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Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp254-259) A Survey of Approximation Algorithms for Multicast Congestion Problems RONGJOU YANG HANN-JANG HO SINGLING LEE Wufeng Institute of Technology Wufeng Institute of Technology National Chung-Cheng University Dept. of IM Dept. of CSIE Dept. of CSIE 117, Chian-Kuo Rd., Sec. 2, 117, Chian-Kuo Rd., Sec. 2, 160, San-Hsing, Ming-Hsiung , Chia-yi Ming-Hsiung , Chia-yi Ming-Hsiung, Chiayi Taiwan, R.O.C. Taiwan, R.O.C. Taiwan, R.O.C. r h s Abstract: Due to the recently rapid development of multimedia applications, multicast has become the critical technique in many network applications. In this paper, We investigate contemporary research concerning multicast congestion problems with the objective of minimizing the maximum sharing of a link. These problems include: multicast Steiner tree and multicast packing problem, etc. Most of these problems have already been proved as NP- complete, thus are mainly formulated as the Integer Linear Programming (ILP). Our objective is to investigate and analyze some of most recently developed approximation algorithms for the optimization of multicast congestion problems. We also discus how they are modelled and solved in the literature. Key–Words: Approximation Algorithm, Multicast packing, Multicast Steiner trees, Randomized metarounding, Integer Linear Programming, LP-Relaxation 1 Introduction optimal packing problem formulation. The delay is a function of the amount of dilation α from the size In multicast routing, the main objective is to send data of the optimal tree obtained for each group multicast from one or more sources to multiple destinations in independently from the others (i.e., in isolation). order to minimize the usage of resources such as band- width, communication time and connection costs. The Priwan [1] proposed both heuristic algorithm multicast congestion problem is to ﬁnd a set of multi- ﬁnding approximate solution and search enumera- cast trees that minimize the maximum congestion over tion based algorithm ﬁnding optimal solution, and all its edges. The congestion of an edge is the number compared the approximate solution with the optimal of multicast trees that use the edge. Given a physical solution in order to lower costs for the subscribers network G = (V, E) with a set V of n nodes, a set E and conserves bandwidth resources for the network of undirected network links and m multicast requests providers. In these algorithms, the connection ap- S = S1 , S2 , . . . , Sm being subsets of V , a solution proach is based on setting multicast tree routes that to the problem is a set of m trees such that the ith each participant (site) has one own multicast tree con- tree spans the nodes of the ith multicast request. The necting to the other participants under two constraints: objective function is to minimize the maximum con- the delay-bounded constraint of source-destination gestion. The problem is formulated as an ILP (Integer path and the available constrained bandwidth for the Linear Programming) and its LP relaxation solution service of links. ﬁnds fractional solutions for each multicast request. In [3], Wang formulated the problem as a tree In the multicast packing problem, the network packing problem with multiple multicast sessions tries to accommodate simultaneously all the multicast under a capacity limited constraint and proposed groups (many-to-many) and avoid bottlenecks on the two heuristic algorithms, Steiner-tree-based heuristic links to achieve higher throughput (i.e., minimize the (STH) algorithm and cut-set-based heuristic (CSH) al- maximum link sharing among the multicast groups). gorithm, for solving this problem. They showed that A shared tree can be considered as the backbone of the STH algorithm can ﬁnd a better approximate solu- a group multicast session. One way to minimize the tion in a shorter computation time compared to CSH. maximum congestion is to increase the size of some The remainder of this paper is organized as fol- multicast trees, but this also increases the delay which lows: Section 2 discusses the multicast Steiner Trees must be considered in the objective function of the problems in general graphs with the objective of min- Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp254-259) imizing