Uniprior Index Coding

0.15 MB·

by  Vijaya Kumar Mareedu

Uniprior Index Coding
Vijaya Kumar Mareedu and Prasad Krishnan
Signal Processing and Communications Research Centre,
International Institute of Information Technology, Hyderabad.
Email: {vijaya.kumar@research.,prasad.krishnan}@iiit.ac.in
Abstract—The index coding problem is a problem of efficient linear code construction for single uniprior IC. It was shown
broadcasting with side-information. We look at the uniprior that the code can be constructed using a polynomial-time
index coding problem, in which the receivers have disjoint
graphical algorithm.
side-information symbols and arbitrary demand sets. Previous
Unlikeintheunicastcase,ageneralunipriorproblemcannot
work has addressed single uniprior index coding, in which each
receiver has a single unique side-information symbol. Modeling be always considered to be a single uniprior problem. In this
the uniprior index coding problem as a supergraph, we focus work,welookatthegeneralunipriorproblem,wheretheside-
7
1 on a class of uniprior problems defined on generalized cycle information sets are disjoint but not necessarily singleton. In
supergraphs. For such problems, we prove upper and lower
0 particular, our contributions are as follows.
bounds on the optimal broadcast rate. Using a connection with
2
Eulerian directed graphs, we also show that the upper and • DemandSupergraphs:We modelthe unipriorindexcod-
n lower boundsare equal for a subclass of unipriorproblems. We
ingproblemusingademandsupergraph,consistingofsu-
a show the NP-hardness of finding the lower bound for uniprior
J problems on generalized cycles. Finally, we look at a simple pervertices and subvertices, and edges. Each supervertex
3 extension of the generalized cycle uniprior class for which we representsa receiverand the subverticesof a supervertex
2 give bounds on the optimal rate and show an explicit scheme are the set of side-information symbols available at that
which achieves the upper bound. receiver. The edges represent the demands.

T] • Boundsfor GeneralizedCycles: We focusonunipriorIC
I I. INTRODUCTION problemsonaspecialclassofsupergraphs(whichwecall
. generalized cycles). We exploit the relationship of such
s The index coding (IC) problem [1] consists of a source
c problems with unicast IC and obtain lower and upper
[ generating messages connected to a set of receivers via a
bounds on the optimal broadcast rate of index codes for
broadcast channel, each of which have some demands and
1 such problems (Theorem 1).
possess some prior side-information. An optimal index code
v • NP-Hardness and Explicit code structures: Bounds for
for a given configuration of demands and side-information is
3 generalized cycles: Using an equivalence between gen-
7 an encoding of the messages with the least broadcast rate,
eralized cycles and Eulerian directed graphs, we show
2 such that the demands are decodable. Different classes of the
the class of generalized cycles for which the lower and
6 IC problem were studied based on the configuration of the
0 upper bounds on the optimal rate are met with equality
side-informationsymbols and the demands. In unicastIC [2],
. (Theorem 3). We also show the explicit structure of a
1 the demand sets of the receivers are disjoint. The unicast IC
0 problem can always be studied as single unicast, where each feasibleschemeforsuchunipriorproblems(Theorem2).
7 Further,wealsoshowtheNP-hardnessofdeterminingthe
receiver demands a unique message. In uniprior IC [3], the
1 lower bound using this equivalence. (Theorem 4). Thus,
side-informationsets at the receiversare disjoint. The general
:
v unlike single uniprior IC, obtaining optimal index codes
multiprior/multicast index coding problem has been studied
i forgeneralunipriorICorevencharacterisingtheoptimal
X in [4]. All such IC problems have been extensively studied
broadcast rate could be NP-hard.
r using approaches from graph theory (see for example [2]–
a [6]). While the optimal broadcast rate for index codes has • Extending generalized cycles: We then generalize the
specialclasstoalargerclassofunipriorproblems,obtain
beencharacterisedpreciselyforonlya relativelysmallclasses
bounds on the optimal rate, and show a feasible index
of problems(single uniprior is one of them), for most classes
coding scheme for the larger class (Theorem 5).
