ITW 2004.San Antonio, Texas, October 24 - 29,20M

On the Advantage of Network Coding for Improving Network Throughput (extended a b s t r a c t )

Aniit .4garwal Moses Chsrihr' Dept. of Computer Science Princeton University Princeton, NJ 08544 {aagarval,moses}mcs.princeton.edu

Abstract - Given a data network w i t h l i n k capacities, w e consider t h e t h r o u g h p u t of the n e t w o r k for a multicast session involving a s o u r c e node and a given set of terminals. It is known that network coding can i m p r o v e the t h r o u g h p u t of the network. We s t u d y the c o d i n g advantage, i.e. the r a t i o of t h e throughput using n e t w o r k coding to that w i t h o u t using network coding. We s h o w that t h e maximum c o d i n g a d v a n tage for a given network is equal to t h e i n t e g r a l i t y gap of c e r t a i n linear p r o g r a m m i n g ( L P ) f o r m u l a t i o n s f o r S t e i n e r tree. T h i s holds for both d i r e c t e d as well as u n d i r e c t e d networks. For d i r e c t e d networks, the c o d i n g advantage is e q u a l to the integrality gap of the d i r e c t e d S t e i n e r tree L P formulation; for undirected networks, the coding advantage is equal to the integrality gap o f the hidirected cut L P f o r m u l a t i o n for S t e i n e r tree. T h i s relates the coding advantage to well s t u d i e d notions i n combinatorial optimization. F u r t h e r , t h i s connection improves t h e k n o w n bounds on the c o d i n g advantage for both u n d i r e c t e d as well d i r e c t e d networks.

I. INTRODUCTION Given a network with capacities on links, a fundamental p r o b lem is t o compute the maximum multicast throughput possible for communication between a source node and a set of receivers. Traditionally, intermediate nodes in the network are allowed t o merely store and forward inforrnation packets. In this model, the multicast throughput problem is computatimally hard t o solve optimally. In fact, tliis problem is equivalent to the problem of maximum fractional steiner tree packing, which is NP-hard [3]. 111 a seminal paper, Ahlswede eta1 [I] introduced the notion of network coding, where intermediate nodes are allowed to encode and decode messages they receive. In this model, they gave a simple characterization of the maximum throughput passible in a directed network for a multicast session between a source and B given set of receivers, and demonstrated that network coding could increase the throughdemonstrated examples where the gap put. Sanders eta1 [i] between the throughput using network coding t o that without using coding was n(logn), where n is the number of receivers. Recently, Li eta1 151 studied the multicast throughput of undirected networks and gave a linear programming formulation to compute the throughput in the undirected c a % Similar techniques were also used recently by Kramer and Savari [4] 'This work wzs supported by NSF ITR grant CCR-0205594,

DOE Early Career Principal Investigator award DE-FG0202ER2554U, NSF CAREER award CCR-U237113 and an Alfred P. Sloan Fellowship.

0-7803-8720- 1/04/$20.00 02004 IEEE

to hound the multicast capacity of undirected networks with network coding. 151 also studied the coding advanloge, i.e. the ratio of the throughput using network coding t o that without using coding. For undirected networks, [5] gave various examples with coding advantage > 1, t h e best value being 918 and proved that it is no more than 2. In this paper we show an interesting connection between the coding advantage and the integrality gap of linear programming formulations far minimum weight Steiner tree. Since the minimum weight Steiner tree problem is NP-hard for both undirected and directed networks, polynomial time solvable LP relaxations of Steiner tree are commonly used t o obtain a lower bound on the optimal Steiner tree weight. The quality of the bound provided by the L P relaxation is mevsured by its integrality gap, i.e. the ratio between the optimum Steiner tree weight and the optimum solution to the L P relaxatiun. We show that for undirected networks, the maximum coding advantage is equal to the integrality gap of the bidirected cut relaxation for the undirected Steiner tree problem [fi]. For directed network-, we show that the coding advantage is equal to the integrality gap of a natural LP formulation of directed Steiner tree. Our r a u l t s improve on the best bounds known for the maximum coding advantage in both directed arid undirected networks. Using an integrality gap example due t o Goemans (presented in [a]), we show a family of undirected networks with coding advantage approaching 817, improving on the 918 hound from 151. For directed networks, using a recent integrality gap example for directed Steiner tree given by Halperin etal. [Z],we show that the coding advantage can be R ( ( l ~ g n / l o g l o g n ) ~where ) n is the number of receivers. It is interesting to note that determining the integrality gaps of these L P formulations for undirected and directed Steiner tree are well studied and interesting open problems in the computer science optimization literature. In the undirected case, it is known that the gap is at most 2, and an upper bound of 3/2 is known for a special class of graphs caled quasi-bipartite graphs [6]. Determining whether t h e gap is strictly lower than 2 is a major open problem. In the directed case, establishing whether the gap is upper bounded by a polygarithmic function is a major open problem. There are interesting parallels between the coding advantage examples obtained in the network coding community and the integrality gap examples produced in the computer science optimization community. Some of the coding advantage examples given irr 161 appear in [6] ils integrality gap examplw. The integrality gap example of Goemans (described in [a]) relies on a gadget which is exactly the same as a simple, well known example t o illustrate the advantage of network coding,