the maximum sharing of a link. We outline Randomized rounding is a probabilistic method and analyze some of most recent approximation algo- to convert a solution of a relaxed problem into an rithms and related lower bounds for these problems. approximate solution to the original problem. Re- In Section 3, we discuss the Multicast Packing Prob- laxation is an optimization problem with an enlarged lem. Discussion is given in Section 4. feasible region and extended objective function com- pared with an original optimization problem. Let G = (V, E) denote a physical network of n 2 The Multicast Steiner Trees Prob- nodes, S1 , S2 , · · · , Sm denote m multicast requests, a binary variable xte indicating whether edge e is cho- lem sen for the tth multicast, to ensure that any solution to the ILP connects all the vertices of each multicast, The Steiner tree problem is solving combinatorial op- the problem is formulated as follows by Vempala and timization problem when adding new vertices is per- o V¨cking in [7] : mitted before ﬁnding the Minimum Spanning Tree (MST). It can be divided into three categories: Eu- Minimize z clidean Steiner tree problem, metric Steiner tree prob- Subject to lem, and Steiner tree in graphs which is focussed on xte ≥ 1, ∀t, ∀S ⊂ V, S ∩ St = ∅, e∈δ(S) this paper. The Steiner tree problem has applications (V \S) ∩ St = ∅ in circuit layout or network design [12, 13, 14, 15, 16]. xte ≤ z, ∀e ∈ E Most versions of the Steiner tree problem are NP- t complete (computationally hard). Some restricted xte ∈ 0, 1, ∀t, ∀e ∈ E cases can be solved in polynomial time. In practice, In the case for each multicast consisting of only heuristics are used. two nodes, a LP relaxation solution was obtained by In this section, we discuss the multicast Steiner relaxing the binary variable to 0 ≤ xte ≤ 1 and any Trees problems including the objective functions and fractional solution of the ILP is decomposed into sev- approximation algorithms for minimizing the sum of eral paths. Each path is associated with a fractional the congestion over all edges. weight so that the sum of the weights of the frac- The Steiner tree problem is to minimize the sum tional paths for each multicast is 1 and the sum of instead of the maximum of the congestion over all the weights of the fractional paths crossing the edge edges in which a single multicast request consists of e corresponds to the weight of that edge. more than two nodes. Finding edge disjoint paths In the case for each multicast consisting of more (the boolean satisﬁability problem) was ﬁrst proved than two nodes, the LP relaxation was described in as a NP-complete problem by Karp back in 1972 [4]. terms of a multicommodity ﬂow between pairs of Finding a minimum Steiner tree is max-SNP hard (see nodes of each multicast. [6] for details) and an approximate ratio solution (1+ ) Let the binary variable ft (i, j) denote the ﬂow be- ( > 0 is a constant) for the special case of edge tween the nodes i and j in the tth multicast and the length equal to 1 or 2 was found in [5]. In Steiner tree xte (i, j) denote the ﬂow on edge e of commodity (i, j) problem, different graphs are formed dynamically as in the tth multicast, the problem is formulated as the different multicast Steiner trees so that the maximum following ILP: ﬂows of the generated multicast Steiner trees are min- Minimize z imized. The congestion of an edge is the number of Subject to multicast trees that use the edge. The problem is to xte (i, j) ∈ ft (i, j), ∀ t, ∀ i < j ∈ St ﬁnd a set of multicast trees that minimize the maxi- ft (i, j) ≥ 1, ∀t, ∀S ⊂ V mum congestion over all the edges. i∈S∩St ,j∈(V \S)∩St xte (i, j) ≤ z, ∀e ∈ E t,i,j 0 ≤ xte ≤ 1 2.1 The Iterative Randomized Rounding Al- An iterative randomized rounding algorithm was gorithm o proposed as follows by Vempala and V¨cking: A Linear Programming (LP) for any Integer Program- Step 1. Decompose the fractional solution into ﬂow ming (IP) can be generated by taking the same ob- paths. jective function and same constraints but with the re- Step 2. Choose one path randomly out of each mul- quirement that variables are integer replaced by ap- ticast node with probability equal to the value of the propriate continuous constraints. The LP relaxation ﬂow on the path, i.e., randomized