finding the optimal rate is NP-hard. Thus, most prior work
focuses on obtaining bounds. Notationsandafewbasicdefinitions: Adirectedgraphalong
The single uniprior ICproblemis a subclassofunipriorIC withitsvertexandedgesetsisrepresentedasG(V,E).Aunion
where the side-information sets at all receivers are singleton, of two graphs is a union of the set of vertices and the set of
and was studied in [3]. In [3], a given single uniprior index edges. A decomposition of a graph G is a set of subgraphs
coding problem is represented as a information flow graph, which are edge-disjoint, and whose union gives the graph G.
containing a set of vertices representing each message (and A trail of a directed graph G(V,E) is a list of distinct edges
thustheuniquereceiverwhichhasitasside-information).The e ,...,e such that the head(e )=tail(e ),1≤i≤L−1.
1 L i i+1
edges of the information flow graph represent the demands Atrailisclosedifitsstartvertexandendvertexarethesame.
made by the receivers. The authors of [3] characterized the A directed graph is strongly connectedif there is a path from
length of an optimal index code, and also gave an optimal any vertex to any other vertex. For a directed graph G, let
ν (G) denote the maximum number of edge-disjoint cycles (whichdemandsx ).Itwasshownin[2]thatthelengthofany
e j
in G. The quantity ν (G) is the maximum number of vertex- optimalscalarlinearindexcode(overF )isequaltoaproperty
v q
disjointcyclesinG.AfeedbackvertexsetofadirectedgraphG of the graph G called the minrank, denoted by mrk (G ).
SI q SI
isasetofverticeswhoseremovalleadstoanacyclicgraph.Let While computing mrk (G ) is known to be NP-hard [8] in
q SI
thesizeofaminimalfeedbackvertexsetbedenotedbyτ (G). general, several authors have given lower bounds and upper
v
Similarly, the size of a minimal feedback edge set is denoted bounds for the quantity, as well as specific graph structures
by τ (G). Fora graphwith vertexset V,we have thatα(G)= for which the bounds are met with equality (see for example,
e
|V|−τ (G) is the number of vertices in a maximum acyclic [2], [5], [6]). From [2], [5], [6], we know that given a single
v
induced subgraph of G. An underlying undirected graph of a unicast IC problem I on G (with n message vertices), we
SI
given directed graph G is the undirected graph obtained by have the following.
ignoring the directions in G. An undirectedgraph H is called
n−τ (G )≤β (I)≤β (1,I)≤mrk (G )≤n−ν (G ).
a minor of an undirected graph G if H can be obtained from v SI q q q SI v SI
(1)
G by a series of edge contractions and deletions. For more
preliminaries on graphs, the reader is referred to [7]. A finite All the above quantities are NP-hard to compute for general
field with q elements is denoted by F . graphs [8], [10].
q
Due to space restrictions, some of the proofs have been
omitted, but made available in [9]. III. UNIPRIOR INDEXCODING: MODELING AND BOUNDS
A. Modeling Uniprior IC using the Demand Supergraph
II. PRELIMINARIES :INDEX CODING AND SINGLE Definition1. ForagivenunipriorICproblemI withmessage
UNICAST set X, we define a supergraph G (V ,X,E ) as follows.
s s s
Formally, the index coding (IC) problem (over some field • For receiver j in I, there exists a corresponding super-
Fq) consists of a broadcast channel which can carry symbols vertex j ∈Vs.
from Fq, along with the following. • Each supervertex j contains subvertices indexed by the
• A set of m receivers side-information S(j)⊂X.
