose that the constant c 0 is arge enough. Then with high probabiity, the foowing hods for every eve i and every group g j : If a vertex u at eve i is such that g j A(u), then the number of eaves in the subtree rooted at u which satisfy g j A(), is at most 3(d/) H i. Proof. Fix a pair (i, j) and a vertex u at eve i such that g j A(u). Let L(u) be the set of eaves of the subtree rooted at u. We now show that if c 0 is arge enough, then Pr [ g j L(u) > 3(d/) H i ] 1/n 3. We then appy a union bound over a (i, j, u) to concude the proof. Let δ = 1/4 < n 3. Let X 1 be the number of vertices v at eve 1 of the subtree rooted at u (i.e., chidren of u) such that g j A(v). Then X 1 has Binomia distribution X 1 B(d, 1/), so by a Chernoff bound on the upper-tai (see e.g. [MR95]), Pr [ X 1 (1 + δ) d ] e δ 3 d. Let X be the number of vertices v at eve of the subtree rooted at u such that g j A(v). Then X has binomia distribution X B(X 1 d, 1/). Suppose that X 1 < (1 + δ)ie[x 1 ] = (1 + δ) d. Then, it is immediate that X (X 1 < (1 + δ) d ) is stochasticay dominated by a random variabe X 6

7 B((1 + δ) d d, 1/), i.e., Pr[X t] Pr[X t] for a t. By appying the Chernoff bound on X we get Pr [X (1 + δ ] )(1 + δ)(d ) e 1 3 ( δ ) (1+δ)( d ) Continue simiary for = 3,..., H i, by defining X to be the number of vertices v at eve of the subtree rooted at u such that g j A(v), and by assuming that X < (1 + δ )... (1 + δ 1 )(1 + δ)( d ). We get by the Chernoff bound that [ Pr X (1 + δ 1 )... (1 + δ ] )(1 + δ)(d ) e 1 3 ( δ δ 1 ) (1+ )... (1+ δ )(1+δ)( d ). (3) The tai-bound obtained in the righthand side of (3) is ceary at most e Ω((d/8)). Appying the union bound on these H i H events we get that with probabiity at east 1 1/n 3 none of these events happen, if the constant c 0 is sufficienty arge; in particuar, X H i (1+ δ )... (1+ δ H i 1 )(1+δ)( d )H i 3( d )H i. (This is because of the foowing. For any δ we have 1 + δ e δ. Thus, (1 + δ )... (1 + δ 1 )(1 + δ) e δ δ +δ e δ < 3.) This concudes the proof of Proposition.. We now upper bound the vaue of LP (1) for the tree T n by exhibiting a feasibe soution for it: Let each edge e at each eve i have vaue ˆx e = 9 (/d) i. Lemma.3. With high probabiity, ˆx is a feasibe soution to LP (1). Its vaue is 9H. Proof. Observe that ˆx satisfies the constraints of LP (1) if (see aso [GKR00]), for every group g j, every cut (S, S) separating r from a the vertices of g j has capacity at east 1, where the capacity of each edge e is ˆx e. By the (singe-source) max-fow min-cut theorem (or, say, weak duaity) it suffices to show that for every group g j, a unit of fow can be shipped from the root r to the vertices of g j whie obeying the capacity ˆx e of each edge e. To this end, fix a group g j and define the fow f as foows. For every vertex v in g j (i.e., for every eaf v such that g j A(v)), ship 3 (/d) H units of fow aong the unique simpe path from r to v. By Proposition.1, the tota fow shipped to g j is at east g j 3 (/d) H 1 with high probabiity. Next, consider an edge connecting a node u at some eve i to its parent. If g j / A(u), no fow is shipped through this edge; if g j A(u), the tota fow shipped through this edge (i.e., through u) is, by Proposition., at most 3(d/) H i 3(/d) H = 9(/d) i with high probabiity. In both cases, the fow aong the edge obeys the edge s capacity. We concude that with high probabiity ˆx is a feasibe soution to LP (1). The vaue of the soution ˆx is H i=1 di 1/ i 9(/d) i = 9H since each eve i contains d i edges of weight 1/ i..3 The integra soution We now show that with probabiity 1 o(1) (over the random choice of the groups), a integra soutions have vaue Ω(H og k). Whenever we say that some T is a subtree of T n, we aow T to be an arbitrary connected subgraph of T n. Since T n is rooted, any subtree T of T n is aso thought of as rooted in the obvious way: the node in T of the smaest depth is the root of T (and is denoted root(t )). Aso, when we say that some T is a subtree of T n with root u, we aow T to be an arbitrary connected subgraph of T n with root u. Let M(c) be the number of subtrees of T n which are rooted at r and have tota weight at most c. Fix g j G. For any given subtree T of T n, et p j (T ) be the probabiity that no eaf of T beongs to the group g j, conditioned on the event that g j A(root(T )). Since by symmetry p j (T ) = p i (T ) for a i, j, we wi 7