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first given in [I]. Finally, the example given to show the coding advantage i n directed networks in 171 is exactly the same as a weil known integrality gap example iur set cover, a special case of directed Steiner tree. I n this extended abstract, we present our results for the undirected case. We omit a detailed exposition of the results for the directed case; the details arc similar to those in the undirected setting.

E , we have two directed edges el and e2 which represent the two orientations of e; both have the same weight as the undirected edge. D = (el,e?,Ve E E } . A valid set C is a subset of vertices such that C contains the source mo and contains a t least one terminai. S(C) = { ( U , " ) E D : Y E C,v C}. The fallowing integer program computes the minimum weight Steiner tree:

c

11. PRELMNARIES

-

co

We represent the input network as an undirected graph G = ( V . E ) . Let c : E R+ be a n assignnient of non-negative capacities to the edges. We use ce to denote the capacity of edge e E E . For given edge capacities e, we will investigate the network throughput for a multicast session between source mo and receivers m l , . . . , m k . The amount of information that can be sent with network coding is denoted by x ( C , c ) . The anmunt of information that can he sent without network coding is denoted by n(G,c) and is the same as the fractional steiner tree packing number for G. Define the Let T he the set coding advantage t o he the ratio -. of all possihie Steiner trees far G connecting the given set ai ternlinals (mo,.. . :mh]. The Steiner packing number is given by the following linear program: (there is a variable x Yfor every possible Steiner tree t E T)

co E (0,

5 ce,

Ve E E

n,

vt E T

2

c0

C Y . 2 1, eei

ya 2 0,

E,,,C,Y, Vt

Let the optimum value of the hidirected cut relaxation for C with given weights w he denoted by B(G, w). The maximum value of the ratio O P T ( G ,w)/B(G, w) is called the integrality gap of the L P relaxation. Note that the coding advantage is invariant under multiplicative scaling of capacities; similarly, the integrality gap is invariant under multiplicative scaling of weights.

Theorem 111.1.

€7

Ve E

E

Note that y, are the variables i n the above LP. As given in [ 5 ] , the failowing L P (called the cFlaw L P in [ 5 ] )computes the optimal throughput achievable with network coding. To give the LP, we use similar notation as in [5] (modified far consistency). max ce,

+

f'

c, 2 0 cez = Cr

Va E D

Ve E E V i E [l.. . k],VaE D

f'(Q)5 c ( a ) =fL(v) f:n(mo) = 0

f:"t(m%)= 0 f' = fk(m.)

-

vi E

2 n,vo E D

111. ANALYSIS FOR THE UNDIRECTED CASE

The dual of the above linear program is as fallows: niin

Va E D

In this section we present the arguments to show that maximuni coding advantage far a given network G is the same as the integrality gap of the hidirected cut relaxation for minimum weight Steiner tree. We establish this by proving illequality in both directions hetween these two quantities.

1sr:ett

X,

I}

Note that co are the variables in the above intrger program; the optiinum value is OPT(G,ur). In the bidirected cut L P relaxation. xve replace the last constraint by the fallowing

" CfETZl

xt

2 1 V valid sets C

oEI(C)

[I.. . k ] , ~Eu v - {mo,m,) W E [i... k] vi E [1 . ..k] V i E [ I ... k]

Proof. Let us consider the Steiner packing number for a graph with capacities ce. Assume, without loss of generalization, that the capacities are scaled so that the value of the objective of the cFiow L P for this graph (i.e. x ( G , c ) ) is exactly 1. Consider the dual t o the Steiner packing LP; by strong duality, the optimum value is equal to n(G,c). We claim that the optimum solution of this dual gives us a gap example for the bidirected cut relaxation, with integrality gap a t least as large as the coding advantage. To see why this is true, notice that we can view the ye's as edge costs. The first constraint of the dual gives us the condition that every Steiner tree under these edge costs should have cost at ieast 1.

Claim 111.2. CIEEceyeis an upper bound on the value of the bidirected cut relazation for the Steiner tree instance with edge costs given by the y. P.

