This paper appeared in: Networks 47:2 (2006), -5 Lower Bounds for the Reative Greed Agorithm for Approimating Steiner Trees Stefan Hougard Stefan Kirchner Humbodt-Universität zu Berin Institut für Informatik 0099 Berin, GERMANY {hougard, kirchner}@informatik.hu-berin.de Abstract. The Steiner tree probem is to find a shortest subgraph that spans a given set of vertices in a graph. This probem is known to be NP-hard and it is we known that a ponomia time 2-approimation agorithm eists. In 996 Zeikovsk [] suggested an approimation agorithm for the Steiner tree probem that is caed the reative greed agorithm. Ti toda the performance ratio of this agorithm is not known. Zeikovsk provided.694 as an upper bound and Gröp, Hougard, Nierhoff and Pröme [6] proved that.333 is a ower bound. In this paper we improve the ower bound for the performance ratio of the reative greed agorithm to.385. Kewords. approimation agorithms, anasis of agorithms, graph agorithms, Steiner tree probem, ower bounds Introduction The Steiner tree probem is to find a shortest subgraph that spans a given set of vertices in a graph. Karp [8] has shown that this probem is NPhard. B a resut of Bern and Passmann [2] it foows from [] that the Steiner tree probem in graphs is even APX-compete. This means that uness P=NP there eists some constant c > such that no ponomia time approimation agorithm for the Steiner tree probem can eist that has a performance ratio smaer than c.
This paper appeared in: Networks 47:2 (2006), -5 It is we known that via the computation of a minimum spanning tree one can get an approimate soution for the Steiner tree probem in graphs that has at most twice the ength of an optima soution [5]. For a ong time no better approimation agorithm for the Steiner tree probem in graphs was found unti in 990 Zeikovsk [0] suggested an agorithm with a performance ratio of /6. Nowadas severa approimation agorithms for the Steiner tree probem are known that have a performance ratio better than two. The current best one is due to Robins and Zeikovsk [9] and achieves a performance ratio ess than.550. In 996 Zeikovsk [] proposed an approimation agorithm for the Steiner tree probem that is caed the reative greed agorithm. He proved that this agorithm has a performance ratio better than.694. Ti toda it is not known what the precise performance ratio of Zeikovsk s agorithm is. Better knowedge of this vaue is of great interest as it might turn out that the reative greed agorithm has a performance ratio beow.550. Gröp, Hougard, Nierhoff and Pröme [6] provided.333 as a ower bound for this vaue. In this paper we wi improve this ower bound to.385. The paper is organized as foows. In Section 2 we introduce some genera notations and in Section 3 we present detais of the reative greed agorithm. Section 4 contains our proof for the new ower bound. We cose in Section 5 with some concuding remarks. 2 Preiminaries Given a graph G = (V, E) with edge engths w : E R + and a subset R V of terminas a Steiner tree is a tree in G that contains a vertices of R. The ength of a Steiner tree is the sum of the engths of the edges contained in it. A shortest possibe Steiner tree is caed a Steiner minima tree. We denote it b SMT and its ength b smt. An approimation agorithm with performance ratio c is an agorithm that for a possibe instances computes a soution that is at most b a factor c arger than the optima soution. A known approimation agorithms for the Steiner tree probem that have a performance ratio better than 2 use the concept of k-steiner trees. This idea was introduced b Zeikovsk [0]. A tree is fu, if a eaves are terminas and a inner vertices are non-terminas. If a tree is not fu it can be decomposed into severa fu subtrees caed fu components which are edge disjoint. If each of the fu components has at most k eaves, these components form a k-steiner tree for this set of terminas. A shortest possibe k-steiner tree is caed a minima k-steiner 2