8 simpy denote it by p(t ). We now define a key vaue f(h, i, c) as foows. Choose an arbitrary vertex u at eve i. Then f(h, i, c) is the minimum vaue of p(t ), taken over a possibe subtrees T that are rooted at u and have tota weight at most c. (If there is no such T, then f(h, i, c) = 1. Aso, it is easy to see by symmetry that f(h, i, c) does not depend upon the choice of j or u.) Let P c be the probabiity that there exists an integra soution of weight c. We wish to show that P c = o(1) for c that is smaer than a certain threshod of the order H og k. Using the independence between the different groups and appying a union bound over a possibe subtrees rooted at r that have tota weight c, we obtain P c M(c)(1 f(h, 0, c)) k. (4) We now have to ower bound f and upper bound M. We empoy the foowing crude bound on M(c). Note that it suffices to count ony subtrees of T n that span distinct sets of eaves (since the groups g i contain ony eaves). Observing that T n has d H eaves, and a subtree of tota weight at most c spans at most c H eaves (since each spanned eaf requires a distinct edge at eve H), we get that ( ) d H M(c) c H d chh. Our goa in the next subsection is to prove the foowing key emma. Lemma.4. For H 1 og k and a constant γ > 0 that is sufficienty arge, we have f(h, 0, c) e γc/h. Thus P c M(c)(1 f(h, 0, c)) k M(c) exp{ k e γc/h }. (5).3.1 Bounding f(h, 0, c). We start with some preiminaries. The main technica resut is Lemma.7 beow. It gives a more genera bound for f than stated in Lemma.4, and hence the proof of the atter woud foow quite easiy. Proposition.5. Let and β > 0. Then the minimum of S {1,...,} i S e βx i over a (x 1,..., x ) with a given i=1 x i is attained when a x i are equa. Proof. The minimum is ceary attained at some point (x 1,..., x ), so assume to the contrary that at this point not a x i are equa, say without oss of generaity that x 1 > i x i/ > x. We wi show that changing both x 1 and x to x 1+x decreases the above sum whie maintaining i x i, which contradicts the assumption that (x 1,..., x ) is a minimum point. Actuay, it suffices to prove that S {1,} i S e βxi > e β S {1,} i S x 1 +x, (6) since mutipying (6) by i S e βx i and summing over a S {3,..., } shows that changing x 1, x indeed decreases the above-mentioned sum. To prove (6), observe that it simpifies to e βx 1 + e βx > e β(x 1+x )/ which foows from the arithmetic mean-geometric mean inequaity since x 1 x. This competes the proof of Proposition.5. It is easy to check (by considering higher derivatives) that for a B 0, e B 1 B + B B3 6. (7) 8

9 Proposition.6. For a B 0 > 0 there exists δ > 0 such that for a B B 0 we have 1+e B e B +δ. Proof. Fix B 0 > 0. We first make sure that the inequaity hods at B 0. By the arithmetic mean-geometric mean inequaity 1+e B 0 > e B0/ (since B 0 > 0) so a sufficienty sma δ > 0 satisfies 1+e B 0 > e B0/(+δ). It now suffices to make sure that for a B B 0 the derivative of the efthand side is at east that of the righthand side, i.e., that 1 e B 1 +δ e B/(+δ). This hods for any 0 < δ < B 0 since +δ = 1 + δ/ 1 + B 0 / e B0/ e B/ e B B/(+δ), competing the proof of Proposition.6. Lemma.7. Let γ be a sufficienty arge constant. Then f(h, h, c) exp( γch (H h) ) for a c > 0 and a 0 h H 1. Proof. Fix B 0 to be an arbitrary positive constant, and et δ > 0 be the corresponding constant from Proposition.6. The proof is by backward induction on h, i.e., we assume that the caim hods for h + 1 and prove it for h, where h H. We wi consider the base case, which is the case that h H 1 6 δ, ater on. In order to bound f, we derive a recurrence reation for f(h, h, c). Reca the definition of f(h, h, c): fix an arbitrary vertex u at eve h, and take the minimum vaue of p(t ), over a possibe subtrees T rooted at u such that the tota weight of T is at most c. We bound f(h, h, c) by considering a possibiities of u having = 1,,..., d chidren and a possibe partitions x () = (x 1, x,..., x ) of the weight c to (the 1 