The optimum of this LP is x ( G ,c ) . Let w : E R+ be an assignment of non-negative weights to the edges. We use we to denote the wcight of edge e E E. For given edge weights w. O P T ( G , w ) denotes the weight of the minimum weight Steiner tree on G. Finding OPT(G, w ) i s NP-hard. One can formulate t h e following hidirected cut integer program t o find OPT(G,w). For each undirected edge e E

Proof. Since the graph G with capacities ce has a cFlow value of 1, every edge can be hidirected and the capacity distributed between the two appositeiy directed copies so that a flow of 1 can he routed separately from the source t o every termiiial according t o these capacities. But then these hidirected capacities give a valid solution to the bidirected cut relaxation. (Note that the directed capacity ~ C I O S Sany cut separating the 0 source and at least one terniinal must be at least I ) .

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Hence, lroin the optirnum solution of the dual, there exists a setting of weights w such that OPT(G,w ) 2 1 and B(G:w ) 5 II(G>C).

[21 E. Hutperin, G. Kortsarz, R. Krauthgamer, A. Srinivdsan and N. Wang, "Integmlity ratio for Group Steiner T k e s and Directed Stciner Ttees," Pmc. 14th ACM-SIAM Symposium on Diacrete Algorithm (SODA), pp. 275-284, 2003.

.

Theorem 111.3.

e:

L

Pioof, Consider a n instance G with weights w for which the bidirected cut relaxation has gap g. From G, we will construct an instance for which the coding advantage is atleast g. Consider the capacities col(one for each orientation of each edge) returned hy the bidirected cut relaxation on G. For each undirected edge e, let ce t o be the sum of the edge capacities for both orient.ations of edge e. Now run the cFlow L P on G with these capacities. C l a i m 111.4. The value of the cFlow L P (i.e. x(G,c)) the graph G with capacities ce is atleast 1.

V

fOT

a0

separating the source and a terminal, the directed capacity is atleast 1. This means that the directed graph can support a flow of a t least 1 from the source t o every terminal. 0

C U L reliLxLL*IuII.

The Steiner tree packing number for this instance is II(G,c). This means for any setting of weights w ,there is a Steiner tree of cost a t most Recall that we started with a gap instance for the bidirected cut relaxation with B ( G , e ) = C w . c . and = 9. But the above statement implies that there is a Steiner tree of cost a t most Hence lI(G, e) 5 l / g , which implies that the coding advantage 2 g, the integrality gap. 0

w.

w.

GAP EXAMPLE

V

FOR THE UNDIRECTED CASE

a

For completeness, we reproduce the hest known gap example for the bidirected cut formulation for Steiner tree (due to Goemans, as described in [SI). The example consists of n I terminals, labeled aa, . . . ,a,. In addition we have 2(;) vertices, which are labeled c;? a n d d,,, for I 5 i , j 5 n. T h e vertices are connected as shown in Figure 1. For every 1 5 k 5 n, wc have a n additional vertex b k . We have edges ( a ; , b ; ) and (b;,ao) for all i of cost 2. For every 1 5 i , j 5 n, i # j , we also have edges (a,,ct,) of cost 2, a n d edges ( b , , d , , ) of cost 1. Finally, for every 1 5 i , j 5 n, i # j we have edges ( c , j , d ; , ) of .cost I. The integrality gap of this example approaches $ as n + m. A feasible solution for the linear relaxation is t h e following: each edge is selected t o extent .; Using these values m edge capacities, we get an example with coding advantage approaching $. T h e example is similar t o the gap example described previously; a portion of the graph is depicted in Figure 2.

+

REFERENCES [l] R. Ahlswede, N. Cai, S. R. Li, and R. W.Yeung, "Network Information Flow," IEEE Tmnsactions on Infomation Theory, 46, no. 4, pp. 1204-1216, 2000.

Fig. 2: Portion of the example with high coding advantage.

[3] K. k i n , M. Mahdian, and M. R. Salamtipour, "Packing Steiner trees," P m . 24th ACbI-SIAM Symposium on Discrete Algon t h m (SODA), pp. 26G274, 2003.

[4] G. Kramer, and S. A. Savari, "Cut sets and information flow in networks of tweway channels," P m . IEEE International Symposium on Infomotion Theory (ISIT), p. 33, 2004 [5] 2. Li, B. Li, D.Jiang, and L. C. L a , "On achieving optimal end-

t e e n d throughput in data networks: theoretical and empirical studies," ECE Technicn( Report, University of Toronto, 2004. [61

S.Rajagopalan, and V. Vuzirani, "On the bidirected cut reluxatinn for the metric Steiner tree problem," Pmc. 11th ACMS I A M Symposium on Discrete Algorithm (SODA), pp. 742751, 1999.

[7] P. Sanders, S. Egner, and L. Tolhuizen, "Polynomial Time A l g e rithms for Network Information Flow," Pmc. 15th ACM Sympos i u m on Parallelism in Algonthms and Archztectvres (SPAA), pp. 286.294, 2003.

181 A. Sinha, "Steiner trees: first summer paper f i x the PhD program at GSIA," manuscript, 2000.

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