This paper appeared in: Networks 47:2 (2006), -5 tree and is denoted b SMT k and its ength b smt k. Borchers and Du [3] have shown that for arge enough k the ratio between smt and smt k tends to : Theorem (Borchers ( and Du ) [3]) For a instances of the Steiner tree probem hods: smt k + og k smt. For a given graph G = (V, E) with edge engths w : E R + and terminas R V we define the termina distance graph as a compete graph with verte set R. The ength of an edge in the termina distance graph is defined as the ength of a shortest path between these two terminas in G. A minima spanning tree in the termina distance graph is denoted b MST and its ength b mst. It is we known that mst 2 smt hods [5]. B repacing the edges of the MST in the termina distance graph b the paths in G that defined the edge engths one gets a Steiner tree in G which is a 2-approimation of the Steiner minima tree in G. 3 The Reative Greed Agorithm The main idea of the reative greed agorithm (see Figure ) is to start with a Steiner tree that is obtained via the minimum spanning tree in the termina distance graph. This soution is successive improved b adding certain Steiner minima trees on at most k terminas of R and destroing the resuting cces. Note that for a constant number of terminas a Steiner minima tree can be computed in ponomia time [4]. Reative Greed Agorithm G = (V, E), R V i := 0 whie mst(r/t,..., t i ) 0 choose t i+ R, t i+ k, so that f i (t i+ ) := i := i + i ma := i return = i ma i= SMT (t i) smt(t i+ ) mst(r/t,...,t i ) mst(r/t,...,t i+ ) is minima Figure : The reative greed agorithm For a subset t i of the terminas R we denote b SMT (t i ) a Steiner minima tree for these terminas and b smt(t i ) its ength. For subsets 3
This paper appeared in: Networks 47:2 (2006), -5 t, t 2,..., t i of terminas we define mst(r/t,..., t i ) as the ength of a minima spanning tree in the termina distance graph to which for each set t j, j i one has added edges of ength 0 compete connecting the terminas in t j. For a tupe t i+ of terminas the difference mst(r/t,..., t i ) mst(r/t,..., t i, t i+ ) is the change in ength that is obtained b adding a Steiner minima tree on terminas t i+. The reative greed agorithm chooses in each iteration a tupe t i+ that maimizes this difference in reation to the ength smt(t i+ ), i.e. it chooses a tupe t that minimizes the function smt(t) f i (t) := mst(r/t,..., t i ) mst(r/t,..., t i, t). () The reative greed agorithm outputs the union of the Steiner minima trees of the chosen tupes t i as the fu components of the k-steiner tree. We denote b RGA k a soution returned b the reative greed agorithm that is obtained b considering tupes of size at most k and b rga k its ength. Zeikovsk [] proved that the reative greed agorithm has a performance ratio of ess than.694 for sufficient arge k. 4 Lower Bounds In [6] a ower bound for the performance ratio of the reative greed agorithm arbitrar cose to 4 3 was presented. First, we present a simper eampe ieding a ower bound of 4 30 ɛ which is etended in a second step to get a arger ower bound. The fist instance G k is shown in Figure 2. It contains 4k terminas w,..., w 4k, three specia vertices T a, T b, and T c, 2k vertices,..., 2k and k vertices z,..., z k. Each termina w 4i 3 is connected to T a and each termina w 4i and the termina w are connected to T b b an edge of ength (the vaue of wi be specified ater). Moreover each verte i is connected to the terminas w 2i and w 2i b edges of ength, the vertices z i are connected to 2i and 2i b edges of ength /2 and the vertices z i are connected to T c b edges of ength for i k. Lemma For ever ɛ > 0 and = 2 rga 0 ɛ it hods im 4k (G k ) k smt 4k (G k ) = 4 30 ɛ. Proof. The vaue of wi be chosen in such a wa that the reative greed agorithm is forced to choose the stars with centers T a and T b in the first two steps. In a further steps it wi seect fu components of size 2 on. The Steiner minima tree consists of the binar tree with root T c and 4