subtrees under) these chidren; since the edge from u to each of its chidren has weight, we get that h+1 i=1 x i = c. We then get that h+1 f(h, h, c) min { min 1 d x () :x i 0, P { 1 i x i=c h+1 S {1,...,} i S f(h, h + 1, x i )}}; since once the chidren of u are chosen, we ony need to consider the subset S of a chidren with g j in their A( ) set. (Each such set S occurs with probabiity 1/.) Pugging the induction hypothesis in, we get that f(h, h, c) min { min 1 d x () :x i 0, P { 1 x i =c h+1 S {1,...,} i S exp( γx i h+1 )}}. (8) (H h 1) For any, we have by Proposition.5 that the righthand side of (8) is minimized when a x i are equa to c 1. We thus get that h+1 ( ( 1 f(h, h, c) min 1 d exp γ( c )) 1 ) h+1 S h+1 (H h 1) S {1,...,} ) ( exp ( γ( c )) 1 i = min 1 d = min 1 d 1 i=0 ( i 1 + exp ( γ( c 1 h+1 ) h+1 (H h 1) ) h+1 )h+1 (H h 1). Fix arbitrariy such that 1 d. Let B = γ( c 1 ) h+1 h+1 (H h 1) and C = γ c h. To compete the (H h) 9

10 induction, we want to prove that ( 1+e B ) e C, i.e., that 1 + e B e C. (9) We have four cases. Case 1: In this case we assume that C B. By the arithmetic mean-geometric mean inequaity we have that 1+e B e B/ e C, which proves (9). Case : In this case we assume that B B 0 and B +δ C B Proposition.6. Then we have from Proposition.6 that 1+e B e B ; reca that δ is the constant from +δ e C, which proves (9). Case 3: In this case we assume that C B and B < B 0. Then by (7) we have 1+e B 1 B + B 4 B3 Since C 0, we have (by Tayor s Theorem) that e C 1 C + C. Thus, it suffices to prove that Since B < B 0 1 B3 we have that therefore suffices to prove that Note that 1 B + B 4 B3 1 1 C + C. 1 B 4, and then since C B, we have that B 4 B3 1 5B 4 5C 6. It C + C 3 B. B C γ h+1 c (H h 1) (H h) γ (H h 1). Pugging in the vaues of B and C and simpifying we get that it suffices to prove that γ h+1 c c (H h 1) (H h) γ γ h + (H h 1) 3(H h) 4. If h+ c H h 1, then the desired inequaity indeed hods since γ h+1 c inequaity hods for any γ 96, since then, γ c h 3(H h) 4 = γ 6 h+ c H h Case 4: In this case we assume that C < + δ B C = c 1 h+1 c (H h 1) (H h) γ c h 1 (H h) 3 16 γ c h 1 (H h) 3 γ h+1 c (H h 1) (H h). B +δ. Note that for h H, (H h) (H h) (H h 1) (H h 1) + 6 H h γ (H h 1). Otherwise, the Thus, h H 1 6 δ. Since δ > 0 is a constant, this is reay the base case of the induction, which we sha prove directy. Consider a subtree T of weight at most c that is rooted at a vertex u at eve h. Since u has at most c h+1 chidren in T, each not having the group g j in its A( ) set independenty with probabiity 1/, with probabiity at east ch+1 the subtree T does not cover g j. Thus, f(h, h, c) e ch+1. Choosing a constant γ (1 + 6 δ ), we get that γ (H h), and thus This concudes the proof of Lemma.7. f(h, h, c) e ch+1 exp( ch γ (H h) ). 10

11 Proof of Lemma.4. Lemma.7 impies that f(h, 0, c) e γc/h. Pugging this into (4), we get that P c M(c)(1 f(h, 0, c)) k M(c) exp{ k e γc/h }..3. Bounding the weight of an integra soution. By Lemma.4 in conjunction with (5), we have P c exp{ch H og d ke γ c H }. Now, suppose that c 1 4γ H n k. Then ch H = O( H H 3 og k). Recaing that H = 1 og k, we have P c exp{õ( k) Ω(k 3/4 )} = o(1). We concude that with high probabiity no subtree of weight at most 1 4γ H og k covers a the groups, and thus an optima integra soution has vaue at east Ω(H og k). Since LP (1) has a fractiona feasibe soution of vaue 9H, we get: Theorem.8. The integraity ratio of the reaxation (1) for Group-Steiner-Tree ( is Ω(og k). In terms of N, k, the integraity gap is Ω(og k og N/ og og N) and in terms of n it is Ω og n ). (og og n).4 Integraity ratio for unit-weight trees The above anaysis gives a ower bound on the integraity gap for Group-Steiner-Tree in HSTs. A consequent interesting question is whether the LP is tighter for unit-weight trees. We show that a sight modification of the trees described above gives the same integraity ratio ower bounds for unit-weight trees. The idea is very simpe reca that in our random construction T n, edges at eve i had weight 1/ i ; repacing each such edge by a path of H i unit-weight edges does not reay change our integraity ratio argument, because the resuting