rounding. of the IP is the LP obtained by omitting all integer Step 3. If the multicast nodes are all connected then and 0-1 constraints on variables. stop and output the solution, otherwise, contract the Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp254-259) vertices corresponding to the connected components dron, the problem of the LP relaxation P of a positive and form a new multicast problem with regarding the ILP along with an r-approximation algorithm is for- contracted vertices as the new multicast nodes. mulated as follows: Step 4. The original fractional solution was derived r · x∗ ≥ λj · xj (1) from the solution of the new multicast problem de- j composed from the fractional solution. Repeat the where λj = 1, λj ≥ 0, for ∀ j. j above steps till all multicast nodes are connected. In order to construct a set of xj ’s which satis- In this way, there are at most log k iterations ﬁes (1), let x∗ denote a feasible solution for P (I), (k is the maximum number of nodes in a multicast). ext(Z) = {xj |j ∈ J}, E denote the index set for With high probability, the congestion of the solution the variables in P (I), and xc denote the solution for found by the algorithm has an approximation bound each non-negative object function c returned by the less than O(log k · OP T + log n). r-approximation algorithm A, the problem is formu- lated as the following ILP and solved in order to ob- 2.2 The General Randomized Rounding Al- tain (1): gorithm Maximize j∈J λj subject to Randomization is a powerful technique in ﬁnding ap- j ∗ j∈J λj · xe ≤ r · xe , ∀e ∈ E (2) proximate solutions to difﬁcult problems in combi- Maximize j∈J λj ≤ 1 natorial optimization by solving a relaxation (usually λj ≥ 0, ∀j ∈ J linear programming relaxations or semideﬁnite pro- The solution λ∗ of (2) provides an explicit con- gramming relaxations) of a problem and then using vex decomposition into points in ext(Z), i.e., r · x∗ ≥ randomization to return from the relaxation to the λ∗ · xj , J := {j ∈ J|λ∗ ≥ 0. The sum- j j original optimization problem. j∈J De-randomization can be applied by using stan- mation in this inequality is a linear combination of dard techniques to yield deterministic polynomial- {xj |j ∈ J } ⊂ ext(Z) dominated by rx∗ and it is time algorithms that yield approximations as good a constructed convex combination if λ∗ = 1. j as those given by the randomized algorithms they j∈J are derived from, even though the process of de- Obviously, (2) has an exponential number of vari- randomization typically takes a relatively simple and ables and in order to obtain an approximate bound on clean randomized rounding procedure and turns it into its dual ILP as follows, it was further solved by em- a complex and generally slower deterministic algo- ploying the r-approximation algorithm: rithm. Minimize r · x∗ · w + z In [8], Carr and Vempala proposed a general ran- subject to domized rounding algorithm in polynomial time for xj · w + z ≥ 1, ∀j ∈ J (3) constructing a convex combination based on the ellip- we ≥ 0, ∀e ∈ E soid method. z≥0 Assume that a polynomial-time algorithm A is an (3) also has an exponential number of variables r-approximation algorithm to a min-ILP problem with and was solved in polynomial time by using the ellip- the LP relaxation P then A ﬁnds a solution with the soid method and has an optimal solution of 1. cost being at most r (integrality gap) times the cost of the optimal solution to the LP relaxation P of the ILP. A min-ILP or max-ILP problem is a minimization 3 The Multicast Packing Problem or maximization problem, respectively, whose set of In the Multicast Packing Problem, there are a num- feasible solutions can be described by a positive ILP. ber of applications which try to use the network for Let P denote a LP relaxation of ILP by employing the purpose of establishing connection and sending the r-approximation algorithm A, x∗ denotes a feasi- information, organized in different groups. Thus, the ble solution of P(I), Z denote an integer polytope, network capacity must be shared accordingly with the and P denote a LP relaxation of Z, then according to requirements of each group based on known heuris- [9], r · x∗ dominates a convex combination of extreme tics for constructing Steiner trees and the cut-set prob- points of Z(I). lem or using