Mission impossible: Computing the network coding capacity region

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by  Terence Chan

1
Mission impossible:
Computing the network coding capacity region
Terence Chan and Alex Grant
Institute for Telecommunications Research
University of South Australia, Australia
terence.chan, alex.grant @unisa.edu.au
{ }
Abstract—One of the main theoretical motivations for the non-polyhedral nature of Γ¯∗, revealed in [9] implies a non-
emergingareaofnetworkcodingistheachievabilityofthemax- polyhedral capacity region (in contrast to the max-flow result
8
flow/min-cutrateforsinglesourcemulticast.Thiscanexceedthe
0 for single sources). To make things even worse, it is also
rate achievable with routing alone, and is achievable with linear
0 known that linear network codes are not sufficient for the
network codes. The multi-source problem is more complicated.
2 Computationofitscapacityregionisequivalenttodetermination multi-source problem [3], [8].

n of the set of all entropy functions Γ∗, which is non-polyhedral. Inthispaper,weshowthatnon-polyhedralcapacityregions
a The aim of this paper is to demonstrate that this difficulty can can occur even in single source scenarios. We demonstrate
J arise even in single source problems. In particular, for single
this phenomenon for single source networks with hierarchical

1 source networks with hierarchical sink requirements, and for
sink constraints, and for single source networks with security
3 single source networks with secrecy constraints. In both cases,
weexhibitnetworkswhosecapacityregionsinvolveΓ∗.Asinthe constraints. Our approach is in the spirit of our recent work
] multi-source case, linear codes are insufficient. [8], which revealed a deep duality between network codes
T andentropyfunctions.Directconsequencesarenon-polyhedral
I capacityregions,theinsufficiencyoflinearnetworkcodesand
.
cs I. INTRODUCTION the importance of non-Shannon information inequalities.
[ Section II provides the basic setup for secure network
Network coding [1], [2] generalizes routing by allowing in- codes,andformallydefinesachievabilityandadmissibilityfor
1
termediatenodestoperformcodingoperationswhichcombine networks with wiretapping adversaries. Section III focuses on
v
0 received data packets. One of the most celebrated benefits of the single source incremental multicast scenario, in which the
3 this approach is increased throughput in multicast scenarios. sinks have hierarchical requirements. Given a function g, we
0 This stimulated much of the early research in the area. One construct an incremental multicast network that is solvable if
0 fundamental problem in network coding is to understand the and only if g is entropic. In Section IV we construct a special
.
2 capacity region and the classes of codes that achieve capacity. single source secure multicast problem which is equivalent to
0 In the single session multicast scenario, the problem is well an insecure multi-source multicast problem. Invoking the du-
8 understood. In particular, the capacity region is characterized ality results from [8] these constructions relate the solvability
0
by max-flow/min-cut bounds and linear network codes are of both single-source incremental multicast and single source
:
v sufficient to achieve maximal throughput [2], [3]. Network secure multicast, to multi-source multicast problems.
i
X codingnotonlyyieldsathroughputadvantageoverrouting,its
capacitycanbeeasilydetermined,andeasilyachieved.Thisis
r II. BACKGROUND
a instarkcontrasttorouting,wherecomputationofthecapacity
region and of optimal routes is fundamentally difficult. Thenetworktopologywillbemodeledbyadirectedacyclic
Significant practical and theoretical complications arise in graph = ( , ). Vertices u correspond to communi-
G P E ∈ P
more general multicast scenarios, involving more than one cation nodes and directed edges e are error-free point-
∈ E
session. An expression for the capacity region is known [4], to-point communication links. The connection requirement
however it is given by the intersection of a set of hyperplanes M (cid:44) ( ,O, ) is specified by three components. The set
S D
(specified by the network topology and connection require- indexes the independent multicast sessions, each of which
S
ment) and the set of entropy functions Γ∗. Unfortunately, isacollectionofpacketstobemulticasttoaprescribedsetof
this capacity region, or even the inner and outer bounds [5]– destinations.Thesession-sourcelocationmappingO :
S (cid:55)→P
[7] cannot be computed in practice, due to the lack of an specifiestheoriginatingnodeO(s)forsessions.Thereceiver-
explicit characterization of the set of entropy functions for location mapping : 2P indicates the set of nodes
D S (cid:55)→
morethanthreerandomvariables.Thisdifficultyisnotsimply (s) which require the data of session s.