instance T n is essentiay equivaent to the instance T n with edge weights scaed up by a factor of H. Formay, it is easy to verify that our fractiona soution for T n naturay yieds a fractiona soution for T n with vaue 9H H, and that an optima integra soution for T n with vaue OPT corresponds to an integra soution with vaue OPT / H for T n. Since we know the atter vaue is at east Ω(H og k), we concude that OPT = Ω(H H og k), and the integraity ratio of T n is Ω(H og k) = Ω(og k) = Ω(og k og N/ og og N). Furthermore, the tota number of vertices in T n is at most H n = O(n og ), and hence the integraity ratio is aso Ω( n ) in terms of the number of vertices (og og n) in T n..5 Integraity ratio for c-hsts Straightforward modifications of our integraity ratio proof for Group-Steiner-Tree in -HSTs ead to the same ower bounds for c-hsts, for arbitrary constant c > 1. Here, we take an aternative approach; instead of going through the whoe proof, we show that our ower bounds for -HSTs impy (in a back-box manner) simiar bounds for c-hsts, where c > 1 is an arbitrary constant. We first consider the case c > and then use it to hande the case 1 < c <. 11

12 An arbitrary constant c >. In this case our -HST instance T n can be modified into a c-hst instance T n as foows. The set of vertices of T n is a subset of the vertices of T n. For every j = 0, 1,,... iterativey (up to about og c H ), et i = i(j) be the (unique) integer such that i c j < i+1, and incude a the eve i vertices of T n as the eve j vertices in T n. For exampe, at j = 0, we incude the root of T n as the root of T n because 1 c 0 <. We may assume that the height of T n is chosen so that at some iteration j 0, we incude in T n the eaves of T n (i.e., i(j 0 ) = H), at which point the iterations are stopped. With this assumption (and since a the groups in T n contain ony eaves), we aso get that T n and T n have exacty the same k groups. Finay, two vertices at two consecutive eves j 1, j in T n are connected by an edge of weight 1/c j whenever, in T n, one of the two vertices is an ancestor of the other one. For exampe, the edges incident at the root of T n have weight 1/c. Notice that T n is a c-hst with height j 0 og c H = H/ og c. A fractiona soution LP for T n, of vaue p, say, naturay induces a fractiona soution to T n with vaue at most p. Indeed, we et the fractiona vaue of an edge connecting a vertex u to its parent v in T n be the equa to the fractiona vaue of the edge connecting (the same vertex) u to its parent v in T n. It is easy to see that whenever this fractiona soution for T n pays (fractionay) 1/c j for an edge, the soution LP pays 1/ i for the corresponding edge in T n, with 1/c j 1/ i. Since the corresponding edges in T n are distinct, the vaue of the constructed soution for T n is at most p. An optima integra soution OPT for T n, of vaue opt, say, naturay induces an integra soution INT for T n with vaue at most O(c) opt. Indeed, simpy take in T n the (minima) subtree that spans exacty the same eaves (that are spanned by the soution for T n). It is easy to see that whenever OPT pays 1/c j for an edge connecting a vertex u to its parent v in T n, the soution INT has to pay for the path between u and its ancestor v in T n. The tota weight of this path is at most 1/ i + 1/ i / i = O(1/ i ) = O(c) 1/ i = O(c) 1/c j. We thus get an integra soution for T n with vaue O(c) opt. Combining these arguments with our bounds on the fractiona and integra soutions for T n yieds a ower bound of Ω( 1 c H og k) on the integraity ratio in c-hsts. Notice aso that the number of vertices in T n is simiar to that in T n because they have the same eaves. For fixed c >, we thus get the same integraity ratio ower bounds in c-hsts as in -HSTs, namey, Ω(og k) = Ω(og k og N/ og og N) and og aso Ω( n ) in terms of the number of vertices in T (og og n) n. An arbitrary constant 1 < c <. Let t be the smaest integer such that c t >, and define q = 1 + c + c c t 1. The above construction then yieds a c t -HST instance T n. Now repace every edge of weight 1/(c t ) j in T n with a path of t edges having weights 1/(qc tj (t 1) ), 1/(qc tj (t ) ),..., 1/(qc tj ). Ceary, the resuting