• AsourcewhichhasmessagesX ={xi,i∈[1:n]},each • An edge (xi,j) ∈ Es with tail node being the subvertex
of which is modelled as a t-length vector over F . x and head node being supervertex j denotes that the
q i
• For each receiver j, a set D(j)⊆X denoting the set of message xi is demanded by the receiver j. All such
messages demanded by the receiver j. demands in I are represented by their corresponding
• For each receiver j, a set S(j) ⊆ X\D(j) denoting the edges in the super graph.
set of s side-information messages available at the jth
j Thenotationxj denotessomearbitrarymessage(subvertex)
receiver.
inS(j).Forxj ∈S(j),wealsousexj ∈j withrespecttothe
For a message vector x ∈ Fnt, the source transmits a l- supergraph.Wealsodenotebyl∗(G )thelengthofanoptimal
length codeword E(x) (the function E : Fnt → Fl, is known s
scalar linear index code for the uniprior IC problem defined
astheindexcode),suchthatallthereceiverscanrecovertheir byG ,andusetheterml∗ insteadwhenthereisnoconfusion.
s
demands. The quantity l is known as the length of the code
We now give the definition of a cycle in a supergraph.
E. The transmission rate of the code is defined as l. If t=1,
t
then the index code is known as a scalar index code, else it Definition 2. A cycle (of length L) in a supergraph Gs is a
is known as a vector index code. A linear encoding function sequence of distinct edges of Gs of the form
E is also called a linear indexcode. The goalof indexcoding C =((xi0,i ),(xi1,i ),..,(xiL−1,i ))
1 2 0
is to find optimal index codes, i.e., those with the minimum
possible transmission rate. For an index coding problem I wherexij ∈ij,∀j =0,...,L−1andthesuperverticesij,0≤
(overF )witht-lengthmessages,letβ (t,I)denotethelength j ≤ L−1 are all distinct. A supergraph without a cycle is
q q
ofanoptimalvectorindexcode.Thebroadcastrate[6]isthen naturally called an acyclic supergraph.
βq(I)=limt→∞ βq(tt,I). Clearly, we have βq(I)≤βq(1,I).
Anindexcodingproblemiscalledasingleunicastproblem B. A special uniprior problem which is also single unicast
if m = n and each message is demanded by exactly one
We now define a class of uniprior IC problems which are
receiver.Anindexcodingproblemiscalleda generaluniprior
also a special case of single unicast IC problems.
(or simply, a uniprior) problem if S(j)∩S(j′) = φ,∀j 6=
j′. The single uniprior problem is then a special case of the Definition3(GeneralizedCycle). Ageneralizedcycledenoted
uniprior problem with sj =1. by Ggc(Vgc,Xgc,Egc) is a demand supergraph satisfying the
AgivensingleunicastindexcodingproblemI canbemod- following properties.
elled usinga directedgraphcalled the side-informationgraph • The message set is Xgc and each message (subvertex) is
[2], denoted by G (V ,E ), where the set of vertices V , demanded exactly once.
SI SI SI SI
identified with the set of message symbols X, represents also • Thenumberofincomingedgestoanysupervertexj ∈Vgc
the the set of receivers (each demanding an unique message). is equal to s (the number of side-information symbols).
j
A directed edge (xj,xi) in ESI indicates the availability of • The supervertices Vgc are connected, i.e., for every two
the message symbol x as side-information at the receiver j i,j ∈V , there is a path from some message in i to j.
i gc
Remark 1. For a single uniprior index coding problem, the
definition of a demandsupergraph specializes to the informa-
tionflowgraphof[3],andthedefinitionofageneralizedcycle
specializes to a cycle in the information flow graph.
Given a uniprior IC problem I on a generalized cycle, it
is clear that I can be looked at as a single unicast problem
also (making s ‘copies’ of a receiver j, each demanding
j
an unique single symbol in D(j)). Hence one can define its
corresponding side-information graph G . It is easy to see
SI
that for each message x demanded by a receiver j in G ,
i gc
thereexistss edgesinthecorrespondingG toeachmessage
j SI
(subvertex) in S(j) from x .
i
Equivalent to definition of ν (G) for a directed graph G,
e
let ν (G ) be the maximum number of edge-disjoint cycles
e s
of G . We now prove a result which shows that edge-disjoint
s
cycles of Ggc are equivalent to vertex-disjoint cycles of the Fig.1. Supergraphcorresponding touniprior ICproblem ofExample1
corresponding G and vice-versa.