D ⊆P
a consequence of the particular formulation of the capacity A network code is identified by a set of discrete random
region given in [4]. It was recently shown that the problem of variables T ,W ,definedonfinitesamplespaces,wherefor
S E
{ }
determining the capacity region for the multi-source problem concise notation, set-valued subscripts denote a set of objects
is in fact entirely equivalent to the determination of Γ¯∗, indexed by the set, e.g. Z = Z ,i . The source
X i
{ ∈ X}
the set of almost entropic functions [8]. Furthermore, the random variables T ,s are mutually independent and
s
∈ S
2
are uniformly distributed on sample spaces whose size will whereW isthemessagesymboltransmittedalonglinkeand
e
be denoted T . The variables W ,e are the messages T is the input symbol generated at source s.
s e s
| | ∈ E
transmitted over link e. Theprecedingdefinitionsconsiderzero-errornetworkcodes
Since the network is acyclic, variables in T and W can and perfect security. Relaxing these requirements prompts the
S E
be ancestrally ordered according to the network topology. following definition.
Causal coding requires that edge messages are conditionally Definition 4 (Achievable): A rate-capacity tuple (λ,ω) is
independent of their non-incident ancestral messages given achievable if there exists a sequence of network codes Φ(n)
their incident source and message variables. and normalizing constants r(n)>0 such that
Definition 1: Anetworkcodeisprobabilisticifthereexists 1 (cid:16) (cid:17) 1
anoutgoinglinkmessagewhichisnotafunctionoftheincom- nl→im∞r(n)H We(n) ≤nl→im∞r(n)log|we(n)|≤ωe, ∀e∈E,
ing source and link messages. Otherwise, it is deterministic.
1 (cid:16) (cid:17) 1
Probabilistic network codes can be implemented via using nl→im∞r(n)H Ts(n) =nl→im∞r(n)log|ws(n)|≥λs, ∀s∈S,
independent random variables Vu (internal randomness) at (cid:16) (cid:17)
each node u such that all outgoing messages from a lim Pe Φ(n) =0,
∈ P n→∞
node are deterministic functions of incoming sources and link 1
messages and the independent randomness generated at the nl→im∞r(n)I(TA(nr);WB(nr))=0, ∀r ∈R.
node. It is easy to prove that all probabilistic network codes In the absence of any security constraints, = 0, these
canbeimplementedinthisway.Accordingly,weshallspecify definitions reduce to the usual ones and the|Rm|ulti-source,
a probabilistic network code by the set TS,WE,VP . multi-sink capacity region is given by [4]. Bounds for the
{ }
Lemma 1: Given random variables X1,X2 and V, if V is multi-source multi-sink scenario with wiretappers were given
independent of X1 and X2, and X2 is a function of X1 and in [10].
V, then X is a function of X alone
2 1
The implication of the lemma is as follows. At the sinks (or
III. INCREMENTALMULTICAST
anyintermediatenode)ofthenetwork,ifreconstructionofthe
In this section, we study a the special case of incremental
source messages is possible, then it can also be achieved in
multicast,meaningthatthesessionindexesaretotallyordered
the absence of “internal randomness”. In fact, in the absence
such that a receiver requesting a particular session also re-
of security constraints, it is known that deterministic network
questsallsessionswithlowerindex.Weconsiderthesimplest
codes are sufficient [6]. This is not always the case for the
incrementalmulticastscenario,withonlytwosourcemessages
wiretapping scenarios considered in Section IV.
and no secrecy constraints (permitting deterministic codes).
