A Survey of Approximation Algorithms for Multicast Congestion

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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
             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-
                                                                          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 = ∅,
   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
                                                                               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.
   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.
   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
   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
   over all edges). In the multicast packing problem, the                           tk · xk ≤ λ, ∀e ∈ E
   minimization of network congestion is defined as the                          k∈K
   total load of the most congested edge and the load                           πk ≥   OP T k
                                                                                              (    we ·xk −(α·OP T k )),
                                                                                                        e                      ∀k ∈ K
   (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
        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.
                                                                          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
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                                                                               LANs,” ACM Transaction on Comp. Syst., 8(2),
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