SI
Proposition 1. Consider an uniprior IC problem with its
demand supergraph being a generalized cycle G , and the
gc
corresponding side-information graph G . For any set C of
SI
edge-disjoint cycles in G , there exist a set of C′ vertex-
gc
disjoint cycles in G of the same cardinality, and vice versa.
SI
Thus ν (G )=ν (G ).
e gc v SI
Proof:We provethetheoremforasetoftwocycles.The
extension to any finite number of cycles follows.
Suppose C ,C are any two edge-disjoint cycles in G ,
1 2 gc
where
C =((xi0,i ),(xi1,i ),...,(xiL−1,i )), (2)
1 1 2 0
C2 =((xj0,j1),(xj1,j2),...,(xjL′−1,j0)), (3)
where xik ∈ik,∀k and xjk1 ∈jk1,∀k1. Fedigg.e2in.comSiidneg-iantfoarnmyaxtiioanndgrfarpohmfsoormthee‘bulnoicpkr’ioirndIiCcaptersobthleamtthoefreFiagr.e1e.dgAens
As the cycles are edge-disjoint, we must have incomingatxi fromeachmessageintheblock
(xik,i(k+1)(mod L)) 6= (xjk1,j(k1+1)(mod L′)),∀k,k1.
Note that this implies xik 6= xjk1 for any k,k1, as each
message is demanded precisely once in Ggc. {x2,x4,x7}, S(2) = {x1,x5,x8}, D(3) = {x3,x5,x9},
Consider the cycles correspondingly in GSI considered as S(3) = {x2,x4,x6}, D(4) = {x6,x8}, S(4) = {x7,x9}.
follows. The demand super graph corresponding to the problem is
C′ =((xi0,xi1),(xi1,xi2),...,(xiL−1,xi0)). (4) a generalized cycle, shown in Fig. 1. The demand super-
1 graph G in Fig. 1 can be decomposed in to four edge-
gc
C2′ =((xj0,xj1),(xj1,xj2),...,(xjL′−1,xj0)). (5) disjoint cycles : (cid:0)(x1,1),(x3,3),(x2,2)(cid:1), (cid:0)(x5,3),(x4,2))(cid:1),
Such cycles clearly exist because xik ∈ ik,∀k and xjk1 ∈ (cid:0)(x9,3),(x6,4)(cid:1),and(cid:0)(x7,2),(x8,4)(cid:1),denotedbyC1,C2,C3
and C respectively. Corresponding to the cycles C , we get
Cjk′1,a∀ndk1C.′AisnxGik 6=axrejkv1efroterxa-ndyiskjo,ikn1t.,itisclearthatthecycles the cy4cles (cid:0)(x1,x3),(x3,x2),(x2,x1)(cid:1), (cid:0)(x5,x4),(ix4,x5)(cid:1),
1The co2nverseSIfollows by picking vertex-disjoint cycles in (cid:0)(x9,x6),(x6,x9)(cid:1), and (cid:0)(x7,x8),(x8,x7)(cid:1) in the side-
information graph shown in Fig. 2.
G as in (4) and (5) and showing that corresponding edge-
SI
disjoint cycles exist in Ggc as in (2) and (3). We leave the Following the definition of a feedback vertex set of a
details to the reader. directed graph, we define the feedback edge set of a demand
supergraph G as a set of edges whose removal leads to an
Remark 2. It is easy to see that Proposition 1 should hold s
acyclic supergraph. The size of a minimal feedback edge set
for all uniprior IC problems whose supergraph satisfies the
is denoted by τ (G ).