integer programming to ﬁnd the min- Also let I denote a min-ILP problem, Z(I) de- imum cost under bounded tree depth and the cost note the integer polyhedron for the ILP, P (I) denotes minimization under bounded degree for intermediate the LP relaxation of Z(I), xj denote an extreme point nodes [3, 1, 2]. of Z(I), x∗ denote a feasible solution of P(I), and In [19], Noronha and Tobagi proposed an efﬁ- ext(P ) denote the set of extreme points for a polyhe- cient solution by employing decomposition principle Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp254-259) to speed up LP of the problem and by enhancing tions might produce high-cost multicast trees. In order value-ﬁxing rule to prune the search space of the IP. to bound the cost of each multicast tree to guarantee Ofek and Yener [11] presented a window-based re- service quality for some cases, the size of some trees liable multcast protocol with a combined sender and may be increased. Assume that, for simplicity, each receiver initiation of the recovery protocol in order to link has the same cost such as unit cost and the cost of combine the multicast operation with the internal ﬂow a multicast tree is the total number of its links. control. They proposed two Min-Max objective func- Given k ∈ K (k is a multicast group and K is the tions: one for delay which is caused by the number set of multicast groups), Let OP T k denote the cost of links needed to connect the multicast group; the of the least-cost multicast tree, if the ratio of the cost other for congestion which is caused by sharing a link of the multicast tree in the solution to the cost of the o among multiple multicast groups. In [20], Gr¨tschel, least-cost multicast tree OP T k exceeds a threshold Martin, and Weismantel considered the Steiner tree α ≤ 1, then a cost based on the dilation from the size packing problem from polyhedral point of view, called of the optimal tree obtained for each group multicast polyhedron, which can be described by means of in- independently from the others is incurred. equalities that deﬁne the facets of the Steiner tree Also given a link e ∈ E, let Ke ⊆ K denote polyhedrons. They presented joint-inequalities for the set of multicast groups that use link e in the solu- these polydrons based on cut-and-branch algorithm. tion, ze denotes the total congestion on link e such that Baldi, Ofek and Yener [21] proposed an approach ze = k∈Ke tk where tk is the amount of trafﬁc gen- based on the embedding of multiple virtual rings, one erated by the multicast group k, λ = maxe {ze } de- for each multicast group, to route messages to all the note the maximum congestion, a binary variables xk e participants of multicast group while minimizing the for all e ∈ E and k ∈ K, tk denoting a trafﬁc load of bound on the buffer sizes and queueing delays so as multicast group k ∈ K, ST k = {xk ∈ {0, 1}|E| : xk to resolve two problems raised from the time-driven induces a Steiner tree spanning M k , e∈E we · xk de- e priority ﬂow control scheme proposed in [22]: one is note the cost of the multicast tree in the solution for the scheduling problem in which how time intervals multicast group k ∈ K, OP T k denote the size of the are reserved to each multicast group and the other is optimal tree for multicast group k in isolation, P k ≥ 0 the adaptive sharing problem in which how the active denote the cost coefﬁcients, and π k ≥ 0 used to mea- (transmitting) participants can dynamically share the sure the threshold, in [10], the tree packing problem time intervals that have been reserved for their multi- u u was formulated by Chen, G¨nl¨k, and Yener as the cast group. following ILP: In this section, we discuss the multicast packing Minimize λ(λ = maxe {ze }) + P k · πk problem including the objective functions and approx- k∈K imation algorithms for minimizing the maximum link Subject to sharing among multicast groups (i.e., the congestion xk ∈ ST k , ∀k ∈ K e over all edges). In the multicast packing problem, the tk · xk ≤ λ, ∀e ∈ E e minimization of network congestion is deﬁned as the k∈K 1 total load of the most congested edge and the load πk ≥ OP T k ( we ·xk −(α·OP T k )), e ∀k ∈ K e∈E (or congestion) of an edge is the total trafﬁc demand summed over the multicast groups using that edge so as to prevent bottlenecks and thus increase