In addition to legitimate sinks, there are adversaries,
|R| We will show that determining the capacity region, even in
which can eavesdrop any message transmitted along a given
such a simple scenario, can be no simpler than solving the
collection of links. Each adversary attempts to reconstruct a
general multicast problem.
particular set of source messages, according to a wiretapping
Our approach is inspired by [8]. Let H[ ] R2N with
pattern. M ⊂
coordinates indexed by proper subsets of a ground set
Definition 2 (Wiretapping pattern): The wiretapping pat- M
with N elements. Points h H[ ] can be regarded as
tern is specified by a collection of tuples (Ar,Br) for r ∈R functions, h : 2M R wi∈th h(M) (cid:44) 0. Given such an
such that Ar ⊆S is the subset of sources to be reconstructed h H[ ] we wi(cid:55)→ll construct a∅special network †, an
by adversary r, which observes only the links in Br. incr∈ementMal connection requirement M† and a rate-caGpacity
For a given network code designed with respect to a
tuple T(h) that is admissible if and only if h is entropic.
connection requirement M, define P as the error probability
e The network topology, connection requirement and link
that at least one receiver fails to correctly reconstruct one or
capacities are defined in Figure 1, which for convenience,
more of its requested source messages. A zero-error network
is divided into several subnetworks. The single source node
code is one for which P =0, and hence the source messages
e is an open circle, labelled with the two available sessions
T can be perfectly reconstructed at desired sinks. The goal
S (this node is repeated for convenience in Figures 1(a), 1(b)
of secure communications is to transmit information such that
and 1(c)). The destinations are double circles, labelled with
any eavesdropper listening to the traffic on all the links in
their requirements. Intermediate nodes are solid circles. The
remains “ignorant” of the data transmitted by the sources
Br source and sink labels define the mappings O and . Each
in Ar. A perfectly secure network code is one for which the capacitated edge is labeled with a pair of symbols dDenoting
information leakage I(T ;W )=0 for all r .
Ar Br ∈R the edge capacity, and the edge message (and corresponding
Definition 3 (Admissible rate-capacity tuple): Given a net-
random variable). Unlabelled edges are assumed to be unca-
work = ( , ) and a connection requirement M, a rate-
capaciGty tuplPe (Eλ,ω) (cid:44) (λ ,ω ) is admissible if there exists pacitated, or to have a finite but sufficiently large capacity to
S E losslessly forward all received messages.
a perfectly secure, zero-error network code Φ = W ,f
{ f ∈ Thefirstpartofthenetwork,showninFigure1(a),contains
, such that
S∪E} the source where there are two independent sessions (i.e., two
messages S and S )available. Thedesiredsourcerates asso-
H(We)≤log|we|≤ωe, ∀e∈E, ciated with0S and1S are respectively (cid:80) h(i) and h( ).
H(T )=log w λ , s , 0 1 i∈N N
s s s There are 2N specific edge messages that are of particular
| |≥ ∀ ∈S
3
interest. Rather than naming all edge variables W ,e , the entropy of X :i α is equal to h(α) for all =α
e i
∈ E { ∈ } ∅(cid:54) ⊆
we label these 2N particular edge variables U and V for .Furthermore,hiscalledquasi-uniformifanysubsetofthe
j j
N
j =1,...,N. Remaining edge variables will be labelled with variables are uniform over their support.
generic symbols W indexed by an integer i. Theorem 1: For the network † and a connection require-
i
G
In Figure 1(a), the source node generates from S ment M†, if a rate-capacity tuple T(h) is admissible, then h
0
and S respectively the sets of network coded messages is quasi-uniform and hence entropic.
1
U ,U ,...,U and V ,V ,...,V whichareduplicated Proof: SupposethatT(h)isadmissible.ByDefinition3,
1 2 N 1 2 N
{ } { }
as required and forwarded to the rest of the network. The admissibility of T(h) on †,M† requires the existence of a
G
remainder of the network is divided into subnetworks of two zero-error network code Φ with source messages S , =
[α]
∅ (cid:54)
types, shown in Figures 1(b) and 1(c). α andasubsetofitscodedmessagesU andV .Given
N N
⊆N
this hypothesis, we will show that h is the entropy function
S0,S1 of VN, and that VN is quasi-uniform.