first property of Definition 3. For the purposes of this work, e s
Proposition 1 is sufficient. Proposition 2. Let a generalized cycle G represent a
gc
unipriorICproblemI,andletG beits correspondingside-
SI
Example 1. Consider a uniprior index coding problem with information graph. Suppose the set of edges
ninemessagesandfourreceivers.Thedemandsets,sideinfor-
mation sets are as follows: D(1) = x1, S(1) = x3, D(2) = {(xik,jk):k =1,..,K,jk ∈Vgc} (6)
is a feedback edge set of G . Then the set of vertices Lemma 2. An Eulerian directedgraphG can be decomposed
gc
into a set of edge-disjoint cycles. In particular, any maximal
{xik :k =1,..,K} (7)
set of edge-disjoint cycles of G is also a decomposition of G.
isafeedbackvertexsetofG .Conversely,if(7)isafeedback
SI Proof: We only prove the second statement as it implies
vertex set of GSI, then for some K supervertices {jk : k = the first statement. Suppose some maximal set C of edge-
1,..,K} in G , the set in (6) is a feedback edge set of G .
gc gc disjointcyclesisnotadecomposition.Assumethatweremove
Thus, τ (G )=τ (G ).
v SI e gc alltheedgesfromG whicharepresentinC,andsubsequently
Proof: We first show that if (6) is a feedbackedge set of also any isolated vertices (which don’t have incoming or out-
G , then (7) must be a feedback vertex set of G . Let G′ goingedgesafterthe removalof C)and lookatthe remaining
gc SI SI
denote the subgraph of G which remains after deleting the graph G′. By our assumption that C is not a decomposition,
SI
vertices{xik :k =1,..,K}(andthe incidentedgesonthem). G′ must have least one edge and thus at least two vertices.
Suppose G′ is not acyclic, then there exists a cycle C′ in Howeveralso note that, for any remainingvertex, the number
SI
GSI thatdoes nothave anyverticesfrom {xik :k =1,..,K}. of incoming and outgoing edges must be the same (since for
ThenbyProposition1,thereexistsacorrespondingcycleC in any removed incoming edge, one outgoing edge must also be
G which does not have the edges in (6). This means (6) is removed). This means that G′ is also Eulerian, which means
gc
not a feedback edge set of G , which is a contradiction. The there is one cycle in G′ (and hence in G) which is edge-
gc
converse can be similarly proved;we leave this to the reader. disjoint to those in C. This contradicts the maximality of C
and concludes the proof.
We now give the main theorem in this section.
Theorem 1. For an uniprior IC problem I on a generalized B. Eulerian Graphs associated with Generalized Cycles
cycle G , we have
gc Definition 4 (Eulerian Graph associated with G ). The
gc
n−τ (G )≤β (I)≤β (1,I)≤l∗≤n−ν (G ). (8) Eulerian graph associated with a generalized cycle
e gc q q e gc
G (V ,X ,E ) is the directed graph G with vertex
gc gc gc gc eu
Proof:AsI isalso a singleunicastproblem(represented set V =V and edge set E defined as follows.
eu gc eu
by,say,G )andbyProposition1andProposition2,wehave
SI • For each edge (xi,j) with message xi ∈i, an edge from
νv(GSI)=νe(Ggc), i to j exists in Geu.
τv(GSI)=τe(Ggc). Remark 3. It is easy to see that the directed graph defined
Furthermore, l∗ = mrk (G ). With all these facts, we can in the above way is indeed Eulerian, by the properties of Ggc
q SI
(using Lemma 1). From the definition it should be clear that
invoke (1) to prove our theorem.
the Eulerian graph associated with a generalized cycle could
have paralleledges. Supposethere exists p edges between the
IV. EXPLICITCODES, HARDNESS RESULTS, AND
vertices i and j in the Eulerian graph G , then we refer to
TIGHTNESS OF BOUNDS eu
those edges as {(i,j) :k =1,2,...,p}.
k
In this section, we show an explicit achievable index code,
and also show the NP-hardness of obtaining the lower bound
in Theorem1, andobtaina specialclassofgeneralizedcycles
for which (8) is satisfied with equality throughout. For this
purpose we use the connection between Eulerian directed
graphs and generalized cycles.
A. Eulerian Directed Graphs
Eulerian graphs [7] are those which contain an Eulerian
circuit, which is a closed trail containing all edges. The
following Lemma is found in [7] (Chapter 1), and will be
used in this section.