instance T n is a c-hst. Notice further that the tota weight of the above t-path in T n is (c t 1 + c t )/(qc tj ) = 1/c tj, i.e., equa to the edge in T n it repaced. Hence, any fractiona soution for T n yieds a fractiona soution for T n with the same vaue, and aso an optima integra soution in T n yieds an integra soution for T n with the same vaue. Since the number of vertices in T n is arger than that in T n by ony a constant factor of t og c 4 = O( 1 c 1 ), we get the same integraity ratio ower bounds in c-hsts as in -HSTs, namey, Ω(og og k) = Ω(og k og N/ og og N) and aso Ω( n ) in (og og n) terms of the number of vertices in T n..6 Integraity ratio for Directed Steiner Tree The above resuts immediatey ead to a ower bound of Ω( ) on the integraity ratio for the (og og n) og Directed-Steiner-Tree probem. Let T n be an instance as described above with Ω( n ) integraity (og og n) ratio for Group-Steiner-Tree, and construct a Directed-Steiner-Tree instance as foows. Orient a the edges of T n away from the root r. Then, introduce new nodes v 1, v,..., v k, and for each j and each u g j, introduce a zero-weight arc from u to v j. This defines a Directed-Steiner-Tree instance I which is essentiay the same as I: fractiona soutions for the two probems map bijectivey, with identica og n 1

13 tota weights, and the same thing hods aso for integra soutions. Observe that the number of vertices in og the resuting graph is n + k n, and thus the ower bound of Ω( n ) on the integraity ratio for T (og og n) n hods aso for I..7 An O(og k)-approximation for Group Steiner Tree in HSTs We now show a tight O(og k)-approximation agorithm for the Group-Steiner-Tree probem on HSTs. This was obtained jointy with Anupam Gupta and R. Ravi, and we thank them for aowing us to incude this agorithm here. Our agorithm uses the rounding procedure of [GKR00] as a subroutine and takes advantage of the geometricay-decreasing-weights property of HSTs. Let T be the HST instance of the Group-Steiner- Tree probem. We assume this tree is a -HST, i.e., the weight of each edge in the (i + 1)-st eve equas haf the weight of its parent edge in the i-th eve, and each edge in the first eve has weight exacty one; the agorithm extends in a simpe way to c-hsts. We aso assume that the height of the tree is H > og k; otherwise, the approximation ratio O(H og k) in [GKR00] aready impies the O(og k) upper bound. It is not difficut to arrange that a members of each group are at the eaves of the HST (with ony a constant factor increase in the optimum vaue). Our agorithm is as foows: 1. Create a new tree T consisting of ony the first H = og k eves of T. For each eaf in T find its corresponding node in eve og k of T, and assign to a groups that appear in the subtree of rooted at in T.. Run the approximation agorithm of [GKR00] on the Group-Steiner-Tree instance T. Let SOL be the vaue of the soution obtained and et OPT be the vaue of the optima soution in T. The anaysis of [GKR00] shows that SOL = O(H og k OPT ) = O(og k OPT ). 3. From the soution SOL (which is a subtree of T ), we construct a soution SOL for T as foows: (a) Find the subtree S in T that corresponds to SOL, and incude S in SOL. (b) For each group g G, repeat the foowing steps. Find a eaf in SOL that beongs to g (there must be such a eaf since SOL covers g), and et be the eve H vertex in T corresponding to. Now find in the subtree under in T a eaf u that beongs to g (there must be such eaf because of the way we assigned groups in T ), and add to SOL the path that connects to its descendant u. It is easy to verify that the above procedure produces a vaid soution SOL to the -HST instance T. We caim that SOL = O(og k OPT), where OPT is the optimum soution vaue in T. Indeed, SOL consists of SOL and at most k paths (one path per group) added in step 3(b). Because T is an HST, each of these paths has tota weight O(1/k) (since its edges have geometricay decreasing weights), and thus, SOL = SOL + O(1). Since the optima soution must contain at east one edge in the first eve (and thus having weight one), we get that OPT 1. It is aso obvious that OPT OPT, and therefore, SOL = SOL + O(1) = O(og k OPT ) + O(1) = O(og k OPT). 3 Improved approximations for certain famiies of trees To better understand the approximabiity of the Group-Steiner-Tree probem, one may consider the foowing question: What are the instances (in particuar, trees) that are difficut to approximate better than within ratio O(og k og N)? We partiay answer this question by presenting a significanty better 13