First focus on Figure 1(a). Applying min-cut bounds, it is
h(i) h( )
!i N Vα{ h(α),W1 straightforward to prove
(cid:88)
h(1),U1 h(2),U2 h(N),UN h(1),V1 h(2),V2 h(N),VN i!∈Nh(i)+h(N)−h(α),W2 H(UN,VN)= H(Ui)+H(VN),
i∈N
S0,S1 S0,S1
H(U )=h(i), i ,
i
(a) Sourcenode (b) Type1subnetworks ∀ ∈N
H(V )=h( ).
N
N
H(V )=h(i), i .
h(i)+h(N)−2h(i) i ∀ ∈N
S0,S1 i!∈N
Similarly, applying min-cut bounds to type 1 subnetworks of
Ui h(i),W1 P0 Figure 1(b), H(V ) h(α), =α .
α
S0,S1 We now focus on≥type 2 ∅su(cid:54)bnetw⊆oNrks of Figure 1(c) and
Vi h(i),W2 aim to prove that H(Vα) h(α) for any = α . In
≤ ∅ (cid:54) ⊆ N
order for the upper receiver to reconstruct S and S ,
0 1
Vi h(α,i)−h(α) h(i) h(i),W3 S0 H(W1,W2)+h( )+(cid:88)h(j) 2h(i) H(S0,S1)
N − ≥
P1 j(cid:54)=i
or equivalently, H(W ,W ) 2h(i). In addition,
{ { { 1 2 ≥
Vα Vα Uj:j!=i H(W ,W ) H(U ,V ,W ,W )
1 2 i i 1 2
≤
(c) Type2subnetworks
=H(U ,V ) 2h(i).
i i
≤
Fig.1. ThenetworkG†. As a result, H(W1,W2) = H(Ui,Vi,W1,W2) which further
implies that V is a function of W ,W . Thus V can be
i 1 2 i
With reference to Figure 1(b), there are 2N 1 type 1 sub- recovered at P . On the other hand, from the lower part of
0
−
networks,oneforeachnonemptyα 2N.Thesesubnetworks the subnetwork,
∈
introduceanedgeofcapacityh( ) h(α)betweenthesource
and a sink requiring S1. There iNs an−intermediate node which H(Ui|W3)=H(Ui|W3,Uj,j (cid:54)=i)+I(Ui;Uj,j (cid:54)=i|W3)
has another α incident edges (from Figure 1(a)), carrying (=a)I(U ;U ,j =iW )
V = V ,j | α| . The intermediate node then has an edge of i j (cid:54) | 3
caαpaci{tyjh(α∈) to}the sink. ≤I(Ui,W3;Uj,j (cid:54)=i)=0
Figure 1(c) shows the structure of type 2 subnetworks, where(a)followsfromthefactthatS canbereconstructedat
0
which are indexed by =α and an element i ,i thelowerreceiver.ThisimpliesthatUi canbereconstructedat
∅(cid:54) ⊂N ∈N (cid:54)∈
α. Each type 2 subnetwork connects the source to the upper P1.From[8],thatP0 candecodeVi andthatP1 candecodeUi
receiver. In addition, there are other incident edges carrying further implies H(V V )=h(α,i) h(α). By mathematical
i α
| −
V :j α and U :j . For notational simplicity, we induction (similar to the proof of [8, Theorem 1]), the only
j j
h{ave wr∈itten}h(α{ i )(cid:44)∈hN(α},i). solution that satisfies all of the conditions above is when the
So far, we hav∪e{d}escribed a network †, a connection entropy function of V is equal to h.