Fig.3. EulerianGraphassociated withthegeneralized cycleinFig.1
Lemma 1. A directed graph (with at least one edge incident
Fig. 3 represents the Eulerian graph associated with the
on each vertex) is Eulerian if and only if for every vertex, the
generalized cycle in Fig. 1. We can thus obtain a Eulerian
numberofincomingedgesis equalto the numberofoutgoing
graphfromanygivengeneralizedcycle.Thefollowinglemma
edges and the graph is strongly connected.
shows that there is a generalized cycle correspondingto each
Thefirststatementofthefollowinglemmaisalsoknownas Eulerian graph.
Veblen’stheoremfordirectedgraphs([7],Chapter1,Exercise
Lemma 3. Let G be an Eulerian directed graph with no
1.4.5). The second statement is mentioned in [11] in passing.
isolated vertices (each vertex has at least one incident edge).
Asaformalstatementorproofcouldnotbefound,weinclude
Then there exists a generalized cycle G whose equivalent
a short proof here. gc
Eulerian graph is G.
Proof: Let G(V,E) be the given Eulerian graph. We vertices xik,1,∀k are well-defined because if an edge
constructacorrespondinggeneralizedcycleG withV =V (i ,i ) exists in G , then at least one message
gc gc k (k+1)(mod L) eu
as follows. Assume that the sets Vgc and V are indexed from xik,1 ∈ik must exist and be demanded by i(k+1)(mod L).
1 to m=|V|. Now with respect to the cycle C′, we pick a cycle
2
Consider a vertex i ∈ G having p outgoing edges equal C in G which is edge-disjoint from G as follows.
2 eu 1
(and thus p incoming edges as well). In the corresponding For some edge (jk′,j(k′+1)(mod L′)) ∈ C2′, if there is no
supervertex i ∈ Ggc, we create p subvertices. For an edge (ik,i(k+1)(mod L)) ∈ C1′ such that jk′ = ik, then we pick an
(i,j)inG, weconstructanedgeinGgc startingfromaunique edge (xjk′,2,j(k′+1)(mod L′)) for some message xjk′,2 ∈ jk′
message subvertex of supervertex i from which no previous whichisdemandedbyj(k′+1)(mod L′) (suchamessageindeed
outgoing edge exists, ending at supervertex j. Clearly, the exists).
number of incoming and outgoing edges at any supervertex Ontheotherhand,ifthereexistsan(i ,i )∈C′
k (k+1)(mod L) 1
i of Ggc is the same as that of i ∈ V. Furthermore, as G is such that ik = jk′, then because the cycles C1′ and C2′ are
Eulerianand hasnoisolated vertices, itis stronglyconnected. edge-disjoint, there must be a message xjk′,2 ∈ jk′ which is
Thus the Ggc so constructed is a generalized cycle. distinct from xik,1 and is demanded by j(k′+1)(mod L′). We
thus choose the edge (xjk′,2,j(k′+1)(mod L′)). This way, we
C. Using Eulerian graphs to show a simple explicit code canconstructC2, correspondingtoC2′, andedge-disjointofC1
in G . This completes the proof of (9). The proof of (10) is
The following proposition relates some properties of G gc
gc similar as that of Proposition 2, hence we skip it.
with G .
eu
Proposition3. LetG be ageneralizedcycle andG be the Theorem 1 gave an upper bound for the generalized cy-
gc eu
corresponding Eulerian graph. For any set C of edge-disjoint cle IC problem and it is clear that an achievable scheme
cycles in G , there exist a set C′ of edge-disjoint cycles in based on the circuit packing bound on G of [5] meets
gc SI
G of the same cardinality, and vice versa. Thus the upper bound. The following theorem makes the structure
eu
of an achievable code explicit. For this, we need the idea
ν (G )=ν (G ). (9)
e gc e eu of encoding messages in a cycle, which is well known in
Furthermore, we also have index coding literature for the unicast problem. For a cycle
C =((xi0,i1),(xi1,i2),..,(xiL−1,i0)) of Ggc, we refer to the
τe(Ggc)=τe(Geu). (10) setofL−1transmissionsxi0−xi1,xi1−xi2,...,xiL−2−xiL−1,
Proof: We proceed in a similar way as in the proof of as the cyclic code associated with C.