14 approximation ratio for a certain famiy of trees, which differs from the trees constructed in Section in a crucia way. This improved approximation aso sheds ight on the instances T n constructed in Section. For exampe, it may be tempting to beieve, at a first gance, that the edge weights pose an unnecessary compication to T n. Notice that the uniform weight version of T n (i.e., a tree simiar to T n, except that a its edges have the same weight), has the same fractiona soutions as T n. Furthermore, it can be verified that for both T n and its uniform weight version, the fractiona soution presented in Section. is, with high probabiity, near-optima, and that appying to it the rounding procedure of [GKR00] yieds an integra soution with vaue arger by a Θ(og k og N) factor than the reaxation vaue. However, in the uniform weight version of T n, the contribution of eve-i edges to the reaxation vaue increases significanty with i; and as we sha soon see, this impies that an approximation ratio better than O(og k og N) is possibe. This expains why the weights in T n are necessary they make every eve have the same contribution to the reaxation vaue. This aso eucidates the disparity between the performance of a rounding procedure for a reaxation and the integraity ratio of the reaxation the uniform weight version of T n exhibits a arge ratio according to the former measure, but a significanty smaer ratio according to the atter. Technicay, fix a Group-Steiner-Tree instance T on a tree of height H, and an optima soution x e to its fow-based reaxation LP (1). Define zi to be the tota contribution of the edges at eve i (of T ) to the objective function of the reaxation. We show that the reationship between the different zi pays a crucia roe in the strength/weakness of the LP: if for some constant α > 1 we have zi+1 αz i for a i, then we can achieve an O(og k og og(kn)/ og α) = O(og k og og(kn)) approximation. This may suggest that instances with zi z i+1 for a/most i are among the worst cases for the reaxation. The foowing emma proves the improved approximation ratio for the case where α =. The argument easiy extends to any constant factor α > 1. We sometimes refer to a vaid integra soution simpy as a cover. Lemma 3.1. If z i+1 z i for a i, then we can find an integra soution of vaue O(z og k og og(kn)), where z = H i=1 z i denotes the optima LP vaue. Before getting into the forma proof, et us outine the main ideas. We separate the tree T into a ower part, that contains the owest Θ(og og(kn)) eves of the tree, and an upper part, that contains the rest of the tree. Let z (U), z (L) be the contributions of the upper and ower parts to the fractiona soution vaue z. Notice that z (U) z /poyog(kn) since zi+1 z i for a i. We can thus take care of the upper part as foows. We use the same randomized rounding as in [GKR00] for the upper part, ony that now we repeat the process about O(og N) more times (mutipicativey) this resuts in a soution that is consideraby more expensive with respect to z (U), but it is sti not too much with respect to the tota fractiona soution z. Since we repeat the rounding procedure more times, we cover each group more times (in a way that is formaized in the proof). Now every eaf of the upper part that we managed to cover, can be regarded, in the ower part, as the root of a subtree. This aows us to appy the agorithm of [GKR00] to some of the subtrees in the ower part, namey, to those subtrees whose root was covered by our upper part soution. By the anaysis of [GKR00] we ony need to pay proportionay to the height of the ower part (times O(og k)), i.e. the ower part soution has vaue O(z (L) og k og og(kn)). In the case where zi+1 αz i for a i, we define the ower part to be the owest Θ(og og(kn)/ og α) eves of the tree, Proof. We may assume that a groups contain ony eaves of T, by adding zero weight edges. Let L i be the set of edges at eve i. Let h = og og(kn). Let U = {e : e L i for i H h} and L = {e : e L i for i > H h}. For every e U, et y e be x e rounded upwards to the nearest power of, increasing the LP vaue by a factor of at most. We first find a cover of U, as foows. Let c 1 > 0 be a suitaby arge constant. For every e U, assign ˆx e = min{1, x e c 1 og k og (kn)}, and use one iteration of the rounding scheme presented in 14