This paper appeared in: Networks 47:2 (2006), -5 T b T a 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 3 4 5 6 7 8 9 0 2 3 4 5 6 T c Figure 2: Lower bound (G k, k = 6) has ength 6k. In the foowing we wi assume > 2. This assumption is justified ater. The dashed edges in Figure 2 form a minimum spanning tree in the termina distance graph and have ength 7k + (k ) 2. Let t a and t b denote the set of terminas that are connected to T a and T b respective. B t c we denote the set of a terminas in G k. We want the reative greed agorithm to seect the star with center T a as the first fu component. This requires that the inequait f 0 (t a ) = hods, which is true for k f 0 (t a ) = k (k ) 2 (k + ) (k ) 2 + 3 = f 0(t b ) (2) 2 2 3. In addition it must be the case that k (k ) 2 6k 7k + (k ) 2 = f 0(t c ) (3) which is true for k 2 2 5 2. Note that for no other subset of terminas the function f 0 can be smaer than f 0 (t a ). This shows that the greed agorithm chooses the star with center T a in the first step. For the second step, using smmetr and monotonicit of the function f i, it is enough to consider the subsets t = {w, w 2, w 3, w 4 }, t 2 = t {w 4i i k} {w 4i i k}, t 3 = t 2 {w 4i 2 i k} and t b. 5
The reative greed agorithm chooses t b in the second step if f (t b ) = (k + ) 3k min (f (t ), f (t 2 ), f (t 3 )) = min ( 5 7, 3.5 k + 2.5 5 k + 2, 5 k + 7 k ) = 7k + 5 0k + 4 for k 5. (4) If ɛ > 0 and is chosen as 2 0 ɛ then for sufficient arge k the reative greed agorithm wi choose t a and t b in the first two steps. In a remaining steps the reative greed agorithm chooses a fu component of size two. Therefore the tota ength of the Steiner tree returned b the reative greed agorithm is k + (k + ) + 2k 2 = 4k + + 2k. (5) Setting = 2 0 ɛ we get as the ratio between the soution found b the reative greed agorithm and the optima soution: 4k + 2 0 ɛ + 2k( 2 0 ɛ) 6k = 4 30 ɛ 3 + 2 0ɛ 6k (6) which tends to 4 30 ɛ 3 for k. A B A B B 0 0 2 Figure 3: Improved ower bound (G k,, k = 3, = 8) 6 This paper appeared in: Networks 47:2 (2006), -5
This paper appeared in: Networks 47:2 (2006), -5 The second ower bound is obtained b constructing an instance G k, which paces the instance G k into a grid. The instance G k, consists of an 4k grid of terminas where the ast terminas of each coumn have been identified as one termina. For each coumn of terminas the graph G k T b T c is added, whie the terminas of each odd row (ecept for row 4k ) and the identified termina are connected aternate b stars with center B i and A i (see Figure 3). The edges which are incident to the vertices with degree k have weight. The identified edges with weight in G k have weight. The weights a i = a i (k,,, ) and b i = b i (k,,, ) of the edges incident to A i resp. B i wi be specified ater but assume here that a i > ++, ma{a i } < min{b i } and the sequences a i and b i are monotone increasing. Ever other edge has weight. Let w i,j denote the termina in the ith coumn and jth row and N(A i ) the eaves of the star a i. The idea is that in the instance G k, the soution of the reative greed agorithm consists of a stars with center A i and B i and a fu components of size two. The optima soution consists of the vertica trees. This wi be proved in the foowing theorem. Theorem 2 The performance ratio of the reative greed agorithm is not better than.385. Proof. Since b j > a i > + + the minimum spanning tree has ength mst = ((6 + 2)k + 2( + + )(k )). (7) The change in ength of a minimun spanning tree after seection of a star A i N(A i ) is 2( + + ). Now we have to find the minimizing vertica component after the stars (A 0 N(A 0 ))... (A i N(A i )) have been seected. For a fied h with h et S h = {w i,h i 4k}. For the ith step, using smmetr and monotonicit of the function f i, it is enough to consider the subsets S i, h = S h\({w 4j,h j i} {w 4j,h j i}) and S i,2 h = {w j,h 4i + j 4k}. For the seection function of N(A i ) we get f i (N(A i )) = + ai(k,,, ) 2( + + ) min(f i(s i, h ), f i(s i,2 h )) ( (4 + 2 + )k (2 + )i = min, (4 + 2)i mst 7