Proposition 1, using the propertiesof the generalizedcycle to
Theorem2. LetI beanunipriorICproblemonageneralized
prove our result. We consider two edge-disjointcycles in Ggc cycle G , and ν = ν (G ). Let C = {C ,C ,...,C } be a
gc e gc 1 2 ν
as in (2) and (3). We claim that corresponding edge-disjoint
maximal set of edge-disjoint cycles of G . The transmissions
gc
cycles can be picked in G as follows.
eu corresponding to the cyclic code associated with each of the
′C1′ =((i0,i1)m0,(i1,i2)m1,...(iL−1,i0)mL−1) (11) ned−geν-d.isjoint cycles in C is an index code for I with length
C2 =((j0,j1)n0,(j1,j2)n1,...,(jL′−1,j0)nL′−1) (12) Proof: By Lemma 2 and by Proposition 3, the set of
where(ik,i(k+1)(mod L))mk is themkth edgebetweenik and cycles in C decompose Ggc, in the sense that all the edges
i(k+1)(mod L) in Geu, and similarly (jk′,j(k′+1)(mod L′))n ′ (and hence all the messages as well) of Ggc must lie in the
k
is the nk′th edge between vertices jk′ and j(k′+1)(mod L′). cyclesinC.Considerthecycliccodeassociatedwithanycycle
To see why such edge-disjoint cycles can be picked, we C ∈ C. This code satisfies all the demands corresponding to
first note that an edge can be common between the two the edges in C. Thus the cyclic codes associated with all the
cycles C1′ and C2′ only when ik = jk′ and i(k+1)(mod L) = cycles in C satisfy all the demands in Ggc. The number of
j(k′+1)(mod L), for some (xik,i(k+1)(mod L)) ∈ C1 (with transmissionsisn−ν (asonetransmissionis‘saved’foreach
xik ∈ ik), and (xjk′,j(k′+1)(mod L)) ∈ C2 (with xjk′ ∈ jk′). cycle, and the cycles in C are edge-disjoint). This proves the
However,in such a scenario,there mustbe two distinctedges theorem.
in G between i and i , because the two edges
eu k (k+1)(mod L) Remark 4. We call the code as described by Theorem 2 as
in C1 and C2 are distinct. We can therefore find mk,nk′ the cyclic code corresponding to the generalized cycle G .
such that mk 6= nk′, thus getting (ik,i(k+1)(mod L))mk 6= gc
(jk′,jk′+1(mod L′))n ′. Continuing this way, we can choose
two edge-disjoint cyckles C and C . D. Tightness ofthe boundsof Theorem 1 andNP-hardnessof
1 2
Conversely consider two edge-disjoint cycles from G as the lower bound
eu
in (11) and (12). Corresponding to cycle C1′ in Geu, it is easy IfGgc consistsofjusta singlecycle,thenitisclearthatthe
to see that there is a cycle C1 in Ggc as follows. lower and upper bounds of Proposition 2 coincide. Since the
generalized cycle G is highly structured and has properties
C =((xi0,1,i ),(xi1,1,i ),...,(xiL−1,1,i )) gc
1 1 2 0 similar to cycles, it may be tempting to think that the same
where xik,1 ∈ ik is some message symbol available holds for all generalized cycles. The following proposition
at supervertex i and demanded by i . The from [11] will be used to show that there exists generalized
k (k+1)(mod L)
cyclesforwhichtheupperandlowerboundshowninTheorem cycle C ∈ C and ends at some supervertex in the same
1arenotalwaysequal.WenotebeforethatthePetersenfamily cycle C.