15 [GKR00] (w.r.t. ˆx e ) to sove the probem in U. The expected tota weight of this soution is at most O(z (U) og k og (kn)) O(z og k), where z (U) = H h i=1 z i is the tota contribution to z of the edges in U. Using arguments simiar to those in [GKR00], we now wish to show that from the perspective of U, every group g is covered sufficienty many times, with high probabiity. Let e 1,..., e m be the eaves of (the subtree induced on) U that ead to g (i.e., g contains at east one of their descendants in T ). A unit amount of fow can be shipped in T from the root to g, under the LP vaues x e (as capacities), because x e is a feasibe soution. Let f 1,..., f m be the corresponding fows on the edges e 1,..., e m. Ceary, j f 1 j = 1. Partition the m fows, etting A i = {j : < f i j 1 }. Let B(g) = {i : i i 1 og N and j A i f j > 1/4 og N}. It is easy to see that the fow in the remaining sets A i is at most ( ) + ( og N) N N 4 og N < 3/4, and thus i B(g) j A i f j 1/4. We can therefore ignore the remaining sets and focus on the fows in B(g). Fix i B(g) and et V i be the set of eaves (of U) e j, for j A i, that are chosen by the [GKR00] procedure in U according to ˆx e. For the sake of upper bounding the ower-tai of V i, we may assume that the capacity x e on every edge e equas the tota fow shipped aong the edge e (since a arger capacity x e just increases V i ). Thus, the expectation of V i is µ i = j A i min{1, f j c 1 og k og (kn)}. If c 1 i og k og (kn) 1, then V i = A i = µ i with probabiity 1. Otherwise, µ i c 1 f j j A i og k og (kn) c 1 og k og(kn) 8, and by Janson s inequaity [Jan90], Pr[ V i µ i /] e Ω( µ i ) + i /µ i, where i = e e Pr[e and e are chosen]; here, the sum is over pairs of distinct edges e A i and e A i whose events of being chosen are not independent. By the proofs in [GKR00, KRS0], it is easy to see that i O(µ i og k), where the constant in the O( ) is an absoute constant that is independent of c 1. Thus, Pr[ V i µ i /] e Ω(µ i/ og k) e Ω(c 1 og(kn)), where the constants in the Ω( ) are absoute constants that are independent of c 1. There are ony k groups, and for each one B(g) O(og N). Thus, by choosing c 1 sufficienty arge, we get by the union bound (over a poynomia in kn number of V i s) that with high probabiity, for every group g and every i B(g), V i µ i /. Reca that at east 1/4 of the tota fow in f 1,..., f m is shipped through sets A i with i B(g), and that the fows among each such set A i are a equa, up to a constant factor. Hence, at east Ω(1) of the unit amount of fow into g must be shipped using the eaves of U chosen by the [GKR00] procedure. We next appy the rounding agorithm of [GKR00] to L with the vaues x e, starting from every chosen vertex of U. Since we know from above that one can ship to any group g, an Ω(1) amount of fow from the eve H h vertices chosen in the soution for U, we get that x e satisfies the LP constraints, up to a constant factor. It is proven in [GKR00], that after O(h og k) iterations of the rounding scheme, with high probabiity a the groups are covered. We now caim that the expected cost of each such iteration is at most z. Indeed, the probabiity to choose and edge e is proportiona to its fractiona vaue x e, and the caim foows by the inearity of expectation. Therefore, the expected cost of this soution is O(z h og k + z og k) = z O(og k og og(kn)). 4 Discussion Our resuts improve the current understanding of the integraity ratio of the fow-based reaxation for the Group Steiner Tree probem, but some very intriguing gaps sti remain. Athough for HSTs our Ω(og k) ower bound is tight, for genera trees there is a sight sack between our Ω(og k og N/ og og N) 15