utilization 3.1 The Multicast Packing Heuristic Algo- (or throughput). rithm Given a physical network G = (V, E) where V is the set of nodes and E is the set of the undirected An approximation algorithm considering each multi- network links, a weight we > 0 (such as cost, delay, cast in isolation and taking the set of optimal multicast and distance) is associated with each link e ∈ E. trees computed independently was proposed by Chen, Let K denote the set of multicast groups, a set of u u G¨nl¨k, and Yener for packing multicast trees with nodes M ⊆ V denote a multicast group, and mi de- minimum congestion is as follows: note any arbitrary member of M , the objective of the Step 1. Solve the optimization problem for each mul- multicast packing problem is to ﬁnd a subgraph of G ticast group independently. that spans M and has the minimum total cost. The Step 2. Compute the congestion for each edge and subgraph is required to be a tree and the cost is mea- rebuild the multicast trees T based on the results from sured as the sum of the weights of the edges in the step 1: the tree T = {Tx : x = 1, 2, 3, ..., m} and a solution. Since the objective of the multicast pack- bound on the tree size α · OP T k . ing problem is to minimize the maximum sharing of a Step 3. Sort all edges e by ze (the total congestion on link instead of individual multicast tree cost, the solu- link e) into an array in decreasing order. Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp254-259) Step 4. Choose an edge e with the maximum conges- 3.2 The Randomized Version of Partition Al- tion Z gorithm Step 5. try each tree Tx ∈ T until a new tree Ty can A randomized algorithm contains some decision that be found such that each edge in Ty has the congestion is based on pure chance, not the inputs to the algo- value no greater than Z − 1 and tree Ty can be rebuilt rithm, or anything else in the environment in which without exceeding the size limit α. the algorithm is executed. The algorithm may deter- Step 6. Update ze values in the array and go to step 3. mine how the result is computed. The partition prob- Stop if the array cannot be updated. lem is an NP-complete problem and solving that given a set of integers, is there a way to divide the set into In [11], Ofek and Yener proposed a way to rebuild two independent subsets such that the sums of the the above disconnected tree by constructing a new tree numbers in each subset are equal. from scratch that does not use the congested link be- Let Π = {S1 , S2 , ..., S|Π| denote a partition of cause it may cause an increase in the congestion on V , ∆(Π)) ⊆ E denote the associated multicut, the some links that are already used by the other multi- 1 edge weights deﬁned as we = |∆(Π)| if e ∈ ∆(S) ¯ cast trees. Thus the links to be used in the new tree Ty 1 or 0, otherwise, Λk (w) ≥ |∆(Π)| · tk · (sk (Π) − 1) can have a congestion value at most Z − 2 (Z is the maximum congestion). by using these weights, in order to improve the qual- u u ity of the partition Π, Chen, G¨nl¨k, and Yener also u u Chen, G¨nl¨k, and Yener [10] proposed a more proposed a constructive procedure to ﬁnd better par- efﬁcient algorithm considering the cut obtained by re- titions of V by merging some of the subsets in the moving the congested link e from the new tree Ty . If partition to obtain smaller partitions and looking for there is a link e with congestion value at most Z − 2 a promising pair of subsets Sm and Sn with small in this cut set, then e can be inserted into the tree to βmn . By using edge weights γ deﬁned as γmn = rebuild the new tree (Ty − e + e). The cost of re- βmn + u · ( tk + tk ) + v with u and v building the tree is bounded by the cardinality of cut k∈K(Sm ) k∈K(Sn ) set (i.e., O(|E|) and thus the total cost is O(m · |E|2 ) being two random perturbations, they proposed a ran- for m trees. An ILP formulation employing Steiner- domized version of partition algorithm that was ap- cut inequalities (branch-and-cut algorithm) [17, 18] to plied repeatedly starting with the same initial partition construct such trees was as follows: as follows: Minimize z = we · xe Step 1. Let Π0 = S1 , S2 , · · · , S|V | with Si being e∈E a singleton. Subject to Step 2. Let λ0 be the initial best bound and i = 0. Step 3. Repeat as long as |Πi | > 2: xe ≥ 1, ∀S ⊂ V, mi ∈ S, M ⊂ S Step 3.1 compute γ ’s for each neighboring subset e∈δ(S) pair xe ∈ {0, 1}, ∀e ∈ E Step 3.2 identify a pair (Sm , Sn ) with the least Let a partition P = S1 , S2 , S3 , ..., Sk of V de- γmn (based on two random perturbations u and v) note a Steiner partition with respect to M if each Si Step 3.3 merge Sm and Sn to obtain a new parti- contains at least one element of M and ∆(P ) denote tion Πi+1 the multicut associated with P , (i.e., the collection of Step 3.4 compute λi+1 , updated the best bound if edges with endpoints in different members of P ), they necessary, and set i = i + 1 extended the above formulation as the following ILP by deﬁning the Steiner partition inequality associated with partition P , partitioning recursively the solution 4 Discussion spaces by branching on the variables to 0 or 1, and by solving the relaxations of the following ILP formu- The multicast congestion problems are critical to lation in order to ﬁnd the optimum multicast tree in many network applications in multimedia streamlin- isolation instead of generating all of the constraints at ing such as multimedia distribution, software distribu- once : tion, and video-conference; groupware system; game communities; and electronic design automation such ∈ ∆(P ) · xe ≥ k − 1 as routing nets around a rectangle and moat routing. e As widely known, the generalization of the problem 1 ≥ xe ≥ 0, ∀e ∈ E for ﬁnding edge disjoint paths is an NP-hard com- They further extended the above idea to partitions binatorial problems, i.e., minimum multicast Steiner of V involving more than two subsets. tree problem, and also a max-SNP hard problem, i.e., Proceedings of the 5th WSEAS International Conference on Telecommunications and Informatics, Istanbul, Turkey, May 27-29, 2006 (pp254-259) there is a constant > 0 such that it is NP-hard to ﬁnd [9] Carr, R. and Vempala, S., ”Towards a 4/3 ap- a (1 + ) approximation. proximation for the asymmetric traveling sales- All the approximation algorithms discussed man problem,” proceedings of the 11th SODA , above have been developed based on the deep analysis pp. 116-125, 2000. of the problems from different points of views, which u u [10] Chen, S., G¨nl¨k, O., and Yener, B., ”The multi- thus have been formulated as a variety of ILPs and cast packing problem,” IEEE/ACM Transactions solved by employing different schemes. However, the on Networking, 8(3), pp 311-318, 2000. main problem with the LP-relaxation is the time re- [11] Ofek, Y. and Yener, B., ”Reliable concurrent quired to solve the LP formulation. To improve the ef- multicast from bursty sources,” IEEE Journal ﬁciency, it is mandatory to make better use of the mul- Selected Areas Communication, 14, pp. 434- ticast congestion parameters in order to obtain a sim- 444, 1997. pliﬁed approximation algorithm with tighter approxi- [12] Ballardie, T, Francis, P., and Crowcroft, J., mation bound. Finding better and fast approximation ”Core based trees (cbt): An architecture for scal- algorithms for speciﬁc classes of multicast networks able inter-domain multicast routing,” Proceed- is worth further investigation as well. ings of ACM SIGCOMM, pp. 85-95, 1993. [13] Deering, S. E. and Cheriton, D. R., ”Multicast Acknowledgement routing in datagram internetworks and extended LANs,” ACM Transaction on Comp. Syst., 8(2), This work was supported in part by NSC, Taiwan un- pp. 85-110, 1990. der grant no. NSC 94-2213-E-274-004. [14] Deering, S., Estrin, D., Farinacci, D., Jacob- son, V., Liu, C.-G., and Wei L., ”An architecture for wide-area multicast routing,” Proceedings of References: ACM SIGCOMM, pp. 126-135, 1994. [1] Priwan, V., Aida, H., and Saito, T., ”The Mul- [15] Patridge, C. and Pink, S., ”An implementation ticast Tree Based Routing for the Complete of the revised internet stream protocol (st-2),” Broadcast Multipoint-to-multipoint Communi- Journal of Internetworking: Research and Ex- cations,” IEICE Transactions on Communica- perience, 4, 1992. tions, E78-B(5), pp. 720-728, 1995. [16] Paul, S., Sabnani, K., and Kristol, D., ”Multicast [2] Chen, S., GAunlAuk, O., Yener, B., ”Optimal protocols for high speed networks”, Proceedings Packing of Group Multicastings,” Proceedings of IEEE ICNP, pp. 4-14, 1994. of IEEE INFOCOM’98, pp. 980-987, 1998. 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