of graphs(shownin [11])are a set ofsevengraphswhichcan
Thus, Definition 5 means that G is demand-decomposable
be obtained by transformations of the Petersen graph. s
by C if no edges exist ‘across’ different cycles in C. We now
Proposition 4. [11] Let G be an Eulerian directed graph, give the main result of this section.
such that its underlying undirected graph has no minor in
Theorem 5. Consider a uniprior IC problem I (with n
the Petersen family. Then ν (G) = τ (G). Also, there exists
e e messages) represented by a supergraph G . Let G and C
Eulerian graphs for which ν (G)<τ (G). s gc
e e be as in Definition 5. If G is demand-decomposable by C,
s
We thus get the following result as a direct consequenceof then we have (for any field size q)
Proposition 4, Proposition 3 and Lemma 3.
∗
n−τ (G )≤β (I)≤β (1,I)≤l ≤n−ν (G ). (13)
e gc q q e gc
Theorem3. LetthegeneralizedcycleG representauniprior
gc
IC problem I with n messages. Suppose the underlying Furthermore, (13) is satisfied with equality throughout if
undirected graphof the equivalentEulerian graphGeu of Ggc the the underlying undirected graph of the Eulerian graph
does not have a minor in the Petersen family. Then, we have corresponding to G has no minor in the Petersen family.
gc
∗
n−τe(Ggc)=βq(I)=βq(1,I)=l =n−νe(Ggc), Proof: We prove the lower bound first. Note that remov-
ing edges (i.e. demands) from G cannot increase the optimal
for any field size q. There also exist uniprior IC problems on s
broadcast rate. Assume that we remove all edges from G
generalized cycles for which νe(Ggc) < τe(Ggc), and hence which are not present in G , and let I′ be the uniprior ICs
(8) is not satisfied with equality throughout. problemcorrespondingto Ggc. Clearly, β (I′)≤β (I). Note
gc q q
Example 2. The generalized cycle G in Fig. 1 has the the that the number of messages in G is still n. By Theorem 1,
gc gc
corresponding Eulerian graph Geu in Fig. 3 which does not we have n−τe(Ggc)≤βq(I′). This gives the lower bound.
have a minor in the Petersen family; and hence ν (G ) = Now the upper bound. Note that cyclic code for G
e gc gc
τ (G ) = 4 (we leave it to the reader to check these correspondingtoCissuchthateverysupervertex(i.e.receiver)
e eu
statements). Thus, the cyclic code correspondingto the cycles in any cycle C in C can decode all the messages in C. Since
picked in Example 1 is an optimal code with length 5. the start-message and the end-supervertex of any additional
edge in G (and not in G ) are both in an identical cycle,
As the final result in this section, we show hardness of s gc
this means that all those demands (denoted by the additional
finding τ (G ). In a recent work [12], it was shown that
e gc edges) will also be satisfied. Hence the cyclic code scheme
findingτ (G)foradirectedEuleriangraphG isNP-hard.Thus
e of Theorem2 is an achievable scheme for G also. This gives
we have the following result, by invoking Proposition 3. s
us the upper bound and thus (13). The last claim follows by
Theorem 4. Finding τ (G ) is NP-hard. Theorem 3 and (13).
e gc
We also remark that finding ν (G) for any graph (directed Example 3. Consider an uniprior extension of the uniprior
e
or undirected) is generally NP-hard (see [13], for example), IC problem of Example 1 by changing only the demands of
and to the best of our knowledge there are no specific results receivers 1,2 and 3 as D(3) = {x ,x ,x ,x }, D(2) =
1 3 5 9
about the complexity of finding νe(G) for Eulerian directed {x3,x2,x4,x7}, D(1) = {x1,x2}. Then the supergraph
graphs. correspondingtothisproblemisdemand-decomposablebythe
set of four cycles described in Example 1. Hence the cyclic
V. BEYOND GENERALIZEDCYCLES code which is optimal for the IC problem in Example 1 is
optimal for this extension also.
Fromourresultsintheprevioussections,wehavedeveloped
a framework for studying the uniprior IC problem with the
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