N
G
requirement M† and have assigned rates to sources and Finally, from type 1 subnetworks, the support of V is
α
capacitiestolinks.ClearlyM† dependsonlyonN,andnotin at most 2h(α). Hence, V is indeed quasi-uniform (this also
α
anyotherwayonh.Similarly,thetopologyofthenetwork † implies that the U are quasi-uniform, via H(U )=H(V )=
i i i
G
depends only on N. The choice of h affects only the source h(i) and the independence of the U ).
i
rates and edge capacities, which are collected into the rate- Theorem 2 (Converse): For the network † and a connec-
G
capacity tuple T(h). Also, we can assume without loss of tion requirement M†, a rate-capacity tuple T(h) is admissible
generality that T(h) is a linear function of h. if h is quasi-uniform.
Definition 5: A function h H[ ] is called entropic if From Theorems 1 and 2, we can follow the approach in [8]
∈ N
there exists discrete random variables X ,...,X such that and easily extend the result to almost entropic functions.
1 N
4
Theorem 3: For the network † and a connection require- Applying a min-cut bound on the set of edge variables
G
ment M†, a rate-capacity tuple T(h) is achievable if and only W ,W , we can also prove that H(W ) = c and
2 5 5
{ }
if h is almost entropic1. H(W X) = 0. On the other hand, the secrecy constraint
5
|
requires I(W ;X)=0 and hence
3
IV. SECUREMULTICAST I(W ;W )=0 (1)
1 3
Linear network codes (for single source multicast) that are
as W is a function of X.
resilient to eavesdropping are considered in [11]. Sufficient 1
Now, we will show that H(W W ,W )=0. First,
conditions for the existence of such codes was also derived. 1| 3 4
This was further generalized in [12] to multi-source cases. A (a)
I(W ,W ;X) = I(W ,W ;W ,X)
similar result was also obtained in [13] which gives necessary 3 4 3 4 1
and sufficient conditions under which transmitted data are =I(W3,W4;W1)+I(W3,W4;X W1)
|
safe from being revealed to eavesdroppers. All of the above- (b)
= I(W ,W ;W )
3 4 1
cited works assume that the wiretapper aims to reconstruct
all sources. Similar results have been obtained where only a where (a) follows from the fact that W is a function of X
1
subsetof sourcesaretobe reconstructed[14].Inner andouter and(b)followsfromtheconditionalindependenceimpliedby
bounds to the secure capacity region were given in [10]. the underlying network topology. Using the same argument,
We will now show that even for a simple single-session we can also prove that I(K,W ;X)=I(K,W ;W ).
3 3 1
securemulticastproblem,determinationofthecapacityregion Since W is a function of X and is thus independent of
5
can be extremely hard. In particular, the problem is at least as internalrandomness,Lemma1impliesthatH(W W ,W )=
5 3 4
|
hard as any multi-source multi-session multicast problem. 0. Together with H(W )=c, we have
5
Figure 2 shows the construction for a network (cid:63). The
G I(W ,W ;W )=I(W ,W ;X)
source message is X whose rate is d. The link capacities are 3 4 1 3 4
parametrized by 0 < c < d. There is a single eavesdropper I(W3,W4;W5)
≥
who only observes the message variable W . Thus Figure 2 =H(W )=c.
3 5
also specifies M(cid:63), and the wiretapping pattern (cid:63), (cid:63).
A B Since H(W ) = c, it implies that H(W W ,W ) = 0
1 1 3 4
|
or equivalently that W is a function of W and W . Sim-
1 3 4
P0 c,W1 X ilarly, using the same argument, we can also prove that
c,K H(W K,W )=0.
1 3
|
Our final aim is to show that H(K) = H(W ) = c
4
c,W3 d−c,W2 and H(K,W4) = 2c. Clearly, both H(K) and H(W4) are
bounded above by c due to the edge capacity constraint. We
obtain a lower bound on the entropy of K as follows.
c,W4 c,W5
X
P1 H(K)≥I(K;W1|W3)
=I(K;W W )+H(W K,W )
1 3 1 3
Fig.2. ThenetworkG(cid:63). | |
=H(W W )
1 3
|
Proposition 1: Given network (cid:63) and connection (and se- (=a)H(W )=c
G 1
crecy)requirementM(cid:63) depictedinFigure2,ifarate-capacity
where (a) follows from (1). Hence, H(K)=c. And similarly,
tuple T(h) is admissible then K is a function of W .