This paper appeared in: Networks 47:2 (2006), -5 ) (k i)(4 + 2 + ) (k i)(6 + 2) + (k i )(2 + 2 + 2) = f i (S i, h ). (8) Therefore we can specif the vaue of a i (k,,, ) = + 2( + + ) ((4 + 2 + )k (2 + )i). (9) (4 + 2)i Notice that a i is monotone increasing. After a A-stars are seected the argument for seecting the stars with center B i is simiar. On the change in ength of a minimun spanning tree after seection of a star B i N(B i ) has changed and is 2 + 2. Again using monotonicit, the minimizing vertica component is mst S i,3 h = {w 4j+κ,h j i < k, κ 2} {w 4k,h, w 4k,h } after ever A-star and the stars B j N(B j ) with j < i < k have been contracted. For the seection function of N(B i ) we get f k i+ (N(B i )) = + f k i+ (S i,3 Therefore we can specif the vaue of b i (k,,, ) = + = bi(k,,, ) 2 + 2 h ) (k i)(2 + + ) + 2 +. (0) (k i)(4 + 2) + 2 (k i) (2 + + ) + 2 + 2( + ). () (k i) (4 + 2) + 2 Notice for the ast star B k N(B k ) the component S k,3 h consists of on four terminas and there is no edge with costs in SMT (S k,3 h ). So we get b k (k,,, ) = 2 + 2( + ) + 3 +. (2) Notice that b i is monotone increasing. After ever star A i N(A i ) and B i N(B i ) is seected the agorithm has to choose fu components of size two. This can on be satisfied if 4k < min((2 + 2 + )k +, (3 + 2)k), because otherwise the agorithm 8
This paper appeared in: Networks 47:2 (2006), -5 chooses the components {w 2i,h i 2k} {w 4k,h } resp. {w 4i+j,h 0 i < k, 2 j 4}. If ever contraint is satisfied we get a ower bound for the performance ratio of 4k + ( + ) k 2 i=0 a i + ( + ) k i=0 b i. (3) (4 + 2 + )k This is the case for the vaues k = 5, 5, = 0.8688 and = 0.063 and so we get a ower bound of.385383. 5 Concuding Remarks Further improvement on the performance ratio for the reative greed agorithm is sti an open probem and the gap between our new ower bound.385 and.694 remains arge. Especia it woud be ver interesting to know whether the reative greed agorithm has a performance ratio better than.550. References [] S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szeged, Proof verification and hardness of approimation probems, J. ACM 45 (998), 50 555. [2] M. Bern, P. Passmann, The Steiner probem with edge engths and 2, Inform. Process. Lett. 32 (989), 7 76. [3] A. Borchers, D.-Z. Du, The k-steiner ratio in graphs, SIAM J. Comput. 26 (997), 857 869. [4] S. Drefus, R. Wagner, The Steiner probem in graphs, Networks (972), 95 207. [5] E.N. Gibert, H.O. Poak, Steiner minima trees, SIAM J. App. Math. 6 (968), 29 [6] C. Gröp, S. Hougard, T. Nierhoff, H.J. Pröme, Lower bounds for approimation agorithms for the Steiner tree probem, In: Proceedings of 27th Internationa Workshop on Graph-Theoretic Concepts in Computer Science, WG 200, Springer LNCS, 2204, 27 228. 9
This paper appeared in: Networks 47:2 (2006), -5 [7] C. Gröp, S. Hougard, T. Nierhoff, H.J. Pröme, Approimation Agorithms for the Steiner Tree Probem in Graphs, In: Steiner Trees in Industr, X. Cheng and D.-Z. Du (Eds.), 235 279, Kuwer Academic Pubishers, 200 [8] R.M. Karp, Reducibiit among combinatoria probems, In: Compeit of Computer Computations, (Proc. Smpos. IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 972). New York: Penum 972, 85 03. [9] G. Robins, A. Zeikovsk, Improved Steiner tree approimation in graphs, In: Proceedings of the Eeventh Annua ACM-SIAM Smposium on Discrete Agorithms, SODA 2000, 770 779. [0] A. Zeikovsk, An /6-approimation agorithm for the network Steiner probem, Agorithmica 9 (993), 463 470. [] A. Zeikovsk, Better approimation bounds for the network and Eucidean Steiner tree probems, technica report CS-96-06, Universit of Virginia, 996. 0