4
Proof: From the capacity constraint on (cid:63), we have we can also prove that H(W4)=c.
G Independence of W and W implies
1 3
H(W ,W ) H(W )+H(W )
1 2 1 2
≤ H(K W ,W )=H(W ,K,W ) H(W ,W )
=c+d c | 1 3 1 3 − 1 3
− =H(K,W ) H(W ,W )
=H(X). 3 − 1 3
=H(K,W ) H(W ) H(W )
3 1 3
− −
Togetherwiththedecodabilityrequirement,H(X W1,W2)= (a)
| = H(W K) H(W ) 0,
0, we have 3| − 3 ≤
where (a) follows from H(W ) = H(K) = c. Consequently,
H(W ,W )=H(W )+H(W ) 1
1 2 1 2 H(K W ,W )=0.
1 3
H(W ,W X)=0 |
1 2| Similarly, H(W4 W1,W3)=0. Finally,
H(W ,W )=H(X) |
1 2
2c H(W ,W )
H(W )=c ≥ 1 3
1
=H(W ,K,W ,W )
H(W )=d c. 1 3 4
2
− H(W ,K,W )
1 4
≥
1Afunctionhisalmostentropicifitisthelimitofasequenceofentropic (a)
= H(W )+H(K,W ) 2c
functions. 1 4 ≥
5
where (a) follows from independence of W and (K,W ). existingboundingtechniquesloose,thenon-polyhedralnature
1 4
Hence, H(K,W ) = c which further implies H(K W ) = of the capacity region suggests that LP bounds cannot fully
4 4
|
H(W K)=0. characterize the region, even with the addition of more and
4
Und|er a regularity condition (that 2c and 2d are integers), morenewlydiscoveredinformationinequalities.Anyfiniteset
the converse of Proposition 1 also holds. of such new inequalities can only further tighter the bound,
Proposition 2 (Converse): Forthenetwork (cid:63)withconnec- but can never yield the exact capacity region.
G
tion(andsecrecy)requirementM(cid:63),thespecifiedrate-capacity Despite the hardness of the problem, there are still many
tupleisadmissibleifasecretkeyofarateccanbetransmitted questions to be answered. It is unclear what makes finding
from the node P to P . the capacity region problem so difficult. In the case of a
0 1
Essentially, Propositions 1 and 2 suggest that the admissi- single session multicast or the case where there are only
bility of the single source secure multicast problem depends two sinks, capacity regions have explicit polyhedral charac-
on communication of a secret key from P to P . Adhering terizations provided by min-cut bounds. On the other hand,
0 1
several copies of (cid:63) together (see Figure 3), we can easily where there are many sinks, the capacity region can be
G
generalize the network such that admissibility implies that extremely complicated to characterize, even if there are only
multiple secret keys must be transmitted across a network. two independent sessions. It will be of great importance to
This turns the single source secure multicast problem into a classify the set of networks and connection requirements that
multi-source multicast. lead to polyhedral capacity regions characterized by min-cut
Theorem 4: For any multicast problem (without secrecy bounds or LP bounds.
constraints), there exists a corresponding secure multicast
problem such that the multicast problem is admissible if and REFERENCES
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1) Capacity regions are not polyhedral2 in general.
2) LP bounds are not tight in general.
3) Linear codes are not sufficient to achieve capacity.
In other words, finding capacity regions for (secure) multicast
problems seems to be a mission impossible. Not only are the
2Thatthesingle-sourcesingle-sessionsecuremulticastproblemhasanon-
polyhedral capacity region is somewhat surprising, since the region for the
sameproblemwithoutthesecrecyconstraintiscompletelydeterminedbythe
min-cutbound